1

The Atomihedron Puzzles

This is a trek into a self-organizing geometry.

The Atomihedron puzzles self-organize from simple to complex. Surprises, and

twists and turns develop into a topological knot-link system currently the subject of

cutting-edge physics opened up by quantum connections with the Jones knot

polynomial. But we do not just use string for the knots. We use precision 3D

geometric models that develop in only one way using 6 dimensions of travel and all

13 symmetries of the cube. Time is the dimensional arbitrator.

By Doug Engel, Littleton, CO, USA.

Rev.0 May, 2020 Contact: dengel99@aol.com

2

The Atomihedron Puzzles

by Doug A. Engel, Discoverer of the Atomihedron

With a degree in Math, (incl. Physics, Chem, Fine Art)

GSI Inc. & Puzzleatomic.com

Research and Development

1st Edition

Includes colored illustrations

& Photos of Physical models,

Drawings & Elementary math with

References and links and commented links

& Showing attempts by others to develop a physics using

tetrahedra and knot theory

Copyright © 2020 by Doug Engel, Littleton, CO, USA

All rights reserved Contact: dengel99@aol.com Other Copyrights ©1966 drawings, etc., thru2016

Other included material is solely owned and copyright

by the respective creator(s) or owner(s)

1st edition Online at Puzzleatomic.com, May, 2020

This book is dedicated to all who love puzzles of all kinds. It is no small stretch to believe

that all our fun, math, technology, invention, and science begins with observing puzzles in

nature, creating our own puzzles and trying to solve puzzles.

My thanks to the many physicists, mathematicians and scientists, puzzlers and

philosophers, quantum theorists, that have written expositions making this stuff exciting

and keeping me interested and involved in the wonders of nature.

Particular thanks to Edward Witten, mathematical physicist, and to Louis Kaufmann,

mathematician, for their separate essays about knots and quantum theory. Many thanks

to Valery Tsimmerman for allowing me to reprint part of his well-known Adomah

tetrahedron periodic table. Please see references and links for more information and

other related materials.

3

Table of Contents (TOC)

Preface and Introduction

Chapter 1 The Atomihedron knot-link Puzzles

Chapter 2 The Adomah Periodic Table

Chapter 3 Restrictions imposed by the Atomihedron System

Chapter 4 Developing the Eh unit link

Chapter 5 Reconnection, Duality, and Space filling

Chapter 6 Ahn Circuits and Rich Clusters Ah2(k^2) (includes emulation of e orbitals with simple quantum numbers)

Chapter 7 Development of truncations & compositions

Chapter 8 Addition of Ah and Nh units

Chapter 9 Puzzle size small circuits are all-space filling

Chapter 10 Some z symmetric non-duals and a fractal

Chapter 11 What is symmetry?

Chapter 12 Ideas, constructions, unsolved problems

Current research with References and links:

Definitions formulas tables and information

Appendix A

Topics of Interest A Knot tying machine Periodic tables Unity solution Ah2, Nh2 solution

Magic # emulation Jones polynomial ref. Nh6 qt emulation Dowel model

Spectral table

4

Preface and Introduction (TOC) When is a puzzle more than a puzzle? The Atomihedron puzzles present new ideas, not

previously considered, ideas alive with possibilities. It all starts with a single special

tetrahedron I call the Protohedron that generates a single link, P24. This link, Eh,

generates an infinity of amazing knot-link structures with many interesting properties.

My problem is that the mathematical complexity of the system is a bit daunting.

Someone who wants to tackle this may well discover new mathematical principles. For

me it takes more than the limited time left in my life to master the mathematics of this

system that employs simple geometry, topology, knots and links, the ribbon twist of knots

and links, the linking twist and writhe twist of knots and links, the space filling duality of

the spin geometry, time slicing to allow 3D to simulate up to 10D, and so on. This is all

the output of one single link the Eh.

With 4D and greater there are no knots. Schrodinger’s wave equation is probabilistic and

is 1, 2 or 3 dimensional depending on its application. It uses time as well. This fact

supports (the fact that it is 3D) our 3D time sliced Atomihedron model as a toy physics

puzzle with interesting possibilities.

This idea started in the late 1960’s in my college days. I have continued to work on it

from time to time since then. The Atomihedron-Neutrohedron was discovered in the

early 1970’s. At first it seemed to be a fabulous discovery. But I soon discovered it

developed into a difficult topological puzzle if I wanted to develop it as pure

mathematics. Unlike quaternions and octonions these links have a 3D thickness and

shape which is the meat of the system. A link connected to another link can go straight

by moving a square root of 2 amount. With two types of rotated connections they can

move a unit amount or a square root of 3 amount. These three numbers appear often in

science and math. But quaternions are unit lengths and generally a multiplication is a

rotation by some amount. The Eh link does not fall into the same place mathematically.

Eh rotations are always by sixty degrees instead of ninety degrees. The Eh link is mirror

asymmetric meaning it has a structural twist, but it is rotationally symmetric. It exhibits

some of the left-right multiplication asymmetry of octonions. It is also worth noting that

quaternions and complex number multiplication is asymmetric. But saying this is

dangerous because I have yet to find an iron clad definition of how to multiply the Eh

link with itself.

Most of what I present is in the form of models that produce a system with some very

interesting properties such as filling space in many different ways, twisting both

positively and negatively when duals are used, and many other properties you will like.

After over about 50 years of fun with little progress and many beautiful self-organizing

models I am hoping someone will take an interest and discover what it is all about. It

would open the world of self-organization as a true science.

5

Chapter 1 The Atomihedron knot-link Puzzles (TOC)

Abstract. history, motivation, and overview

A study of Z-metry (or co-symmetry) and ‘SORG’

The most incredible amazing thing, the Eh link. One chain link does all this. It is not just

lines, points, and rotations like most vector systems. Eh has a 3D shape yet acts like an

overly complicated symmetrical-asymmetrical vector, full blooded in 3D and with time

cycling or slicing, and highly mechanical, deeply puzzle based. It gets so complex that it

is beyond my humble ability to turn it into pure mathematics. Yet it begins so simply,

you will want to at least have a look at it in what follows. Eh is a lifelong attempt to

understand some of the principles of ‘SORG’, self-organization. Hopefully someone will

want to write it down in terms of pure mathematics. Z-metry comes from the z spin axis

of the Atomihedron and is meant to also represent the duality of symmetry combined with

asymmetry.

My purpose here is to present a very interesting puzzle made with a space filling

tetrahedron. The puzzle has gotten interesting as a mathematical object with some

mysterious properties that will be explained. Many questions about these puzzles remain

unanswered. There will be some simple quantum number patterns shown. This is

presented to point out a strong pattern of quantum number analogies but has none of the

wave mechanics required for a full quantum mechanics. Therefore, it can only be

regarded as a toy model at present. However, the knot-link and twisting topologies of the

Atomihedron, Neutrohedron system become fantastically complicated. Yet the way the

system is built up is very easily understood, just that it’s knot-link-twist-topological

properties become difficult to understand once built up, The reader will need to decide if

this could have any use or significance for physics. It is very relevant by forming a truly

ground up self-organizing, ‘SORG’ topological geometry.

The Atomihedron was derived by trying to imagine how you could create a system that

naturally self organizes itself. It should get more complex and organized as it develops to

each higher level using some basic unit. Basic units could be combined to form a new

more complex basic unit one level higher. Thus, it would form a hierarchy of self-

organization.

This idea started back in my 1960’s, college days after reading a Scientific American

column by Martin Gardner about hexaflexagons. Long before that, plumbing pipes

formed into a flexible circuit were also an inspiration to me, as well as hinged right

triangles.

After a lot of experimentation with hinged circuits of polyhedrons, a set of principles was

formed:

1. Use the simplest symmetrical space filling polyhedron, call it a Protohedron, Ph, the

identity element of the system. The Ph chosen is the 90,90,60,60,60,60 (These are the six

dihedral angles of Ph) tetrahedron which can also fill space using 8Ph to form a larger

Ph.

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2. Connect Ph units at the opposite 90-degree edges in a closed hinged circuit.

3. The Ph circuit must be given a twist, before making the final connection, in order to

reduce randomness to some ideal amount. Thus, Ph can flex inside out in the manner of

a ‘digital torus’. It does this in a precise manner being forced into self-organization by

having the perfect form with the perfect amount of twist.

4. Find the best, or the perfect, precision Ph flex position, call it Eh.

Later on it was found that:

Eh is a precision link that forms circuits called Ec, by linking together into a chain of

links. Ec2 (an Ec linked with itself) is the identity of the Ec level of self organization and

is self dual, i.e., it is its own (spin)dual. Dual definition: A dual formed of Eh links is

produced by rotating each Eh ¼ turn about each main axis of Eh, hence a dual by ¼ spin.

The main axis is the axis passing thru the perpendicular center of the Eh ‘torus’ link (not

the circular axis going around the torus).

Still later it was found that:

5. Eh forms circuit systems that split into two precise dual topological tetrahedrons of

linked circuits and knots, Ec circuits, called Ah and Nh that can be made any size with

more complexity of linked knots and a greater number of linked knots as the size

increases.

Each Ah, or Nh has one or more Ec circuits of various sizes and symmetries and amounts

of twists. Ah is twisted negatively, Nh is twisted positively but has a single negatively

twisted circuit that surrounds it. This single negative circuit could be equated with the

Electron that makes the Neutrohedron have a zero charge.

This may seem too simplistic to result in anything useful, beyond a clever puzzle system.

However as will be explained, it produces a precise tetrahedron of topological linked

knots with many interesting mathematical properties. This tetrahedron, the Ah, exists in

two ways, Ah and Nh, that are duals of each other. Spin every Eh link 90 degrees about

its main axis in Ah and you get the Nh and vice versa. All kinds of other dual knotted

structures exist that are truncations (symmetric subtractions) or compositions (symmetric

additions) of Ah, and/or Nh that also have the duality property.

The Eh link is six dimensional. Eh can move in six separate directions, and all duals fill

infinite space by stacking alone, or if they cannot fill space alone, then with each other in

various combinations. The space filling can become quite complex, especially with

oddly shaped duals. It generally requires use of other duals and use of Ec2, the identity,

or self-dual circuit of two links. At present there is no proof that all duals can fill space in

some combination, but no counter example has been found.

7

Chapter 2 The Adomah periodic table (TOC)

The tetrahedron is important in chemistry. It has also been used as a basis for a new

periodic table called the Adomah periodic table.

There are many different periodic tables and a database on the internet lists them at this

link: https://www.meta-synthesis.com/webbook/35_pt/pt_database.php?PT_id=642

Here is a description of the Adomah periodic table introduced by Valery Tsimmerman.

Valery Tsimmerman, Maryland, USA email: Valery@perfectperiodictable.com phone: (410) 442-4658

8

Here is a description of the Adomah PT reprinted from Valery’s website.

“On Tetrahedral Character of the Periodic System. (link):

ADOMAH Periodic Table is built strictly in accordance with the quantum num-bers n, l, ml and ms. It comprises four rec-tangular blocks: s, p, d and f, containing 16, 36, 40 and 28 ele-ments respectively and it closely fol-lows electron config-urations of atoms.

Surprisingly, when el-ements were placed in rectangular

boxes , instead of traditional square boxes, perimeters of s,p,d and f blocks became equal!

(refer to "Description" page for more discussion on proportions of the

blocks). Since electrons tend to form pairs and each element corre-sponds to one characteristic electron, it is logical to place elements in 1/2x1 rectangular cells, so two such cells would form 1x1 unit. This adjustment re-sulted in block perimeters equal to 18 units:

s-block is 1 unit high and 8 units long (or wide) (1+8 = 9, is half of the perimeter);

p-block is 3 units high and 6 units long (3+6 = 9);

d-block is 5 units high and 4 units long (5+4 = 9);

f-block is 7 units high and 2 units long (7 +2 = 9);

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Therefore, perimeter of each block: P=2x9=18 units.

Those are the only four possible rectangles that could have perimeter of 18 (if only natural numbers are used). What can it possibly mean? Could it be just a coincidence?

Apparently not, this is not a coincidence. There is one 3D geometric shape that, if sliced in a certain way, would produce rectangles with the same proportions, orientation, alignment and order as spdf blocks of AD-OMAH Periodic Table. This shape is Regular Tetrahedron.

If a regular tetrahedron with edge E is intersected by a plane that is paral-lel to two opposite edges, cross section will always be a rectangle or a square with perimeter P=2E.” Valery Tsimmerman

The below figure is from the website: Valery@perfectperiodictable.com

It shows the alternating planes of balls where only the red balls correspond to atomic or-bital quantum numbers.

10

Chapter 3 Restrictions and Advantages (TOC)

The Adomah tetrahedron has been widely admired on the internet and has been used in

some student chemistry textbooks in a planar form. It is a new and intuitive way to look

at the periodic table. The way it uses tetrahedral planes by skipping every other plane is

like the linked planes of the Ah, Nh knot circuit puzzles. Every other plane of linear links

is at right angles in Ah, Nh, with odd orders not producing a simple central quantum

number emulation (this will become clear further on).

The Ah tetrahedral system has many possible duals, and duals where Ah is composited

(knot addition) with Nh to form an Ah # Nh dual where [Ah # Nh] has a [Nh # Ah] dual.

The knots in Ah are different than the dual knots in Nh and of an opposite type of twist.

The Ah system uses only one link, the ‘hyper’ cubic Eh link, to construct the infinity of

duals. In addition to duals, random circuit systems and fractal circuit systems are

possible. When the dual operation is performed on a non-dual it can break into some

collection of open chains, single links, and non-dual Ec circuits. Infinite duals are also

possible where duality only exists if infinite space is stacked with Ec’s in some

combination and structure. An example is an infinite plane stacked with 3D lines of

links. When the dual operation is performed the infinite lines are rotated by 90 degrees.

Our system has severe restrictions over standard knot-link theory and topology. We can

only use the discontinuous Z-metric Eh link to construct the system. We are limited to

the duality spin operation. In other words, if a unit is not dual it is not considered to

belong to the Atomihedron set. Instead it belongs to the Electrihedron, Ec, circuit set and

may or may not fill space by stacking. By unit here we mean a circuit or a system of

linked circuits. A knot can be continuously transformed in an infinity of ways. Dual Ec

units can only be added and subtracted in a finite number of discontinuous ways to form

duals.

Advantages of the Atomihedron system

The dual Atomihedron knot units form a beautiful and precise mathematical sequence

with powerful self-organizing properties. The knots and links form topological and twist

system symmetries never studied before. The twist calculations, though not yet solved,

are thought to be built automatically into the system.

Standard knot theory of a string or rope is allowed to neglect ribbon twist while the Ah

system requires it. Hinged loops of Ph units can form knot circuits of smaller Ph units,

called Pc, that can be infinitely complex or built up to an infinity of twist with infinite

volume or a minimum twist with a Ph volume (8^n). Thus, we can control how much

twist a (Ph, Ec, or Ah or Nh) unit contains.

The Ah system incorporates linear and rotational movement in 6 + 3 dimensions by using

only 3 dimensions. With time this makes 10 dimensions. Each of these space

dimensions are axes of rotational symmetry of the cube. The axes defined by the 4 pairs

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of opposite vertices of a cube are also used by being the linear directions of the 4

equilateral triangular bars of the Eh unit making 13 dimensions. The Eh unit has 6

orientations corresponding to the 6 orientations of a cube if one face has 2 diagonals

drawn on it.

Time is used to make the system dynamic. This is done either by time slicing using time

states, or allowing the circuits to incorporate discontinuous flow so that only one point of

any circuit is occupied in one instant. While it is theoretical it uses well known

principles.

An important discussion about twists

If you move a nut along a helix (bolt thread) with a right-handed twist you will be

rotating the nut in a clockwise direction. But you must twist a spring in a

counterclockwise direction to get this effect. So normal right-handed threaded bolts have

a counterclockwise twist. If you twist a ribbon in a clockwise direction to make a

moebius strip and then you move along the strip you rotate in a counterclockwise

direction. So do you call the strip positively or negatively twisted? For the Eh we will

call it a positive twist if before connecting into a loop we make a clockwise twist, which

will result in a left-handed thread type convention but will be regarded as mathematical

(topological) positive twist for our purposes. This turns out to be similar to the

counterclockwise rotation effect of multiplying imaginary units together such as i^2=-1.

12

Chapter 4 Developing the Eh unit link (TOC)

Figure 4.1

Ph tetrahedron pattern

Shown here is the

pattern used to make the

Ph unit. The two long

edges are the edges that

form the two 90-degree

dihedral angles while

the short edges form

four 60-degree dihedral

angles

Figure 4.2

Here is the Eh

unit, a bi-

rotationally

symmetrical right

angled

tetrahedron. It is

well known as a

space filling

tetrahedron.

13

Figure 4.3

This shows a hinged

chain of 4 Ph units using

the right angles as the

hinge edges.

Figure 4.4

Here is a complete Ph

circuit, a folded Eh,

or Pc24 of 24 links in

one of the folded

positions of the Eh

link. It has a total of

2 twists, 720 degrees.

Figure 4.5

This illustrates the way

twist and writhe twist is

contained in a pair of

maximally twisted,

hinged Ph8 units of 8

small Ph each. If they

were oppositely twisted,

then they could be

pulled apart into two Ph

chains of 8 links with

no twist in either. The

total twist added per

Ph8 tetrahedron here is

+1 twist.

14

Figure 4.6

Elementary Ph twist

calculations

Here we have twist

calculations for Ph0 thru

Ph4. For Ph0, a single

link, the volume is 8^0 = 1,

to Ph4 having a volume of

8^4 = 4096 Ph0 links.

Calculations of Phn are

done by adding twists of

Ph(n-1) to the single twist

of Phn=1 in an 8 unit link.

which gives this reiterative

formula:

Phntt=Ph(n-1)tt*8 + 1

(tt stands for total twist)

The Ph0 unit is itself

carrying twist by rotating

back edge 90 degrees to the

front edge. This is ¼

rotation, but we know that

twist produced is 1/8 since

8 Ph0 create exactly 1 twist

of Ph1. This enables us to

add the 8Ph0=1 innate

twist to Ph1 unit twist of 1

for a total twist of 2 for

Ph1.

15

Figure 4.7

Eh link writhe

twist

The Eh link=24Pc,

and Pc stands for a Ph

closed toroidal

circuit. It exhibits

both symmetric and

asymmetric features.

The two arrows here

show the twisted

structural appearance

of the negative writhe

twist. This

symmetrical

asymmetry is part of

the reason that this

link becomes

automatically self-

organizing. Ph twist

is normally hidden

inside the Ph unit

system while Eh twist has exposed structural ‘writhe’ twist. The two arrows show how

the next link in line will have a ½ twist added to the line of links. The infinite space

around the Eh has the opposite, or complementary, twist so that total writhe twist + space

structural twist is zero.

This shaped or structural twist becomes important since an Eh circuit will twist easily in

the negative twist direction but an Eh circuit will twist with writhe in the positive twist

direction. You can see this effect on the old-style telephone cords that have a coiled

shape. When they get twisted in the same direction as the coil, they wind round

themselves opposite to the twist of the coil. In knot theory this is called writhe twist.

The same thing happens in the Nh unit, while the Ah circuits twist in the same direction

as the Eh writhe so are much easier to form.

16

Figure 4.8 Eh built from 4 identical triangular bars

This figure shows how the Eh is built as an abstracted unit from 4 triangular bars. The

bars have a perpendicular cross section that is an equilateral triangle. Here are the

necessary dimensions to make models with wood or other materials. These details can be

used to produce computer graphic models as an aid to help analyze the Atomihedron

system.

17

Figure 4.9

The Eh as a

flexahedron

Here we see

the Eh

undergoing

one cycle of

flexation in 8

rotations of

groups of

links. All

rotations are

180 degrees

except the 90

degree 1st and

last rotation.

For this to

happen in

three

dimensions

while linked

to another Eh

would require

time states,

where time

substitutes for

the extra

dimensions

required. It

acts like a

‘quantized’

digital torus.

18

Figure 4.10

Ec2 from two

linked 24 Ph

circuits.

Since the Eh

link has a

helical

symmetry it is

natural to link

them together

in various

ways. The

method of

linking here

produces a

maximum link

twist where

only two links

link together

into a single

closed circuit

that also fills

space. They

fill all available

link holes. This model is made of cardboard Ph tetrahedra hinged at their right angle

edges. Each Ph24 circuit has been given two twists before connecting into a circuit..

They can link like this when both of them are flexed into this symmetrical position that

opens up two opposite holes in each link allowing this linkage.

19

The self-dual identity element Ec2

An Ec2 self dual identity circuit showing how the triangular hyper-axes align exactly

with the vertex diagonals of a

cube. It is the identity under the

dual operation because it turns into

itself. It is also a Ec unity element

since it is used to accomplish space

filling in combination with other

duals and is the smallest possible

Eh circuit. Also shows that Eh

links move in straight lines in the

directions of the lines connecting

the centers of the 6 pairs of

symmetrically opposite cube

edges, designated by composite

x,y,z axes as a,b,c,d,e,f. In this

graphic the ‘travel’ is in the

direction of x,y=a and

-x,y=b. For a dual there is always

a main spin axis, which we choose

to be z, Zs as a default.

Figure 4.11

Figure 4.12

Ec2 two piece puzzle solution using two

gapped Eh links.

By providing a gap in an Eh link in two

different ways it is possible to solve

puzzles of two, six and more links. Here is

shown the assembly of the smallest Eh

circuit, Ec2, the identity element of the Ah,

Nh system. For circuits larger than Ec2 all

puzzles must have six or more links in

them. For instance, the smallest Ah and

Nh each have six links. No circuits of 3, 4

or 5 links are possible.

As seen here the green piece snaps into the

blue piece producing Ec2, which is its own

dual.

20

The Six dimensions of Eh Travel

Figure 4.13

Simulating 6 dimensions with a 3D cubic

symmetry based Eh link.

Top view shows one way of viewing a 4

dimensional cube in a 3 dimensional

projection.

A cube has 13 axes of rotational symmetry. The

four vertex axes, t, u, v, w, shown by the red

lines have triangular symmetry, meaning that

1/3 turns of the

cube about these axes restore it to look

the same. The three x. y. z face axes have ¼

turn symmetry. The six axes a, b, c, d, e, f,

connecting the centers of opposite edges have ½

rotational symmetry and are parallel to the

edges of a regular tetrahedron. Thus we have

½, 1/3 , ¼, 6+4+3=13 cubic axes of symmetry

giving 26 vector directions.

We can make the internal blue cube smaller and

smaller until it becomes a

point and disappears. This allows us to compare

the Eh link and fit it into this modified 4D cubic

projection. Eh can move in 6 directions even

though it is 3 dimensional. By squeezing away

one cube of the hypercube we reduce the six

face cubes to 6 cubic prisms and are left with a

hybrid inverted cubic octahedral structure.

The lower figure shows how this structure fits

around an Ec2 unit (E circuit of 2 Eh links).

The center of Ec2 coincides with the center of the projected cube and the 4 dotted line

vertex axes of the cube align with the triangular inner edge axes of Ec2. There is one red

and one blue Eh unit. All Eh link changes of direction when connecting to another link

are multiples of 120 degrees about the triangular bars mimicking the 4 triangular vertex

axes of the cube. Ec2 has only 3 possible main axis orientations, x, y and z.

The Eh link is not 6 dimensional in the sense of 6 axes all at right angles to each other.

Instead it has 3 pairs of perpendicular axes ((x, y), (x, z), (y, z)) defining 3 hyperplanes

21

with the six cubic edge to edge axes (a, b), (c, d), (e, f). Each plane is at right angles to

the other two giving a total of 6 (axis to plane) right angles, equivalent to 3 right angles

by rotating. Planar pairs and the duality property of this system produce a single main

spin axis about one planar pair orientation, designated Xs, Ys or Zs. Zs being the one we

will use.

This is a different way of looking at a higher dimensional system. Since it is embedded

in 3 dimensions it can form knots and links, which is not possible in 4 or more

dimensions. This knot-link forming property is part of what makes the system self-

organizing but challenging to reduce to mathematics. For instance, quaternions and

octonions do not have a solid geometrical appearance but are complex unit vector

operations with point-to-point unit lengths. As we will show with Eh links, two links can

be rotated about their linking axes in different ways.

Figure 4.14

Symmetries of Eh

For a given orientation of Eh

there are three 180 degree

rotational symmetries which

are the a, b, hyper axes and

the z axis and ten non

symmetries which are

360 degree rotational, and are

the hyperplane axes c, d, e, f,

and the cubic vertex axes

(parallel to the Eh triangular

bar axes) v1, v2, v3, v4 and

the remaining two standard x,

and y axes.

22

Figure 4.15

Rotations of a connected link

When connecting another Eh link to a link in

the ‘a’ position like this one the 4 tri-bar

connector rotation axes and tri-bar holes are

parallel to the 4 vertex axes of the cube, v1, v2,

v3, v4. For Eh these rotation axes are the 4

bars, p|, q|, r|, s| and the two bar holes, p, q,

shown here all and are positive clockwise, cw,

rotations looking in the direction of the link.

We call them p| (p bar), q, p, q| (q bar), r| (r

bar), s| (bar).

When x=y we move in the +a direction which is

a linear only motion connection. Thus for m =

+integer links we move a distance a = m(root 2)

Connecting to the +a direction link if you perform a rotation of p| of the connected link

you have a motion and new link z=+1/2, or do -p| where z=-1/2. A double rotation is

possible with qs| where z=+1 or -qr| where z=-1.

We could dispense with the hole rotations by only looking at links that are bar rotated.

However a much bigger simplification is possible by only considering three connection

types which will be developed in the next couple of pages. It makes possible very simple

chemical like formulas for circuits that share the Ah-Nh type duality.

For building an understanding of the Ah, Nh duals the simplfied (z default oriented) z, a,

b, system works fine. To develop the full mathematics x and y spin axes are needed as

well but are not used in this writeup.

Figure 4.16

The four basic pair connections of

Eh.

The left one is Ec2 the self-dual

identity and unity element, type U =

unity for filling space in combination

with other duals, I = the identity Ec2

circuit. It is the only circuit that has a single letter type formula, type U and is the

smallest circuit. Second from left is the type L linear connection. Second from right is a

23

double rotation type symmetrical M connection. The rightmost is a type V single bar

asymmetrical rotation. M and V only appear on the surface of Ah and Nh.

The M and V are referred to as z connections when the main spin axis is z (or Zs) because

they move the circuit in the z axis direction.

Figure 4.17 The 15 possible pair-oriented connections

The Figure above shows the possible pair-oriented connections where the blue Eh (a axis)

is held in place while the yellow Eh is linked to it. The Ec2 is a circuit with extra

symmetry. Ec2 has only 3 orientations, x, y, z, and is not available to create new circuits.

This leaves 14 possible pair connections 7 for + and 7 for – directions. Six orientations

are possible for Eh and this makes 14 times 6 equals 84 possible pair connections. The

main spin axis Zs is the orientation default for Nh and Ah duals to simplify things. The

linear ‘a’ (and ‘b’) travel connection occurs in the interior of the Ah, Nh units in

alternating a and b parallel stacked planes. The 4 connections to the immediate right of

leftmost 2a occur only on the 4 surfaces of the tetrahedral Ah, Nh shapes. Nh designates

the connection occurs only in an Nh unit, Ah only in an Ah unit. and the two rightmost

connections designated ‘s’ occur at two opposite tetrahedral edges of Ah and Nh (but not

in the orientations shown. Thus we have 2 linear connections, for ‘a’ travel, 8 surface

connections, and 2 edge connections = 12 pair connections. For Ah for the default z, a, b,

orientation: We will have two for linear travel a and b. We have four for +a and -a Ah

type V ‘a’ travel and four for ‘b’ type V travel and two for Ah type M ‘a’ travel. This

makes a total of 2+4+4+2 = 12 pair connections in Ah and likewise 12 pair connections

24

in Nh for the default z, a, b, orientation. For all six orientations this makes 72 in total for

Ah and 72 for Nh.

Thus, with having only three pair connections L, V, M, we have a versatile Eh link

connector type designation system. Some of these formulas will appear below.

Figure 4.18

Regular tetrahedral travel

This shows that travel along the edges

of a regular tetrahedron is possible.

This uses the 6 edge or hyperplane axes

of the cube (a,b), (c,d), (e,f). However,

in general, no finite duality results from

tetrahedral travel. Type formula is

2(4L,V,5L,V). Note that by counting

the number of letters used you can

determine the circuit length. This gives

2(4+1+5+1) = 22 links.

25

Figure 4.19

Travel along a and z axes

Here we show how travel only

along a and z axes is possible

using a circuit of 24 links.

Travel along a is always in

units of the square root of two

while travel along z is in units

of 1.

This the system using 9

different axes, a, b, c, d, e, f,

linear axes and x, y, z as pair

linkings.

The Ah and Nh units only use a,

b, and z travel, because we only

orient Ah and Nh in their

default Zs spin orientation (we

make Zs the main spin axis).

This circuit is not dual.

A type formula is

2(5L,V,5M,V ) with 24 links.

26

Figure 4.20

Eh circuit showing a and b travel

Here we see a planar circuit, Ec10, of 16

Eh links. This shows the versatile Eh link

can form a flat plane of links.

Using the connection types M and L, a

formula for this is 4(2M,2L)

Figure 4.21

Ah Nh travel loop

Here we see a planar circuit, Ec16, of 16

Eh links. This shows how the green Ah

connectors on the left can combine with

red Nh surface connectors on the right to

form a loop. Connections like this can

be used to add Ah to Nh units along a

and b (or x, y) axes. The connections

produce opposite twists for a total of

zero circuit twist. A type formula is:

3L,2V,2L,2V,3L,2V’,4L,2V’.

27

Chapter 5 Reconnection, Duality, and Space filling (TOC)

Figure 5.1

The Eh link pairs reconnect

by link rotation.

The two top figures, type V, and

V’, show how the dual spin

operation of the red and blue

left links produce a linked red

and blue pair on the right. The

left type V link connection

occurs only on the surface of

Ah. The blue link is the surface

link. The right type V’ link

connection occurs only on the

surface of Nh.

Looking at the Ah surface the

middle left connection is of the

Ah single circuit outer edge

type, M facing outward. The

middle right connection is of

the Nh edge type, M’ facing

inward.

The bottom figures show the L

type which only works if a

group of 2x2 or greater

rectangular array of links are

adjacent in a plane array. Other

considerations are also needed

for them reconnect, but the

general idea is conveyed.

Duality only happens if Eh

links of the line connection type L move in a direction perpendicular to the main spin

axis, Zs (Zs by default but spin axis can be Xs, Ys, or Zs). The ends of linear links must

connect with a surface link (z motion link). For order 3 or larger Ah, and Nh the interior

is filled with alternate a and b type L linear layers. The fourth connection type (see

figures and text), U or I, produces Ec2, the unity or identity circuit. U is not a connection

type, per se, but a connection circuit. It can be referred to as a (special) connection type

28

because it uses only a pair of links satisfying the definition of a type but not the spirit of a

connectable structure.

Figure 5.2

Illustration of type connections on surfaces

of Ah and Nh.

These additional type illustrations will make

the idea of the type connections more obvious.

The top figure shows the surface type V and

interior L connections that occur on all 4

surfaces of the Ah unit and its truncations. It

shows a ribbon simulating the negative twist

produced.

The next figure shows the surface type V’ and

L connections that occur on all 4 surfaces of

the Nh unit and its truncations. It shows a

ribbon simulating the positive twist produced.

The third left figure shows the Ah2 unit with

the M and V type connections indicated.

There are no L connections.

The fourth left figure shows the Nh2 unit with

the M’ and V’ type connections indicated.

There are no L connections.

The two bottom right figures show the type M

connection and a simulated negative ribbon

twist.

29

Given a dual unit consisting of a set of closed-knotted-linked circuits then the dual

operation of spinning every link 90 degrees about its spin axis produces its dual, a set of

closed-knotted-linked circuits. These basic connection principles can be used as the

initial starting point for a proof that all duals fill infinite space in combination with other

duals. One rule that appears to hold true for a unit to be dual is that a double z link, type

M, can only connect to the same linear type L travel link such as an ‘a’ travel link.

Rules of E duality:

1. Unit must have a main spin axis (we use z by default, but it could be x, y, or z.)

2. Unit must be a closed hole doughnut, i.e.. unit must be compact

3. Every dual fills space with itself or in combination with other duals.

4. Units in simplest form are Ahn or Nhn meta tetrahedra.

5. Units can only be added along the main spin axis heteromorphically to form a

new dual, such as Ahn+Nhm+Ahq+… aligned along z.

6. Each unit, Ah, Nh, Ah, … in the addition must have z rotational symmetry but the

entire sum may not need to have z rotational symmetry.

30

The Eh link solves itself into a self-organizing Ah, Nh dual circuit system with a precise

topology of knotted links that evolves from the bottom up in more complex

organizational units. This only works like this way because the system combines both

symmetrical and asymmetrical advantages to produce an integrated set of units, we call

Z-metry.

Figure 5.3

The left figure is Ah2

comprised of six Eh

links. The right

figure is Nh2 also

made up of six links.

If each link is rotated

90 degrees about its

main axis, shown by

the six axis lines then

Ah2 turns into Nh2

and vice versa.

This duality feature works for any order Ah and Nh and all duals derived from them. No

duals smaller than Ah2 and Nh2 are possible except for Ec2.

The next duals are Ah3 and Nh3 each with 22 links and one circuit.

Figure 5.4

The left figure is Ah3

with one circuit of 22

Eh links. The right

figure is Nh3 also an

Ec22. With spin

duality, either one

turns into the other, by

rotating all 22 links 90

degrees about their

main axes.

The two circuits are color coded as 5G, 2R, 2W, 2Y, 5G, 2Y, 2W, 2R for 22 links. If you

draw the circuits with straight line segments you see them following different writhe

twist paths. Type for Ah3 is 2(6V,MV,L,V,M) and for Nh3 is 2(6V’,M’,V’,L,V’,M’).

The formulas are similar, but the shapes are quite different. For n>2 the formulas are no

longer the same for a given n.

31

Figure 5.5

Ah2 puzzle

solution (with

gapped links)

With three pieces of

one kind of gap and

three pieces of the

other kind you can

build Ah2 as

shown. Assemble a

and c as shown.

Now do a with b

and c with d finally

position as

shown and snap

together as shown

on the bottom right

figure for the final assembly of Ah2. Type formula is 2(M, 2V)

Figure 5.6

Nh2 puzzle solution

(with gapped links)

With three pieces of

one kind of gap and

three pieces of the

other kind you can

build Nh2 as shown.

Assemble the two a,

b and c as shown.

Now do a with b and

finally position as

shown and snap

together as shown in

the bottom right

figure for the final

assembly of Nh2.

Type formula is 2(M’, 2V’).

32

Figure 5.7

Example start of a

planar space filling

using a combo of

Ec2, Ah2, 2 Ah3,

Nh2, 2 Nh3 duals.

A simpler filling

would require equal

numbers of Ah3,

Nh3 and a larger

number of

Ec2. Once a unit

cell has been

established, the unit

cells can be

repeated infinitely to fill the plane and planes can stacked to fill 3D space. It is thought

that all duals can fill 3D space, if not alone then in some combination with other duals.

Figure 5.8

Ah4 and Nh4 models

This shows Ah4 and

Nh4 as duals.

Spinning each link 90

degrees about each z

axis of one produces

the other. Thus all 54

links need to be spun.

Ah4 has negative

twist and Nh4 has

positive twist. The

main Zs spin axis is

also shown.

33

Figure 5.9

Comparison of Ah2 and Nh2

circuits

Ahn always follow a simple

surface connected system of

knotted links while Nhn is much

more complicated and contorted.

Both Ah2 and Nh2 have one

negative circuit twist.

This clearly shows that the Nh2

circuit is writhe twisted about

the z axis and is different from

Ah2. The colored dots show

approximate centers of the Eh

links.

34

Figure 5.10

Nh3, Ah3 circuit diagrams

The Nh3 top diagram here shows the z axis

writhe twist we started in Nh2. This edge

circuit occurs in all Nhn and always moves

along the same 4 edges and is negatively

(ccw, counterclockwise) twisted. Nh4

develops a single separate internal circuit

that is the twin of Ah2 but contains 14 Eh

links in a positively twisted form. From

Nh4 on the internal circuits of Nh(n+2)

follow a path identical to the circuits of Ahn.

This produces very extreme positive twist

since all other twists (link, circuit, writhe) of

Ph and Eh links are all positive. Thus the

reason it is called a neutrohedron is because

the edge circuit twists in the opposite

direction of the internal circuits.

The lower diagram shows Ah3 and you can

easily see that it also twists negatively at top

and bottom edges. The other negative twists

along the side edges are from writhe and the

-1/8 twist added by all the surface links

which also happens for Nh3.

35

Chapter 6 Ahn Circuits and Rich Clusters Ah2(k^2) (TOC)

Figure 6.1

Ah2 details

Here are details

for Ah2

showing

development of

abstracted a, b

planes with the

red circuit.

Dots show link

presence. The

yellow sketch

is a way to find

the number of

circuits in an

Ah order

number. More

info for the

sketch will be

shown for

higher order

Ah below.

Volume

formulas are

given as well

for link

volume,

surface volume

and a,b traverse

volume.

36

Figure 6.2

Ah3 details

Here are details for

Ah3 showing

development of

abstracted a, b

planes with the

circuit. Dots show

link presence. The

yellow sketch is a

way to find the

number of circuits

in an Ah order

number. More info

for the sketch will

be shown for higher

order Ah below.

Volume formulas

are given in the Ah2

details above.

37

Figure 6.3

Ah4 details

Here are details

for Ah4 showing

development of

abstracted a, b

planes with the

circuits. As

noted the dots

show link

presence. The

yellow/gray

sketch shows two

circuits A and B.

More info for the

sketch will be

shown for higher

order Ah below.

Volume formulas

are given in the

Ah2 details

above. This is the

first full dual that

produces a linked

knot. As order

increases the no.

of linked knots

will proliferate.

38

Figure 6.4

Ah5 details

Here are details for

Ah5 showing

development of

abstracted a, b

planes with the

circuits. The

yellow sketch

shows two circuits

A and B. More

info for the sketch

will be shown for

higher order Ah

below. Volume

formulas are given

in the Ah2 details

above. This Ah

dual produces two

trivial links and a

pair of alternating links. As order increases the no. of linked knots will proliferate.

Count interior crossing length in all interior a or b traverse layers (not both a and b) and

divide by two to get total linking # of Ahn (or use (crossings of a circuit/2) for an

individual circuit).

39

Figure 6.5

Ah6 details

Here are

details for Ah6

showing

development

of abstracted a,

b planes with

the circuits.

The

yellow/gray

sketch shows

only two

circuits A and

B.

40

Development of orders 2(k^2) (filled a, b, z central circuit groups)

Ah2 and Ah8 circuits Figure 6.6

Electron orbital simple quantum no.

emulation for 1st and 2nd shells.

This is still hypothetical but worth

checking out as to why there is this

correspondence with orbitals in such a

natural manner. The left figure is Ah2

which would represent Hydrogen with

one electron and Helium with two

electrons. The right figure shows Ah8

emulating the 2p electron orbitals holding

up to 8 electrons. The analogy is strictly

structural in these simple models. The emulation is quite interesting because the

calculated twist of the circuits has many similarities to the energy of electron orbitals.

This is complicated and will need to await the development of a computer program to

calculate twists. The 2p1 symmetrical orbital occurs in Ah6. The 2p2 and 2p3 orbitals

occur in Ah8 and are elliptic, and this can be seen in the link model (Figure 6.61below)

clearly, where 2p3 is symmetrical to 2p2 but alone they are not symmetrical.

The analogy gets even more interesting when you look at the Neutrohedron, a dual of, Ah

and with internal reversed twist of the Atomihedron. In Nhn a negatively twisted edge

circuit surrounds the entire structure. The inner Nhn circuits rapidly build positive twist

because their circuit twist is positive and Eh structural twist is positive. Thus, if twist is

related to energy one could say mass increases rapidly in the Nh inner circuits. The Ah,

Nh system is like the helical structures of biology. It builds and self organizes itself in a

natural manner. Speculatively this is an idea that only matter exists because even

antiparticles are twisted the same way (positive energy) at a very deep structural level.

Thus, Antimatter is not the opposite of matter but has positive energy or twist at an

undetected sublevel that is the same as matter. Currently among theoretical physicists

there is a suspicion that a handedness exists at a deep level of matter and energy.

41

Ah8 central circuit cluster, n = 2(k^2), k=2 Figure 6.61

These

three

figures made of Eh

links show the Ah8 center cluster of circuits.

The four circuits are orange = 2s, (z spin

symmetry) copper = 2p1, (z spin symmetry)

blue = 2p3, elliptic, silver = 2p2, elliptic.

Top left is a top view, right is a side view.

The bottom view is a front view same as

Figure 6.6 plot view. The orange circuit is

the most symmetric being a stretched

version of Ah2 with 18 links. The entire unit

is 3 axis (a, b, z) spin symmetric. The orange

circuit contains 18 links and the other three circuits

have 50+ links each, for a total of 170+ links. In the top right figure, you can see four

link holes near the center silver and blue links through which the remaining circuits

would pass if the top and bottom of the tetrahedron were completed. Spin symmetry here

refers to a ½ rotation or 180 degrees. This unit is not dual because of the four unfilled

holes. The figures appear skewed due to camera angle perspective.

42

Ah 18 circuits Figure 6.7

This shows Ah18 emulating the

3s, 3p and 3d electron orbitals. It

is filled at the n = 2(k^2) Ah order

which emulates orbital size and

filled orbital structure as predicted

by quantum theory. Each orbital

can contain two electrons for a

total of 18 electrons. Elliptic

orbitals also appear as a further

emulation. The 3s, 3p1 and 3d1

orbitals are not elliptic but are symmetric.

Ah32 central circuit cluster Figure 6.8

The complete 4th shell of electron orbital emulation is shown as Figure 6.8 here. Since it

can hold 32 electrons when filled and the d shell can hold 18 electrons, the p shell 8

electrons and the s shell 2 electrons making a total of 60 electrons in this emulated shell.

The with the s = 2, p = 4, d = 9 we have (2) + (2 + 8) + (2 + 8 + 18) + (2 + 8 + 18 + 32) =

100 possible elements. However the way the shells are filled causes the 5th, 6th and 7th

shells to be partially filled providing the full scope of possible elements. A video of this

can be found here: https://www.youtube.com/watch?v=2AFPfg0Como. This figure

shows the 4 symmetric, centered circuits, 4s, 4p1, 4d1, and 4f1. All the elliptic circuits

crowd around the symmetric circuits. The most ellipticity is represented by the most

offset from the symmetric circuits (such as 4f6, 4d4, etc.). Of course this looks nothing

like the wave mechanical solutions. I conjecture that if this system could formulated

dynamically (but I don’t know how!) it would be equivalent to the wave mechanical

solutions. (My belief is that every analog solution in math is basically digital.)

43

Diagram of circuit levels of Ah50

Figure 6.9

This figure does not

show individual circuits

instead showing filled

circuits including those

with fractional levels

above and below the

central cluster of

orbitals. Circuits in

between the levels

(white space) are

generally random

looking and do not

appear in grouped

clusters like levels do

and do not repeat

periodically as order

number increases. Here

level is measured from top and bottom to the center, level 1/1.

44

Diagram of circuit levels of Ah72

Figure 6.10

Ah 72 center 1/1 level

would add 72 electrons

making for a total of

110+72=182 elements

far beyond the known

stable elements. A

level occurs for orders

n=2(k^2) if level/n

reduces to a fraction of

level/k with

denominator equal to

or less than k. Here

level is measured from

top and bottom to

center.

45

Figure 6.11

Ah288, Nh290 main

circuit clusters

The top right figure is

Ah288 presented as a

tetrahedral model. The

circuit clusters are

precisely plotted. Ah288

has a small total negative

twist. The wide red front

edge and wide red hidden

back edge are all type M

connections. The surfaces

are all type V, the interior

are type L, a and b travel.

The main clusters shown

are all the reduced

fractions of layer#/288,

and the number of a, and

b layers is 2(288)-1.

Clusters will all repeat to

infinity for Ahn, where n

is even. The center cluster

or 1/1 is the simplest

(repeating polynomial) n =

2(k^2) producing 12^2 =

144 circuits.

The top left figure is

Nh290. Its interior circuits

mimic Ah288 by being 2

orders larger but are

positively twisted by using writhing. The green edge is a single circuit negatively

46

twisted. When the duality operation is performed the entire set of circuits turn 90 degrees

as shown by Nh290. Topological complexity occurs if you remove the in between spaces

(representing non clustering, mostly non repeating in larger order n, circuits). Then the

twist calculations become more involved. Also, circuits can be removed from various

clusters increasing the complexity. You could also consider some clusters to be in a

different x, y or z spin orientation using the idea of time slicing or introducing interleaved

dimensions. As mentioned, in Nh290 the green edge forms a single negatively twisted

circuit while its interior has a large positive twist. The number of different cluster circuits

grow as n increases and the total number of circuits for Ahn become greatest whenever

n=2(k^2). The Total volume for Ah 288 is 8,044,128 Eh links.

47

Chapter 7 Development of truncations & compositions (TOC)

Figure 7.1

Nh4 with the top truncated

It is possible to truncate Ahn and Nhn in

different ways that preserve 1/2 z spin

symmetry, and duality. Here Nh4 has its top

level or edge truncated. This creates a

negatively twisted path using one circuit. It

has a volume of about 36 links. It also has

the space filling properties of duals.

Truncation formula is Nh4(1,5) or simply

(1,5) meaning a truncation of Nh4 starting

at ‘a’ layer one, and continuing thru to ‘a’

layer five, producing a z height of five a, b

layers.

Figure 7.2

Here is another view of Nh4 with its top

truncated. The black line shows the general

path of the single circuit.

48

Figure 7.3

Ah4 with its top truncated

By lopping off the top double layer of

Ah4 you get this Ah4(1,5). This

structure is the dual of Figure 7.1 and has

z rotational symmetry. The red links

form a single circuit while the single

green circuit pass’s thru the red circuit

twice.

Figure 7.4

Ah5 with the top and bottom

truncated

It is possible to truncate an Ah or

Nh dual symmetrically if the n order

is five or larger. This one can be

designated Ah5(3,7), meaning a

truncation of Ah5 starting at ‘a’

layer 3, continuing thru to ‘a’ layer

7, producing a z height of five a,

and b layers. It is two equal

negatively twisted circuits and the

paths roughly follow the colored

links. Each circuit has 32 links.

There are three axes of 180-degree

rotational symmetry. It is dual.

49

Figure 7.5

Ah5 top and bottom

truncated showing the

two circuits in a layer

drawing.

This is the circuit

diagram of Ah5 (Figure

7.4 shown above as a

photo) with top and

bottom truncated. The

black and red dots show

two linked circuits

having a crossing

connection. Its link

diagram is just two

linked circles.

50

Figure 7.6

Ah5 with

top 4 layers

truncated

This

Ah5(1,5) is a

single circuit

of different

color links.

This unit has

z rotational

symmetry. It

is a dual.

51

Chapter 8 Addition of Ah and Nh units (TOC)

Figure 8.1

Composition or addition of

duals

Adding Ah and Nh units is

possible. This can be done in

numerous combinations. Some

of the space filling might

involve units trapped inside

other units that form holding

cavities that are filled by duals.

It might be possible to find a

situation where some kinds of

oddly shaped duals do not fill

space alone or in combination

with other duals.

Space filling by duals that have

rotational symmetry about z is

thought to be always possible. Addition of duals is possible in several ways but so far

duals have only been constructed by heteromorphic addition along the z axis (adding Ah

to Nh is heteromorphic and adding Ah to Ah is homeomorphic). Homeomorphic addition

along z is possible but no dual has been found this way. Homeomorphic or

heteromorphic addition along a and b hyperplane axes is probably never dual or space

filling. We conjecture that duality only results when addition along the main z ½ spin

axis is heteromorphic and the added units are Ah, Nh species (species could include

truncated units). It is known that space filling for units greater than order 2 uses duals

from both Ah and Nh and requires Ec2, the unity element in multiple numbers. As the

order number increases more possible ways to fill space are available. For instance, with

Ah4 and Nh4 you could have Ah3, Ah2, Nh3 and Nh2 along with several Ec2 to make a

space filling cell.

It is also possible to stretch by adding links along a, b and z axes leaving gaps or holes

inside the units. These holes could be filled by various kinds of added circuits.

Calculating the twist of circuits when holes occur can be involved.

The truncations combined with additions show that a huge number of both dual and non-

dual z symmetric units are possible.

52

Figure 8.2

Nh added to Ah heteromorphs

This dual shows a green Nh3 on the

bottom with two yellow Ah2 units added

to the top of the Nh3. It has composing

formula Nh3(5)#2(Ah2)(1)) where

parentheses show ‘a’ layers used by each

unit in the addition. Since one layer is

shared to produce the addition it has a

total of 7 a, b layers.

It forms a single circuit traveling along

edges of all three units. It has z axis

rotational symmetry. It has 30 links.

Type formula is:

2(4V’,M’,V’,L,V’,2M’,M,V,V,M,V’)

Figure 8.3

Here we have a yellow Nh2 at the bottom

with a blue Ah2 added to the top. Turning

it into a dual flips it upside down, thus it

is its own pseudo self-dual. It has 10

links.

This unit forms a single circuit and has z

axis rotational symmetry. Its formula is

Ah2(3)#Nh2(1) and it fills space by

stacking and has 5 a and b layers.

Type formula is 2(V’,2M’,M,V)

53

Figure 8.4

Ah4 truncated

then added to

itself

Here we have

truncated Ah4

top and bottom

layers. Each is

then migrated ½

layer before

adding as

shown. The

green one is on

top and identical

to the yellow

bottom one. It

is a single

circuit with ½ z

rotational

symmetry. It is

not dual.

54

Figure 8.5

A skewed triple z heteromorph

Add Nh3 at the bottom to Ah3 on one

corner then add two Nh2 units on top of

Ah3 to make this unit. It is dual.

Because the method of adding Nh3 to

Ah3 is not symmetrical about the z axis

I initially thought this unit could not fill

space. Later I found that with the

identity, Ec2 and other dual units it

does fill infinite space.

It forms a single circuit.

55

Chapter 9 Puzzle size small circuits are space filling (TOC) Showing circuit sizes of 8 links or less. Requires compactness (no holes in circuit).

Figure 9.1

Ec2 fills space

Ec2, the unity dual of the

Atomihedron puzzles, fills

space and is its own dual. It is

the smallest possible Eh circuit.

Ec2 has five rotational axes of

symmetry, a, b, x, y, z. The a,

b, x, and y axes are 180 degree

rotational while the z axis is 90

degree rotational.

As shown Ec2 fills space by

stacking in planes. It also can

be used to in combination with

Ah and Nh duals and many of

their compositions to fill space

by filling in gaps.

Type= U (or I)

56

Figure 9.2

Nh2 fills space

This shows that the Nh2

circuit easily fills space by

stacking in planar arrays.

It consists of six Eh links and

has symmetry axes a, b and z.

Type=2(V’, M’, V’)

57

Figures 9.3

Ah2 fills space

Using three Ah2 circuits you can see how this

unit fills space by stacking in planar arrays

very similar to Nh2. It consists of six Eh links

and has symmetry axes a, b and z.

Not shown here is a way in which Ah2 and

Nh2 can fill space in combination with each

other.

The type sequence is 2(V, M, V)

58

Figure 9.4

Ec6 fills space

Consisting of 6 Eh links, the

Ec6 circuit fills space but is

nondual by itself.

The top figure is two Ec6

placed adjacent.

The bottom figure is a single

Ec6 top view.

It has a, b, x, y and z 180-

degree rotational symmetry

axes. It is a regular

tetrahedral expansion of Ec2,

the dual unity element of the

Eh system. Type formula is

2(L, 2V’).

59

Figure 9.5

Ec7 fills space

Ec7 fills space and

completely encloses a

small hidden internal

space. The three Ec7

units at top are displayed

with the uppermost one

inverted to show the two

planar cell arrangements

needed to fill 3 space.

The Ec7 circuit fills space

by stacking in double

planar arrays. An upside-

down orientation of a

unit, as shown by the red

and green and blue unit,

fits on top. Ec7 has

symmetry about the z

axis, similar to many

duals, but it is not a dual.

Type formula is

4M’,V’,L,V’

60

Figure 9.6

Ec8 is nondual but fills

space with over twist

A special Ec8 has a

symmetrical circuit as

shown here that has 90

degree rotational

symmetry about the z

axis and 180 degree

rotational symmetry

about the a, b, x, y axes,

and fills space in

interlocking planar

groups. It is nondual.

It has all double rotation

type M’ Eh link

connections and encloses

a void that is the exact

shape of a rhombic

dodecahedron.

A rhombic dodecahedron

that fills the void in Ec8

can be made with a

circuit of 12 Ph units

with zero twist, so Ec8 in

this form is a

demonstration that Ec8

over twists space.

It stacks like staggered

bricks that interlock in

both a and b planar

directions as the stack

builds up.

Type formula is 8M’.

61

Figure 9.7

This Ec10 circuit

fills space in

combination with

one U element

(Ec2) per Ec10.

Type formula is

2(L, 2V’, V, V’)

62

Chapter 10 Some z symmetric non-duals and a fractal (TOC)

Figure 10.1

Nh2 added to itself along

the z axis, Nh2- helix.

This forms a binary helix

winding in a clockwise

direction. The left unit

repeats Nh2 3 times along

z using 14 links while the

right unit repeats Nh2 2

times along z using 10

links. These units are

nondual. No dual

holomorph with z axis

addition has been found.

Nh2-helix can fill space

only by interlocking to

itself in planes. Type

formula for left unit is

2(V’, M’, L, M’, L, M’,

V’) and right unit is

2(V’, M’, L, M’, V’).

Ah2 added to itself along

the z axis, Ah2-helix

Ah2 can also form a

binary helix along the z

axis. It winds in a

counterclockwise

direction. The left unit

repeats Ah2 4 times along

z, using 24 links while the

right unit repeats Ah2 2

times along z using 12

links. Both are nondual. It does not fill space. Type formula for left unit is 2(V, M, 2V,

M, 2V, M, 2V, M, V). Right unit is 2(V, M, 2V, M, V)

63

Figure 10.2

Ec6 infinite linear duality

The regular tetrahedral unit is Ec6 with type

formula 2(L, 2V’). It fills space as shown on the

left by linear stacking.

What happens when one of these stacks gets the

dual operation? It then becomes an infinite

spiral as shown with a starting version on the

right. It has type formula 2spiral(inf(2V, L)). It

is only a dual at infinity.

Figure 10.3

Larger 4 edge tetrahedra surrounding smaller

tetrahedra produce spirals around spirals when

duality is performed. Larger spirals can be

constructed going around this spiral as shown here

by the partial silver spiral. A binary spiral fits

adjacent to the silver spiral.

64

Figure 10.4

Miniature Eh links in the form

of a fractal like circuit

This model called Nhf was made

many years ago before I

discovered the duality of Nh and

Ah units.

The links are made of very small

wood pieces with each link 8mm

by 14mm in width and length.

And while not very precise it is a

true model. The total dimensions

are 2.25 x 2.25 x 1.5 inches. It

weighs about 2 ounces.

Nhf consists of a single circuit of

about Eh 120 links. It could be

made larger by adding it to itself

iteratively. It is not dual and not

space filling.

Nhf has a, b, and z axes of

rotational symmetry. This

structure shows that many more

interesting Eh structures remain

to be defined and understood

mathematically.

65

Chapter 11 What is symmetry? (TOC)

What is symmetry?

A helical coil in the form of a torus has both symmetrical and asymmetrical properties.

You can rotate it about its main axis one helical turn to return it to its previous position so

that all points look the same as before the rotation. Say it has 100 coils. Then its

minimum symmetry rotation is 3.6 degrees. You could also rotate it around the circular

torus axis and because it is a spring it could be rotated correctly about both torus axes by

any increment and still look the same. Of course, you can also rotate it 180 degrees about

100 axes embedded in and passing through the centered plane of the torus and through

the center of the torus. Yet it has a basic asymmetry due to its helicity. Its mirror image

is its anti-torus. In this sense a particle such as a photon or electron could look

completely symmetrical yet have an inner structure that is basically asymmetrical.

Figure 11.1 Many life forms have bilateral

symmetry. Animal life forms have a natural

need to be bilaterally streamlined and able to

face to the right or left and move either way

quickly and with equal agility. Yet the inner

structure of these symmetrical life forms is

very asymmetrical. The heart is offset from

center, the gut curls in a winding manner, the

blood vessels look like a fractal, and many

other features are only partially symmetrical.

Life uses both symmetry and asymmetry as the

need arises. Externally we must be able to

run, jump, fight, sleep, make love, see

binocularly, smell and hear in all directions,

these being the things requiring bilateral

symmetry. Our senses are generally bilaterally

symmetric because we must exist in a space

that is equal in all directions. But our insides

must deal with providing energy by digesting

food and intaking oxygen. Our inner structures must deal with process, with the passage

of time in a stately manner. Food and air are our energy inputs. They are absolute

necessities. These energies are obtained externally then consumed and processed

internally to be able to obtain more energy. In this sense time requires a necessary

internal asymmetry for life forms, while space demands symmetry.

I am deeply convinced that mathematics is also both symmetric and asymmetric. The

physical world must follow similar rules as well. It is easy to develop a set of rules or

conventions of reality. In math this stuff is so automatic and ingrained that you will have

a hard time convincing anyone that math has basic asymmetries that are just as important,

66

if not more so, than the multitude of symmetries in math. Math and physics are subjects

that cry for symmetry above all else. Yet asymmetry is every bit as important.

Mathematics has a built in self organizing twist or right-handed system. Math would not

work without these asymmetric laws. Usually the asymmetry has a symmetric

component so that the two principles act almost as one. Also, it is worth noting that

many asymmetries have to do with rotation while translation is more symmetrical.

Examples of asymmetry in mathematics: (This is a truly short list of some of the

simplest.)

1. Integers point in one direction which is to greater positivity. Linear asymmetry.

2. Planar Cartesian coordinates are right-handed.

3. Imaginary numbers multiplied always rotate ccw.

4. 3D coordinates are right-handed.

5. Math develops in a time asymmetric, self-organized way, always solving never

un-solving.

6. Addition combined with subtraction is asymmetric. If +a and +b >0 then: (+a

+b) > a or b, (-a -b) < -a or -b, (+a -b)< a, (-a +b) < b. (one > than, and three <

than used)

7. Multiplication of numbers is asymmetric: (1*1) = 1, (-1 *-1) = 1, (-1*1) = -1.

8. Division of numbers is asymmetric: (1/1) = 1, (-1/-1) = 1, (-1/1) = -1.

9. Mathematicians created complex numbers out of imaginary numbers. They

should have been called simplex numbers because they simplify many math

problems. Complex numbers have a built in self organizing asymmetry and this

produces useful properties. Of course, their symmetry properties are equally

useful. They are co-symmetric.

10. Quaternions have asymmetry properties that make them useful. Multiplication of

two quaternions is noncommutative. Click this link for some uses for

Quaternions.

11. Octonions are more asymmetric than quaternions or complex numbers being both

non-commutative and non-associative. For this reason they have not been used

much by physicists but important work is being done by Cohl Furey and others

finding connections of these three number systems to quantum theory.

Notes on time and its mysteries.

Time is global. The only proof of this is nonlocal quantum entanglement. Particles can

be entangled non locally because global time cycles globally simultaneously in all of

space. Local time cycles result from different periodic repeating cycles of global time

such as atomic orbitals, earth years, etc. Rapid motion of matter through space slows

down the local cycles in matter by using up some of the available global cycles as linear

motion local time cycles. Nonlocal entanglement is the result of a local cycle being

stretched in space. The stretching does not happen instantaneously but only as fast as the

speed of light. The entanglement can be stretched any distance since the cycles the

particles use for entanglement are global.

You can not put your finger on your own finger. If time continuously recreates

everything in a single global time cycle, GTC, then you cannot sense the cycle because

when it is off you have no way of being aware of it, when it is on you only know of

67

continuous existence. Since time cannot be proven you may become convinced that it

does not exist

Hypotheses of Z-metry or co-symmetry. One reason why the

Atomihedron system self organizes.

Once there are integers there is no such thing as mirror or anti integers. Negative integers

are not anti-integers. Otherwise (-1)^1/2 would not be imaginary. Also (-1)^2 would be

-1, not +1. Call anti 1 1. Then 1 + 1 = [ ] , the empty set. Does consideration of anti or

mirror numbers produce no usable result? Once a system is chosen the handedness

remains built in after that. Does the creation of mathematics automatically produce the

anti-mathematics of mistakes, bad logic, infinite meaningless chaos, failure, mental

instability, wrong headedness, paradox and the antithesis of all the things that numbers,

equations, precision and exactness, rule systems, algorithms, etc. produce? Or does it

produce its space complement? Thus, the number 1 is surrounded by infinite space, its

complement. This is more like how matter produces space, its true opposite.

If there is no such thing as anti-mathematics, why should there be antimatter? We know

that particles always get created in pairs, particle and antiparticle. But their total energy

is positive meaning that antimatter is not true antimatter. If it were, the two particles

would annihilate to zero energy or not interact at all. We know that the positron is stable,

so it must really be a matter particle, not an anti-electron with positive energy. There

must be an internal twist or handedness to positive energy stored inside these particles

when they are created. Not so farfetched when you realize that we have an internal

(hidden) twist always going the same way in our chemistry and DNA. The production of

an electron produces always also produces a positron, its antiparticle. Yet both electrons

and positrons exist, so they both must be true matter particles.

This produces an organized type of overall symmetry. The right-handed system of

coordinates and positive numbers exists in a surrounding space. So every mathematical

object produces a complementary metric that contains the opposite handedness in the

infinite space surrounding every mathematical object. When everyone agrees to use the

same handedness, the system becomes universal and more efficient. If some DNA

twisted the other way those living forms could not eat forms with regular DNA. The

entire system of life overcomes this obstacle by conforming and using the predominant

DNA.

The number line points to positive right. The x, y, z axes are right-handed. Once a

system is chosen it stays that way.

Mechanical systems also follow these principles. In most countries’ cars drive on the

right because Henry Ford came out with the first mass produced cars that drove on the

right. Other countries could save money by importing cars from the U.S. that drive on

the right. They made their road systems to follow suit. But some other countries also,

68

producing cars were making them to drive on the left with roads to conform, so rather

than changing to right driving they saved money by remaining left driving. When an

American must rent a car in England that person quickly gets used to driving on the left.

Another example is the English system of units versus the metric system of units. This

has produced lots of incompatible products. Metric wrenches do not work on English

bolts as an example. Americans have retained many of the English derived products but

lots of imported items use metric sizes. This results in a boon for tool makers selling

both tool types to thousands of American users. How much more efficient and

convenient it would be to conform to the metric system saving lots of time, energy and

money, leaving room for more tools in our overcrowded toolboxes.

Time and space along with matter-energy and gravity act in a similar way. Right-handed

matter and energy(time) create gravity and space which are its opposite or its

complement.

69

Chapter 12 Unsolved problems, constructions, ideas (TOC)

Recently, with the help of a friend, I was able to finally produce Atomihedra circuits as

wonderful crystal-like translucent models. These make fabulous displays and excellent

puzzles. Only a few photos in this writeup show these units, as the beautiful translucence

is difficult to convey with a photo. So, I have opted to also use opaque photos where

appropriate from units in my collection.

Having spent over 40 years working these Atomihedron puzzle models and developing a

partial mathematical system I am always surprised. Every time I take another look at it, I

discover new stuff. Someone interested might finally shine a light on these self-

organizing structures. Perhaps produce a fine PhD thesis and maybe some new

mathematics. If code could be written to animate the simultaneous rotations of each link,

that turns a unit into its dual, it would be fascinating to watch. The following material is

in no particular order and is meant to show material relating to the Atomihedron puzzles.

Unsolved Problems and Questions

1. Prove that the only duals are the Ah, Nh units and their spin conforming

truncations, and heteromorphic compositions?

2. Find an example of non Ah-Nh duality or prove none is possible. This would

prove (1.) false by counter example.

3. Prove all duals have some infinite space filling solution in combination with other

duals.

4. As Ahn, Nhm orders grow larger they can with fill space more efficiently if n=m

in combination with the unity element. Show it is also possible to fill space if n

does not equal m, in some combination with different orders and the unity

element.

5. Why do duals and dual-like (not fully dual) units twist oppositely.

6. Devise a mathematical system, perhaps something like quaternions, that

symbolically models the Atomihedron or Eh link system, making it possible to

model opposite twists.

7. Write a program that animates a dual turning into its dual.

8. Why does Nhn, where n>=4, always have a single negatively twisted edge circuit

with a positively twisted set of one or more internal circuits? This seems to be an

analogy with the neutron having a zero charge.

9. Write a program that calculates the twist of the individual circuits in a unit.

Include ability to recalculate when any one circuit is removed leaving holes. Also

show order of filling the circuits as the Ah or Nh approaches filled cluster orders.

10. Write a program that plots the surface points of the circuits. This should have the

appearance of a fractal as order increases without limit.

Ideas about time, space, matter, energy, forces, physics, and math:

Some thought experiments.

70

This is mainly about philosophy. In a very real sense modern physics, science and math

are philosophies that use the machinery of mathematics. Isaac Newton was first to

automate a philosophy of movement by creating his calculus machinery. This made the

continuous digital and the digital continuous. Gottfried Wilhelm Leibnitz also did this,

but Newton had a simpler set of rules of force momentum and reaction that got us into a

new age of steady progress. On the other hand, Leibnitz symbolic calculus was easier to

use.

Einstein used thought experiments as a way to blaze a path to mechanize relativity and

gravity. Philosophers have been using thought to analyze stuff since thinking began. But

Einstein knew that he would need a precise machine to make real progress, so his thought

experiments were stripped down and made as simple as possible and he was able to

derive precise equations to describe properties of the universe. Physicists use these kinds

of tools originally commandeered by philosophers. These experiments need not result in

any kind of mathematics but can still point out a trail to pursue.

Experiment 1, can information be lost?

Information to a physicist has to do with the physical state of a system. Quantum theory

implies that a system is exactly described by its wave state (its information).

Assume that some information is lost. This is not allowed since the wave state already

contains all the information of the system. But if some information were lost or the wave

state changed by imperceptible amounts how could you ever prove it? By its very nature

information is always assumed to be complete so any loss would not leave any trace of

ever having been there in the first place. Perhaps information is always being lost but at a

rate that cannot be measured. Entropy is a kind of measure of loss of access to certain

types of physical information. We say the information is there in principle but if there is

no physical means of accessing that information then it should be regarded as lost.

Our experiment goes beyond entropy and posits a loss of quantum information. The

paradox is that you can never prove it. This could mean that information loss is

happening all the time, just in a way that can never be detected. If that were the case the

information loss would need to be small, just enough to keep itself undetectable.

Distant galaxies recede from us faster than the speed of light due to the expansion of

space. This represents permanent loss of information from our location in the universe.

To get quantum information from a closed system you must perform a measurement. Due

to the uncertainty principle of Heisenberg part of the measured information is uncertain

causing a tiny loss of information as far as the person doing the measuring is concerned.

An electron and a positron collide and annihilate into two entangled photons that move

away from each other at the speed of light. The entanglement conserves spin momentum

and photons moving apart conserve local momentum. When one is measured the other

takes on a matching measurement that conserves spin momentum even though the spin

measurement is a partly random selection of how the measurement is done. Thus physics

laws are conserved by information being both lost and gained by the random and non

random parts of the measurement. So physics remains stable if information that is lost is

71

replaced by new information that replaces the lost information in a manner that obeys

conservation laws.

A conjectured conclusion

Conservation of information is not important in an open system, but it is important in a

closed system and becomes more important the smaller the closed system is.

72

11.

Figure 12.1

Making models of Ahn or Nhn

You will need to plug n into these

formulas to find how many of the

pieces shown to assemble an n order

Ah. Either of the two gapped pieces

will work, but the assemblies shown

here are the most stable, especially

for a model with lots of links such

as Ah4 or larger. The two top

figures are z travel, while the

bottom four are a, b linear traverse

links. If you assemble per these

formulas, then you just have to snap

them together.

However, if you want to distinguish

different circuits by different

colored links you will need to

determine this as you build up the

model.

73

Figure 12.2

Quaternions as vectors twist only one way

The i, j, k Quaternion rules of multiplication

were first described by Irish mathematician

William Rowan Hamilton in 1843.

No matter how you multiply any pair such as

ij=k or -ij=-k you get the same vector twist.

This is also known as the right-hand rule with

fingers wrapping around your thumb, twist is

always clockwise.

This is easy to prove by first drawing the

result of multiplying the pair to get the third

vector. Just bend a piece of wire into the

shape shown in the top right figure. Now you

can make sure that the three legs of the wire

line up with all the 8 drawings shown and any

others you care to try.

There are 24 ways of doing this and each one

will obey the right-hand rule. This means that

the vector twist or spin is always in the same

direction. A piece of wire with one leg bent

the opposite way will not line up with any of the ones shown here.

The only way to get negative twist is to change to a left-handed coordinate system and

use a left-hand rule

This shows that the I, j, k Quaternion mathematics is asymmetric and that is what makes

it so useful and self-organizing.

Ah Nh system can produce opposite twists with the dual operation by rotating every Eh

unit about its main axis to make a change in the twist and produce a different sort of

organization. In this sense Atomihedron duality is a brand-new system.

The dual operation could be performed in any order on individual Eh, but all Eh are

rotated simultaneously as a mathematical procedure.

74

Eh spin states Figure 12.3

Here are eight states of motion and twist of a vortex with L being translation, V is torus

vortex motion around the toroid axis, G is geometric twist, and S is spin motion. This

increases to 16 states if negative L is included. However reversing L just turns the 8

states shown upside down. Therefore we only consider G, S and V as reversible.

If EH is allowed to spin and flex inside out in a higher dimension or in partitioned time

slices (like a movie film with individual frames) then it can emulate a vortex.

G is geometric twist and can only go one way so must stay positive in the Eh, Ah, Nh

system. This drops the number of possible states for a dynamic Eh to four. These are SV,

-S-V, S-V, -SV. Two of these could be seen as a hypothetical emulation of the plus and

minus spin of an electron and the other two as a hypothetical emulation of the plus and

minus spin of a positron. Rotation about the main axis, Zs (L), is the spin, S.

75

Figure 12.4

Ah4 Construction This set of four photos shows Ah4 being constructed with Eh links. Top left shows two sets of blue links passing thru the complete circuit of orange links. Top right, bottom left shows progress of the blue links. The white link is the last piece added to complete the construction. The orange links are a separate circuit surrounded by a

single circuit of blue links. The orange links are just an ‘a’ travel stretched circuit of the Ah2 unit. Thus the orange links will repeat by stretching for each 2k order of Ahn where k >=2.

76

Figure 12.5 Nh4 Construction This set of four photos shows Nh4 being constructed with Eh links. The red link was the last piece added to complete the construction. The silver links are a separate circuit surrounded by the single edge circuit of orange links. This edge circuit repeats for all Nhn. The silver links are the first appearance of a central ‘b’ travel circuit similar to the orange ‘a’

travel circuit of Ah4 above. Thus the silver links will repeat by stretching for each 2k order of Nhn where k >=3. The positive twist of Nh causes so much writhing that Nh must grow to Nh(n+2) internal circuits to match with circuits of Ahn. The silver circuit requires 14 links instead of the 6 links required by Ah2.

77

Current research with References and links: (TOC)

(research by mathematical physicists, some involving tetrahedra)

I have tried to make these references readable by anyone and accessible on the internet

where possible. Subjects references are about tetrahedra or knots in physics. The list is

not in any particular order. Where the math gets difficult you can usually get the gist of

what is being said by reading the text. Also includes some links and reprints of my own

work.

Geometrical Patterns of 200,000 Spiral Galaxies Suggest the Universe Has a Defined

Structure TOPICS: Astronomy Astrophysics Kansas State SciTechDaily (excerpt here w full credit)

UniversityMathematics By KANSAS STATE UNIVERSITY JUNE 1, 2020 (CLICK ON IMAGE FOR LINK)

This imge shows an all-sky mollweide map of the quadrupole in the distribution of galaxy spin directions. In this image, the different colors mean different statistical strength of having a cosmological quadrupole at different points in the sky. Credit: Kansas State University

Research reveals asymmetry in spin directions of galaxies and suggests the early universe could have been spinning.

78

Special Elemental Magic: Japanese Scientists Announce a

‘Nuclear’ Periodic Table Kyoto University-Mathematics

By KYOTO UNIVERSITY MAY 29, 2020

https://scitechdaily.com/special-elemental-magic-japanese-scientists-announce-a-nuclear-

periodic-table/

A nuclear periodic table K. Hagino & Y. Maeno Published: 21 April 2020

Foundations of Chemistry (2020) https://link.springer.com/article/10.1007/s10698-020-

09365-5 More explanation of the nuclear periodic table reference.

Periodic Table of Elements Los Alamos National Laboratory (reprinted here

with permission for quick reference) (TOC) Group

1 2 3 4 5 6 7

8 9 10 11 12 13 14 15 16 17 18

Period

1

1 H 1.008

2 He 4.003

79

2 3 Li 6.94

4 Be 9.012

5 B 10.81

6 C 12.01

7 N 14.01

8 O 16.00

9 F 19.00

10 Ne 20.18

3

11 Na 22.99

12 Mg 24.31

13 Al 26.98

14 Si 28.09

15 P 30.97

16 S 32.06

17 Cl 35.45

18 Ar 39.95

4

19 K 39.10

20 Ca 40.08

21 Sc 44.96

22 Ti 47.88

23 V 50.94

24 Cr 52.00

25 Mn 54.94

26 Fe 55.85

27 Co 58.93

28 Ni 58.69

29 Cu 63.55

30 Zn 65.39

31 Ga 69.72

32 Ge 72.64

33 As 74.92

34 Se 78.96

35 Br 79.90

36 Kr 83.79

5

37 Rb 85.47

38 Sr 87.62

39 Y 88.91

40 Zr 91.22

41 Nb 92.91

42 Mo 95.96

43 Tc (98)

44 Ru 101.1

45 Rh 102.9

46 Pd 106.4

47 Ag 107.9

48 Cd 112.4

49 In 114.8

50 Sn 118.7

51 Sb 121.8

52 Te 127.6

53 I 126.9

54 Xe 131.3

6

55 Cs 132.9

56 Ba 137.3

*

72 Hf 178.5

73 Ta 180.9

74 W 183.9

75 Re 186.2

76 Os 190.2

77 Ir 192.2

78 Pt 195.1

79 Au 197.0

80 Hg 200.5

81 Tl 204.38

82 Pb 207.2

83 Bi 209.0

84 Po (209)

85 At (210)

86 Rn (222)

7

87 Fr (223)

88 Ra (226)

**

104 Rf (267)

105 Db (268)

106 Sg (269)

107 Bh (270)

108 Hs (277)

109 Mt (278)

110 Ds (281)

111 Rg (282)

112 Cn (285)

113 Nh (286)

114 Fl (289)

115 Mc (289)

116 Lv (293)

117 Ts (294)

118 Og (294)

Lanthanide Series*

57 La 138.9

58 Ce 140.1

59 Pr 140.9

60 Nd 144.2

61 Pm (145)

62 Sm 150.4

63 Eu 152.0

64 Gd 157.2

65 Tb 158.9

66 Dy 162.5

67 Ho 164.9

68 Er 167.3

69 Tm 168.9

70 Yb 173.0

71 Lu 175.0

Actinide Series**

89 Ac (227)

90 Th 232

91 Pa 231

92 U 238

93 Np (237)

94 Pu (244)

95 Am (243)

96 Cm (247)

97 Bk (247)

98 Cf (251)

99 Es (252)

100 Fm (257)

101 Md (258)

102 No (259)

103 Lr (262)

Alkali metals

Lanthanides

Alkaline earth metals

Actinides

Transition metals

Nonmetals

Post-transition metals

Halogens

Metalloid

Noble gases

An interactive periodic table https://www.ptable.com/ © 2017 MICHAEL DAYAH

Royal Society Active Periodic Table https://www.rsc.org/periodic-table

Physics laws cannot always turn back time (excerpt reprinted with permission from Spacedaily.com) by Staff Writers Amsterdam, The Netherlands (SPX) Mar 24, 2020

80

https://www.spacedaily.com/reports/Physics_laws_cannot_always_turn_back_time_999.h

tml A description of a simulation of the movement of 3 black holes showing that the

math does not time reverse. A quote is included in the text body. (Mathematically proven

Failed Time Reversal, appears to challenge Einstein’s concept of a 4D STC) (excerpt

reprinted with permission from Spacedaily.com (more details also appear above in main

text body)

by Staff Writers, Amsterdam, The Netherlands (SPX) Mar 24, 2020

https://www.spacedaily.com/reports/Physics_laws_cannot_always_turn_back_time_999.h

tml Research Report: "Gargantuan Chaotic Gravitational Three-Body Systems and their

Irreversibility to the Planck Length" “…. Tjarda Boekholt (University of Coimbra, Portugal), Simon Portegies Zwart (Leiden University, the

Netherlands) and Mauri Valtonen (University of Turku, Finland) calculated the orbits of three black holes

that influence each other. This is done in two simulations. In the first simulation, the black holes start from

rest.

Then they move towards each other and past each other in complicated orbits. Finally one black hole

leaves the company of the two others. The second simulation starts with the end situation of two black

holes and the escaped third black hole and tries to turn back the time to the initial situation.

It turns out that time cannot be reversed in 5% of the calculations. Even if the computer uses more than a

hundred decimal places. The last 5% is therefore not a question of better computers or smarter calculation

methods, as previously thought.

The researchers explain the irreversibility using the concept of Planck length [1.6 x 10^-35 m]. This is a

principle known in physics that applies to phenomena at the atomic level and smaller. Lead researcher

Boekholt: "The movement of the three black holes can be so enormously chaotic that something as small as

the Planck length will influence the movements. The disturbances the size of the Planck length have an

exponential effect and break the time symmetry.z

Co-author Portegies Zwart adds: "So not being able to turn back time is no longer just a statistical

argument. It is already hidden in the basic laws of nature. Not a single system of three moving objects, big

or small, planets or black holes, can escape the direction of time."

….I have had this idea for many years but never was able to simulate it. My idea was that the Planck length

is subject to irrational rounding off when oblique motion occurs, causing errors to build to up and producing

irreversible time effects. This process must be going on everywhere in the universe. It is like Edward

Lorentz’s butterfly effect. My idea is similar but here they mention disturbances the size of the Planck

length, not just rounding off the Planck length. So, my idea may not have been refined enough to ever succeed as a proof. Time is partly like an arrow with a natural twisting movement/shape. D Engel

ORDER BY SINGULARITY Perimeter Institute for Theoretical Physics

https://www.perimeterinstitute.ca/seminar/order-singularity

“We present a paradigm for effective descriptions of quantum magnets. Typically, a

magnet has many classical ground states — configurations of spins (as classical vectors)

that have the least energy. The set of all such ground states forms an abstract space.

Remarkably, the low energy physics of the quantum magnet maps to that of a single

particle moving in this space.

This presents an elegant route to simulate simple quantum mechanical models using

molecular magnets. For instance, a dimer coupled by an XY bond maps to a particle

moving on a ring. An XY triangular magnet maps to a particle moving on two disjoint

rings. We can even simulate Berry phases; when the spin has half-integer values, the

particle sees a pi-flux threaded through the rings.”

81

Klee Irwin - The Tetrahedron

https://www.youtube.com/watch?v=xTN9tQGgN6Q Quantum Gravity Research SUBSCRIBE

118K subscribers

Klee Irwin, director of Quantum Gravity Research, talks about the tetrahedron, the most

fundamental building block of our 3D reality according to emergence theory.

….This is a short you tube video about building a code that is used to describe physics

with the simplest ‘information’ bit of 3d geometry, the regular tetrahedron.

Pirsa: 19030113 - 3d Quantum Gravity: from tetrahedra to holography

Speaker(s): Etera Livine video> http://pirsa.org/displayFlash.php?id=19030113

Abstract: 3d quantum gravity is a beautiful toy-model for 4d quantum gravity: it is

much simpler, it does not have local degrees of freedom, yet retains enough complexity

and subtlety to provide a non-trivial example of dynamical quantum geometry and open

new directions of research in physics and mathematics. I will present the Ponzano-Regge

model, introduced in 1968, built from tetrahedra “quantized" as 6j-symbols from the the-

ory of recoupling of spins. I will show how it provides a discrete path integral for 3d

quantum gravity, related to topological invariants and loop quantum gravity and other ap-

proaches to quantum gravity. It is also a perfect arena to investigate boundary theories

and holographic dualities, with a beautiful duality with the 2d Ising model realized

through a supersymmetry, and more. ….A YouTube discussion of a tetrahedral model

for 4d quantum gravity.

The Peculiar Math That Could Underlie the Laws of Nature, Quanta

Magazine, 07-20-2018, An article about Cohl Furey’s work linking particles to numbers

like octonions.

https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-

20180720/

From QM “…To reconstruct particle physics, Cohl Furey uses the product of the four di-

vision algebras, R⊗C⊗H⊗O (R for reals, C for complex numbers, H for quaternions

and O for octonions) — sometimes called the Dixon algebra, after Geoffrey Dixon, a

physicist who first took this tack in the 1970s and ’80s before failing to get a faculty job

and leaving the field. … Furey began seriously pursuing this possibility in grad school,

when she learned that quaternions capture the way particles translate and rotate in 4-D

space-time. She wondered about particles’ internal properties, like their charge. “I real-

ized that the eight degrees of freedom of the octonions could correspond to one genera-

tion of particles: one neutrino, one electron, three up quarks and three down quarks,” she

said — a bit of numerology that had raised eyebrows before. The coincidences have since

proliferated. “If this research project were a murder mystery,” she said, “I would say that

we are still in the process of collecting clues.” ”

82

In the Atomihedron system 6 dimensions of Eh link travel exist. In addition, there are the

three cubic spin axes x, y, and z and the four vertex axes of regular tetrahedron travel

producing a total of 13 simplex axes. With time slicing, embedding the Ph and Eh links

in three dimensions works very efficiently to produce all 13 cubic dimensions.

Quantum Braiding its all in and on your head https://quantumfrontiers.com/2016/05/22/quantum-braiding-its-all-in-and-on-your-head/

An interesting descripting of quantum or time like braiding with some insight to how to

prove the Yang-Baxter relation.

“Life is a braid in spacetime”: http://nautil.us/issue/9/time/life-is-a-braid-in-

spacetime An essay by Max Tegmark, MIT Physicist.

….After the essay there is a lively blog about determinism and free will with the one side

expressing Einstein’s view that everything is predetermined in advance and the other side

vehemently opposed to this as a static boring world devoid of free will or any kind of free-

dom at all.

….QM’s uncertainty principle seems to contradict GR in this respect at least. Tegmark

thinks the universe is just mathematics. This is controversial since math itself is incom-

plete and subject to error and paradox.

….Rounding off errors are built into math due to the proof of irrationality. Thus the but-

terfly effect is a true effect and Tegmark seems to have gotten a bit too smitten with the

power of mathematics. It is bound to be powerful. The invention of the number 1 in-

stantly creates an infinity of successor numbers. Thus it would behoove us to beware of

this power leading us to believe it is more than just our weak grasp of logic.

Knots and Quantum Theory _ Institute for Advanced Study.html

By Edward Witten · Published 2011 (TOC) https://www.ias.edu/ideas/2011/witten-knots-quantum-theory

A very interesting writeup about how the Jones polynomial has created a connection of

knot theory to quantum physics, by a leading theoretical string theory physicist. This a

short essay that is very readable, not inscrutable mathematics. It explains the amazing

connections between knot theory and quantum physics. Only about 15 pages.

Holomorphic Factorization for a Quantum Tetrahedron, L.

Freidel, K. Krasnov and E. R. Livine, Communication in Mathematical Physics, 297

(2010) 45, arXiv: 0905.3627

“…For the case n=4, the symplectic manifold in question has the interpretation of the

space of "shapes" of a geometric tetrahedron with fixed face areas, and our results pro-

vide a description for the quantum tetrahedron in terms of holomorphic coherent states.

We describe how the holomorphic intertwiners are related to the usual real ones by com-

puting their overlap. The semi-classical analysis of these overlap coefficients in the case

of large spins allows us to obtain an explicit relation between the real and holomorphic

description of the space of shapes of the tetrahedron. Our results are of direct relevance

83

for the subjects of loop quantum gravity and spin foams, but also add an interesting new

twist to the story of the bulk/boundary correspondence.”

….This is way beyond our puzzle Atomihedron but shows that tetrahedra do figure into

modern physics in some ways.

Also by L. Feidel, Seminar: "twisted geometry, a geometrical description of spin

networks" Talk given at the GR18 Conference in Mexico city Mexico

Knots and Physics 3rd Edition https://doi.org/10.1142/4256 | July 2001 Pages:

788 By (author): Louis H Kauffman series on knots and everything (University of Illinois,

Chicago) World Scientific

…. A very comprehensive, often cited, treatise on knot theory and its relation to quantum

theory, DNA and much more. Requires a minimum mathematical understanding of Knot

theory and QM. For sale in digital or book form.

The Knot Book, by Colin C. Adams, Originally published by W. H. Freeman, 1994.

Here is a link to a 27 page preview provided by the current publisher The American

Mathematical Society, 2000.

https://www.google.com/books/edition/The_Knot_Book/RqiMCgAAQBAJ?hl=en&gbpv

=1&printsec=frontcover

I have a very worn copy of the original edition. It has been one of my best references,

explaining knot theory in a very simple and clear manner. You could not go wrong on

this. It covers many aspects of knot theory including the Alexander and Jones

polynomial, biological applecations, topology, higher dimensions and includes an

illustrated beginning table of knots and links. Colin Adams did a superb job.

Hofstadter’s amazing butterfly Nature Published: 15 May 2013 pro-

duces periodic patterns quite similar to the filled circuit clusters produced by Ah2(k^2) orders. However

the Ah clusters are complete knotted-linked, circuits not spectrum/gap energy’s. This fractal has become

an immensely popular and important image, in many versions. A big difference is there are no simple re-

peats like in Ah, 1/3, /12 etc. due to the fractal infinities. Ah is finite but could be made to look fractal for

large Ah. Doug Engel Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices

• C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao,

• J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi,

84

• K. Watanabe, K. L. Shepard, J. Hone & P. Kim Nature volume 497, pages598–602(2013)Cite this article

• 6940 Accesses 749 Citations 102 Altmetric Metricsdetails Abstract

Electrons moving through a spatially periodic lattice potential develop a quantized en-

ergy spectrum consisting of discrete Bloch bands. In two dimensions, electrons moving

through a magnetic field also develop a quantized energy spectrum, consisting of highly

degenerate Landau energy levels. When subject to both a magnetic field and a periodic

electrostatic potential, two-dimensional systems of electrons exhibit a self-similar recur-

sive energy spectrum1. Known as Hofstadter’s butterfly, this complex spectrum results

from an interplay between the characteristic lengths associated with the two quantizing

fields1,2,3,4,5,6,7,8,9,10, and is one of the first quantum fractals discovered in phys-

ics. Image credit: creative commons

85

3-D quantum spin liquid revealed Phys.org

https://phys.org/news/2020-05-future-technologies-d-quantum-liquid.html MAY 11, 2020 Future information technologies: from by Helmholtz Association of German Research Centres

Two of the four magnetic interactions form a new three-dimensional network of corner-sharing triangles, … An interesting digital triangle point hinging model concerning crystal like behavior in a liquid. Image credit: from Phys.org.

86

KnotRobot Page 1 (TOC) (Knot machine or Re-combinator) by Doug Engel 2008, reprinted, edited from Puzzleatomic.com

I reprinted this because it was inspired by the Atomihedron and demonstrates magical knot properties.

The KnotRobot consist of two interacting semi cylinder loop

recombinators. Thus it can tie knots, create links, create

knotted links, create the above knots and links with twists in

the ribbons if ribbons instead of strings are used for the loops.

It can be used to investigate some aspects of knot theory. It is

useful as a puzzle when tying to the handle of a coffee cup and

many other kinds of knot situations as the user decides to

dream up and pose for anyone to solve. It represents many

years of research trying to understand the mysteries and self

organizing properties of knots.

"The Knot Book" by Colin C. Adams, 1994, W. H. Freeman and

Company, was a helpful aid in learning some of the elementary math knot nomenclature and in deriving a preliminary

attempt at a simple KnotRobot system notation. The reader can refer to it for any of the knot terms such as the 9(9)

knot, amphicheiral knots, the figure 8 knot, etc.

There are many different kinds of knot machines used by industry.

One early example is the device invented to tie knots in twine, used

by a farmer's hay baler. These devices generally tie two ends of

thread, string, rope or wire together. The KnotRobot is different. It

was designed as a puzzle and a tool to investigate mathematical knots.

Thus, it never ties two ends together. It is always in the form of a

pure mathematical knot-link system with no open ends. As a puzzle

many different problems and demonstrations can be executed with it,

limited only by one’s imagination. The following explanation of the

KnotRobot uses the KR2 binary system, called K for short.

The KnotRobot is normally worked by making alternate rotations

about two axes. We can call them A and B for simplicity, and to not

confuse them with regular x and y axes. Refer to Fig. 1 to see these

axes. If more than two rotors are involved, they can be referred to as

Kp, Kq, Kr, Ks, ... etc. To get even more specific an upper rotor can

be called Kpu, a lower rotor Kpd, etc.

To make the knot rotations always begin and end your rotations about

the vertical, or A axis. For instance, to make a single loop from two

loops bring the two 1/2 cylinders, P and Q, fully together then make a

180 degree clockwise rotation, (looking into the end you are rotating).

This is a positive rotation. Do anticlockwise for a negative rotation

about the A or B axis. Figure two shows the beginning of this 'A'

rotation.

Now pull them apart and you have the simplest possible knot known

as the trivial knot, or 0 knot, or the unknot. We started with two

trivial knots and ended up with one trivial knot that combines the two

we started with. The rotation about A axis can be written as (1) where

the (1) stands for the 180 degree rotation about A Thus K(0)=(0,0)(1).

The right side of the equal sign means to start with the two trivial unknots, perform the single A rotation

and you get the unknot (0) or K(0). This is shown in Fig. 3. Since the (0,0) is normally implied we do not

usually write it down. We might write K(x)=(1). then note in the text which knot K(x) is.

87

Page 2

Figure 4 shows that the two starting loops can be

linked together by making two 180-degree rotations

about A or (2). When you pull P and Q apart you have

a pair of linked loops. The formula for this could be

written K(x)=(0,0)(2), where the x would be the

Alexander and Briggs notation for the simplest link of

2 components, or K(2 )=(0,0)(2). The big 2 means

that the simplest two dimensional drawing of the links

(called a knot or link projection) shows two places

where the loops cross. The small 2 means it is a link

system of two components(links), and the small 1

means it is first one listed in the classification.

Figure 5 shows that the two starting loops can be

made into a trefoil knot by making three 180-degree

rotations about A or (3). When you pull P and Q apart

you have the simplest nontrivial knot or the trefoil

knot. We write K(x) = (3) for convenience.

Figure 6 shows the how a single 180-degree rotation

about B is made. This brings our formula to (3 1). It

is not yet a complete formula because as we stated

above a complete knot or knot-link formula must

begin and end with a rotation about the A axis, and

therefore must have an odd number of A and B axis

rotation numbers listed.

In Figure 7 we have made one anticlockwise, or a -

180-degree rotation about A. This brings our formula

to (3 1 -1). The resulting knot has been rearranged

to show the usual depiction of the figure 8 knot, or the

Alexander and Briggs 4(1) classified knot. It is its

own mirror image, also known as amphicheiral. In

the application of the knot robot formulas this

rearrangement is never allowed. Our formula can

now be written K(4,1)= (0,0)(3 1 -1), or K(x)=(3 1 -

1), as we would normally write.

The 4(1) means the first four crossing knot.

Surprisingly there is only one 4(1) knot since it is its

own mirror image. However, there are two 3(1) knots

since the trefoil knot has a mirror image!

88

An Interesting property of the KnotRobot page 3

Fig. 8

above

shows

the

9(9)

knot

and

its

mirror image. Below are four formulas for 9(9) and their negative inverses or mirror images. The

negative inverse, -\K(), means reverse the rotations and order of the moves that created the knot.

This creates the mirror image with KR ending up at the opposite side of the knot. Another way to

create the mirror image is to negate all the operations that create the knot,= -K(). Thus one

formula always gives four formulas for creating the same knot: 1. create the knot with the original

formula,=K(), 2. reverse the rotations,= -K(), 3. reverse the order of the moves, back to front, the

inverse=\K(), 4. reverse the rotations in reverse order, the negative inverse=-\K(). Using the

distinctly different formulas below for making 9(9), you can first do the one on the left then

follow with any one of those on the right to cancel the whole operation and produce the two (0,0)

loops you started with. Or you could do one on the right then any of those on the left to get (0,0)

1. K(m) = (-4 -1 2 1 -3) -\K(m) = (3 -1 -2 1 4)

2. K(n) = (-3 -1 -1 -2 -1 1 -4) -\K(n) = (4 -1 1 2 1 1 3)

3. K(o) = (-3 1 -1 -1 1 1 -3) -\K(o) = (3 -1 -1 1 1 -1 3)

4. K(p) = (-2 1 3 -1 1 1 -2) -\K(p) = (2 - ,-1 1 -3 -1 2)

A knot that is its own mirror image can have a surprising KnotRobot formula. The surprise is that

the formula executed twice erases the knot back to two trivial loops (0,0). This is because, in a

very real way, the formula is its own negative inverse. Here are three different sets of rotations

and their inverses that produce the 8(12) knot, which is an amphicheiral knot. Anyone of these

formulas followed by itself produces the trivial loops if you started with them. Let K(i) = K(f)

+ \K(g) + K(h). Then K(i) + K(i) = (0,0).

1. K(f) = (3 1 -2 -2 -3) \K(f) = (-3 -2 -2 1 3)

2. K(g) = (2 -2 1 -1 -3) \K(g) = (-3 -1 1 -2 2)

3. K(h) = (1 -1 -2 1 2 -1 -3) \K(h) = (-3 -1 2 1 -2 -1 1)

89

KnotRobot formulas and Problems Page 4

Since a KnotRobot formula can be written in 4 different ways, any given formula can be combined with

itself in 16 different ways, KK, K-K, K\K, K-\K, -KK, -K-K, -K\K, -K-\K , \KK, \K-K, \K\K, \K-\K, -

\KK, -\K-K, -\K\K, -\K, \K. Four of these produce, (0,0). These are the four that are combined with their

negative inverse, K-\K =-K\K = \K-K = -\KK = (0,0).

The four that are combined with themselves, KK, -K-K, etc. all make the same knot with two being mirror

images of the other two and with the KnotRobot ending up at two antipodal positions in the four knots.

The four that are combined with their negatives, K-K, -KK, etc. all make the same knot with two being

mirror images of the other two and with the KnotRobot ending up at two antipodal positions in the four

knots (see Figure 8).

The four that are combined with their inverses, K\K, \KK, -K-\K, -\K-K all make the same knot with two

being mirror images of the other two and with the KnotRobot ending up at two antipodal positions in the

four knots.

A knot formula can be combined with a different knot formula in 4x4x4 = 64 different ways. Many of thes

will produce different knots. There is 16 for K(x)K(y), 16 for K(y)K(x), 16 for K(x)K(x), and 16 for

K(y)K(y). With three different formulas the number of combinations in pairs and with themselves in pairs

is 4x4x4x6=384.

The KnotRobot is similar to the John H. Conway system of rational tangles. It differs in the sense that it

always remains a mathematical knot-link system, while Conway’s tangles are only made into a knot or link

structure at the end of the operations. See "The Knot Book" by Colin C. Adams, 1994, for an easily

grasped explanation of the rational tangle system. The KnotRobot, as presently realized is preferentially

polarized around the A axis by only allowing the first and last rotations to be on the A axis and requiring the

final split along the B axis. Thus it is probably more restrictive than the Conway system. However it is

truly robotic in not requiring the user to keep track of which loose ends to combine at any point in the

execution of the rotations since the ends are never allowed to come apart. The KnotRobot makes

investigating elementary knots and knot puzzles mechanically easy. The KnotRobot forms a mathematical

group.

Unsolved Problems (A knot theorist may already know the answer to some of these)

1. Most of the knots made with the KnotRobot, using only the two rotors, P and Q, are prime knots. Are all

such knots prime knots? See "The Knot Book" by Colin C. Adams, 1994, W. H. Freeman and Co., for a

simple definition of prime knots and composite knots.

2. Can composite knots be made with just a two rotor P and Q KnotRobot, KR2? Composite knots are

knots obtained by combining two prime knots. Probably another KnotRobot would be needed, R and S, to

make a composite knot, K(x)#K(y). Making a composite knot multiplies the two knots. For instance

K(0)#K(x)=K(x), so that K(0) is like multiplying by the number 1.

3. Assuming problem 2 is false, can a composite knot be made using only two rotors P and Q, of KR2 by

breaking the rearrangement rule? Thus after a knot has been made you would move the rotors about in the

knot before continuing your rotations.

4. One problem is to find the simplest knot robot formula for producing a given knot.

5.Using only two rotors, P, Q of KR2 can the KnotRobot produce any kind of prime knot?

6. Can the two rotor KnotRobot generate all the same kinds of knots as the Conway system of rational

tangles?

7. How can we tell what kind of knot a given KR formula will produce?

8. What happens to the character of a knot when each element of the rotation formula is multiplied by a

whole number? There might be some simple rules or a mathematical formula for the changes in crossing,

linking and ribbon twist. Would this be special for amphicheiral knots?

9. What happens to the character of a knot when each element of the rotation formula is squared?

10. When you sum the absolute values of the operations of a formula it is usually close to the crossing

number of the knot produced. Is there a way to tell the crossing number by looking at the formula?

(Perhaps why a knot that is its own mirror image will undo itself when the knot robot performs the tying operation

twice is due to the mirror symmetric operations. For instance, numbers that are their own mirror image might be 5,

-2, 2, -5. Then doing the numbers twice reverses all the moves back to the start twice!!)

90

Triangular and higher symmetry KnotRobots Page 5

By dividing the rotor sub cylinders into 1/3rds, 1/4ths, 1/5ths,

etc. we can obtain KnotRobots with higher symmetry and

greater complexity as to the kinds of knot s and links that can

be produced with less effort the higher the symmetry. For

instance the trinary system, KR3, or K3 for short is depicted

in Figure 10. The unit rotations are 120 degrees so that

K(0)=(0,0,0)(1). This says that one rotation(120 degrees) of

KR3(0,0,0) produces a single unknot. So with a single unit

rotation we make the three unknots into a single unknot. With

two unit rotations a trefoil knot is produced. With three unit

rotations first three component link is produced having 6

crossings.

Given a KR4 system, a unit rotation of 90 degrees produces

the unknot. Two unit rotations produces two linked links

known as the first 5 crossing two component link. Three unit

rotations produces a nine crossing knot. Four unit rotations

produces a multi-crossing four component link.

In general for a KRn system the first unit rotation always

produces a single unknot. If n is a prime number, then every

unit rotation that is not a multiple of 360 degrees produces a

single knot of some kind and every whole multiple of 360

degrees produces a link system of n links. Of course at this

point we have not even discussed unit rotations about all the

possible horizontal axes. Although the complexity produced

is high the organizational properties of these higher order

KnotRobot knot and link systems ought to be high, as well, so

that a full analysis for the simpler kinds of formulas could

probably be extended, and made general with simple

algebraic formulas.

For any KRn the property where K + -\K = (0,0...0) should be

true if the two different K formulas are developed and no

rearrangements of the rotors are made within the knot. Is the amphicheiral property true for

n=3 or greater? Thus, if K is amphicheiral, then K + K = (0,0...0). If so it would be very

surprising.

A Circle Puzzle KnotRobot extended system

In a moment of fantasy this strange idea for a knot system using circle puzzles jumped to the

fore. The rotors can not be lifted away from the device, as contemplated. You could make

rotations then try to determined what knot links were produced. One problem would be to create

different sets of moves to produce the same structure, then determine if the negative inverse of

one cancelled the moves of the other. You could try out the amphicheiral property as well. Other

than that it might be quite interesting just to knot it up then solve it. the strings are removable in

case it gets too tangled up. You can also experiment with fixed ribbons to see how ribbon twist

91

tracks knot systems, for additional moebius-like diversions.

92

THE SET OF MAGNECURVES AND THEIR PROGENEY, IN UNIMODE,

MULTIMODE, ETC. (a special look at cycle double covers)

Short paper by Doug Engel 2012, Originally published by Gathering for Gardner 2012

in the commemorative book only given to attendees(should be viewable online at G4G).

It was inspired by the Atomihedron and the way the paths always point the same way

when parallel. Also published in the unsolved problems section of American

Mathematical Monthly, Jun-Jul,2000, p563. Click this link to read the complete article:

https://www.puzzleatomic.com/MAGNECURVES.pdf

The Atomihedron and the Theory of Replication, Self-Pub. Loose-leaf book, 1989, on

the Atomihedron puzzles by Doug Engel, 1989. Orig. abt. 90 pages. Contains my

Philosophy & /thoughts, inspired by the Atomihedron, describing the universe as a series

of replication instants where each time instant recreates everything globally with motions

and forces all in a new incarnation, i.e. things have all moved by the amount expected.

This has some similarity to an idea of universal evolution, currently part of a theory being

investigated by theoretical physicist Lee Smolin. Leibnitz proposed that nature at the

smallest scale has monads, the smallest organic structures that build up the universe.

F igure 3 F igure 4

93

Definitions formulas tables and information (TOC)

Ph is the symbol of the Protohedron a space filling tetrahedron 24 of which are used to

construct the Eh link. Phn refers to a closed twisted circuit of n hinged Ph tetrahedra.

Ph24 with 2 twists is used to derive the Eh link.

Circuit Is called a circuit only if it is a closed loop of Eh links.

Ecn referred to as a closed circuit of n Eh links.

Dual could also be referred to as Edual. The duality of the Ahn and Nhn meta-tetrahedra

is a main theme of this work. It does not seem to be valid for any other Eh circuit units.

It does create some finite to infinite structures. But no simple unit has been found where

it works to make duals where the duals are not Ahn-Nhn conforming. To make Ahn turn

into Nhn and vice versa perform a ¼ spin of all the Eh main axes, x, y and z.

Zs is the main spin axis. All Ahn, Nhn have a main spin axis we refer to as Zs as a

default to simplify things. You could also use Xs or Ys as main spin axes for instance to

animate the system with a computer. Spin around Zs is always ½ or 180 degrees.

Eh is the link used to construct all Ah, Nh and their additions and truncations. It is called

and Electrihedron or the Ehedron. Eh is self dual and also has a 2 link solution known as

the unity or identity element of the system called U or Ec2, c standing for closed circuit.

Type connections. There are 3 different Eh pair connections possible, type M, type V

and type L. Refer to the table of contents for the chapter explaining the use of the types

The types are a kind of chemical-like way to write a formula for how to construct small

Eh circuits.

Ah is the symbol for the Atomihedron or the Ahedron a meta-tetrahedron of linked knots

where n = 2, 3, 4, … is the order of Ahn. Ahn concentrates mainly on even order Ahn.

Nh is the symbol for the Neutrohedron or the Nhedron and is the dual of the Ah. Nh is a

meta-tetrahedron of linked knots where n = 2, 3, 4, … is the order of Nhn. Nhn is more

complex than Ahn.

twist is a twist given to a chain of Ph or Eh before connecting into a closed circuit.

Ahn( ) formulas for volume, twist, n = 2, 3, 4, ...

Ahn(Lv), Lv = link total volume (M, V, L type conn.), Lv = (2(n^3) + 3(n^2) - 5n)/3

Ahn(Lsv), Lsv = link total surface vol. (M, V type conn.) Lsv = 2(n^2) - 2n

Ahn(Labv), Labv = link total traverse vol (L type conn.) or a, b links vol.

Ahn(Labv) = (2(n^3) -3(n^2) + n)/3 = (Lv-Lsv)

The Ph24 circuit (as presented in this book) is given a clockwise, cw twist of 720 degrees

= 2pi before connecting into a closed circuit to create the Eh link.

If you moved along the circuit you would rotate ccw because of the cw twist.

The Eh link also has a ccw external writhe twist of 90 degrees = pi/2. Each link in a Ehn

circuit therefore, adds ¼ positive link writhe, Lw, twist. Thus Lw twist is Lw = Lv/4.

However the physical twist given to an Eh circuit can be positive or negative. This

94

allows us to give a straight line of connected or L type links a zero twist per link.

Twist calcs. Some of these twist formulas are tentative, not proven (as indicated).

tlktw = total linking twist. Ahn(tlktw) = (n^3 – 3(n^2)- 4n +12)/6

tctw = total circuit twist. Ahn(tctw) = n^2 - 2n + 1

twrtw = total writhe twist Ahn(twrtw) = (n^2)/2 + n/2 +1

ttw = total twist =

AHn(ttw ) = AHn(tlktw) + AHn(tctw) + Ah n(twrtw) = (n^3 +6(n^2) - 25n)/6 +4

ttwpl=total twist per link = Ahn(ttwpl ) = Ahn(tv)/Ahn(ttw)

Lvsv = Link visible surface volume z, a, b Ahn(Lvsv) = 4n^2 - 4n – 2

Lhtrv = hidden traverse links vol (L type).

Ahn(Lhtrv) = Ahn(Lv) - Ahn(Lvsv) = (2(n^3) -9(n^2) + 7n)/3 +2

L4ev = Link 4 edge volume NHn edge vol Ahn(e4v)=16n-26

(all Ahn twists are negative)

Ahn Lv Lsv Labv tlktw tctw twrtw ttw ttwpl Lvsv Lhtrv L4ev

n=2 6 4 2 0 -1 0 -1 -1/6 6 0 6

n=3 22 12 10 0 -4 -1 -5 -2/11 22 0 22

n=4 52 24 28 -2 -9 -3 -14 -11/52 46 38

n=5 100 40 50 -7 -16 -6 -29 -23/100

n=6 170 60 110 -16 -25 -10 -51 -41/170

n=7 266 84 182 -30 -36 -15 -81 -66/260

n=8 392 112 280 -50 -49 -21 -120 -99/392

n=9 552 144 408

n=10 750 180 570

n=11 990 220 770

n=12 1276 264 1012

n=13 1612 312 1300

n=14 2002 364 1638

Ahn Circuit data

nc=total number of circuits Ahn(nc) = approx. formula

ncc=total number of circuit clusters Ahn(ncc) = a function of n

nabl = total number of a, b layers in z direction Ahn(nabl) = 2n-1

nal = tot number of ‘a’ layers in ‘b’ direction Ahn(nal) = n-1

95

nbl = tot number of ‘b’ layers in ‘a’ direction Ahn(nbl) = n-2

tatw=tot ‘a’ entangling twists in a (a,b) winding layer Ahn(tatw) = func of loc.

a,b layer sequence bottom to top n=2, a,(),a n=3, ab, a, ba n=4 ab, aba, ba …

for instance at n=2 a()a is 3 layers tall with no b link, and n=3 has a non self

winding layer in the middle of the metatetrahedron. (it winds both up and down)

The front right most triangle of the meta-tetrahedron is F front left most is F’, back

right is B, back left is B’.

Nhn initial formulas, n = 2, 3, 4, ... for volume, twist, …

Lv = link volume(#links) An(Lv)= Nhn(Lv) = (2(n^3) + 3(n^2) - 5n)/3

Lsv = linl total surface vol. M’, V’ typw conn. Vol. Ahn(Lsv)= Nhn(Lsv) = 2n^2 - 2n

Labv = link total traverse vol. or a,b links vol._ Ahn(tv)= Nhn(Labv) = (2(n^3) -3n^2 + n)/3

L4ev Link 4 edge circuit vol Nhn(L4ev) = 16n – 26

Lnev = non edge volume (+twist) (= Lv – ev4) Nhn(Lnev) = (2(n^3) + 3(n^2) - 53n +78)/3

Lisv total internal surface vol Nhn(Lisv) = 2(n^2)-10n + 12

Lesv total edge circuit surface vol Nhn(Lesv)= 8n - 12

Letrv total edge traverse (a,b) vol Nhn(Letrv) = 8n - 14

Litrv total intern’l trav. vol Nhn(Litrv)= Nhn(trv) - Nhn(tetrv)

= (2(n^3) -3n^2 -23 n)/3 + 14

Nhn Lv Lsv Labv L4ev Lnev Lisv Lesv Letrv Litrv

n=2 6 4 2 6 0 0 4 2 0

n=3 22 12 10 22 0 0 12 10 0

n=4 52 24 28 38 14 4 20 18 10

n=5 100 40 50 54 46 12 28 26

n=6 170 60 110 70 100 24 36 34

n=7 266 84 182

n=8 392 112 280

n=9 552 144 408

n=10 750 180 570

n=11 990 220 770

n=12 1276 264 1012

n=13 1612 312 1300

n=14 2002 364 1638

96

ettw edge circuit total twist NHn(ettw)= 2n – 3 - (4n-7)?

titw = total internal positive twist____________ NHn(titw) = -AH(n-2) - (ettw

ettw titw tlktw tctw ttw ttwpl

n=2 -1 0

n=3 -5 0

n=4 -9 5

n=5 -13 14

n=6 -17 29

These formulas and tables represent a beginning effort to formalize the Ah-Nh

system.

Nhn Circuit data

nic=total number of interior circuits Nh(n+2)(nic) = Ahn(nc)

nicc=total number of int. circuit clusters Nh(n+2)(nicc) = Ahn(ncc)

nabl = total number of a, b layers in z direction Nhn(nabl) = 2n-1

nal = tot number of ‘b’ layers in ‘a’ direction Nhn(nal) = n-1

nbl = tot number of ‘a’ layers in ‘b’ direction Nhn(nbl) = n-2

Orientaton of clusters in Nh(n+2) turned 90 degrees about Zs from orientation in

Ahn

97

Appendix A Additional information, diagrams, and illustration (TOC)

(A Spectral table of rich cluster levels)

Figure A.1

Comparison of

filled levels shows

that these Ah orders

produce cluster

levels with

increasing

frequency as n

increases.

Figure A.2

A simple way to refer to a

portion of a Ph chain is

shown. The first number is

the q edge length of P.

Thus 3P refers to a P with

edge 2^3 and volume of

8^3. The number after P is

the P chain length. Thus

the q length in ( ) is

redundant since it can be

found with the first two

numbers.

98

Figure A.3

Figure A.3 shows the approximate overall shape of the Ah, Nh central filled clusters of

circuits. Similar shapes could be drawn for off center level clusters such as ½ level.

These levels occur symmetrically in pairs.

99

Figure A.3.1 (TOC)

Model of Nh6

This is the first

Neutrohedron that

displays two internal

circuits. These two

circuits follow mirror

image paths of the two

circuits that form Ah4.

Of course Ah4 does

not have the

surrounding negatively

twisted edge circuit.

The yellow and red

circuits are each

strongly positively

twisted. This structure

emulates a Hydrogen 2

nucleus where the

surrounding circuit

provides one negative

charge and the two

internal circuits

provide two positive

charges for a total

nuclear charge of +1.

The two internal

circuits provide an atomic mass of 2 units.

100

Figure A.4

For the Atomihedron puzzle models to be taken seriously the system must be developed

mathematically. There are enough quantum number emulations to indicate that

101

something interesting is shown and deserving of more investigation. The waves of

quantum mechanics are probability waves. The figure above conjectures possible

tetrahedral like Ahn waves produced by the electron orbital system.

Figure A.5

The tetrahedron toy puzzle model is conjectured here concerning possible waves in the

atomic nucleus. This concerns the Nh puzzle which emulates the Proton-Neutron system

by having a negatively twisted edge circuit with positively twisted internal circuits.

Thus, negative, and positive like charge pairing to create the same neutral and positive

102

charges of the nucleus. This is a real puzzle, why the emulation would carry so far (both

in Ah and Nh and their duality) if there is no connection, only coincidence.

Discussion of magic numbers: (TOC)

The semi magic numbers are 2, 8, 20, 28, 50, 82, 126.

Each of these numbers is a sum of 2(k^2) + 2(j^2) + 2(i^2), where k, j, i are integers.

Thus k=1 yields 2, k = 2 yields 8, k=1, j = 3 yields 20,

k=1, j = 2, i = 3 yields 28, k = 5 yields 50, k = 1, j = 5, i = 6 yields 126.

It is easy to find these numbers in the Nh central and off central circuits for certain rich

cluster orders. For instance, a cluster at level ½, -1/2, with 2 circuits, would be balanced

when occupied by 8 particles, 4 at the ½ and 4 at the -1/2 level. With 8 circuits you

would have balance with 32 particles (k=4). Many other balancing magic number

scenarios can be found for magic numbers not exactly conforming to the 2(x^2) sums.

After a great deal of thinking but with no mathematical connection to wave mechanics to

back it up I can only put forth a possible scenario of how it might work.

Scenario: The Nh forms a single nucleus structure. It looks like a group of protons and

neutrons when bombarded. This is because the circuits of the internal cluster levels each

carry one positive charge. The edge circuit being very adjacent and negatively charged

can carry multiple electrons. The negative and positive charges cancel except for

additional internal positive circuits more than the negative electrons in the edge circuit. It

looks like a set of individual protons and neutrons when bombarded to break up the

nucleus. Thus, one part of the conjecture is that both the central cluster and non-central

clusters in the nucleus are occupied circuits and contribute to the nuclear structure. It

could also explain why some nuclei are unstable, for instance if non central clusters are

not symmetrically occupied, thus the loss of magic of the magic numbers that produce the

more stable nuclei.

A great deal is known about the atom since the math is proven and makes exactly

accurate predictions, and we have here only a toy puzzle with some intriguing properties

still needing a great deal of mathematical development. If anyone wants to risk it there

could be a huge payoff if particles can be proven to have an internal mechanical like

structure, a self-organizing system at all scales.

103

Figure A.5 (TOC)

(see below figs.)

This Shows dowels with

45 degree cuts along with

the numbers of each

piece needed to build any

n order Ahn, n= 2, 3,

4, …, Atomihedron. If

the formula gives a

negative answer that

piece is not used.

These pieces cannot be

used to build the dual

Nhn unit.

104

Figure A.6

This Shows the dowels

of Figure A.1 used to

make Ah2, Ah3, Ah4

and Ah5.

Since these are not

made with Eh links the

geometric self

organizing and Eh

structural twist

properties are lost.

Also the fun of building

these units by solving

the Eh gapped puzzle

pieces is gone.

However it does

preserve the knotting

and linking structure

and could be used in

that regard. It also

makes a great display if

done in fine woods

showing the different

circuits for n > 3.s

105

Figure A.7

Here are diagrams meant to show space

filling for U, Ah2, Nh2, Ec6, Ec8. This

is reprinted from previous publications

by the writer.

Each diagram shows the standard Zs

spin axis vertical orientation for U, Ah2

and Nh2.

The 4 top units fill space by forming a

line of units which then can form planes

of units which can then stack to fill 3

space.

Ec8 below stacks by interlocking in 3

dimensions.

106

Figure A.8

This shows Ah5 with 4 circuits and 100

total links. As an odd order unit, not

nearly as much research has been done

with odd order units. Odd order units do

not have a centallly locted top or bottom

Eh link and do not have a centally located

middle circuit while all even order

unitshave the Ah2 ‘a’ travel stretched

circuit.

107

Figure A.9

Plot of Ah72, n =

2(6^2). It shows

specific letterings

indicating how the

different level clusters

are derived by

reflecton across the

main diagonals.

108

Figure

A.10

Shows all 5

plots of the

first five

shells plus

the other

circuits

filling each

meta-

tetrahedron.

It shows

how the in

between

circuits

look

random and

non-

repetitive.

109

TABLE 2 THE FIRST 32 ATOMIHEDRA, Ahn, n=1,2,3,… and showing k, n,

where n = 2(k^2), k = 1, 2, 3,…

Ah Nh

k n no. link kn edge no. int. link

circuits volume circ. circ. vol

1 2 1 6 1 0 6

3 1 22 1 0 22

4 2 52 1 1 1 52

5 4 100 1 1 100

6 3 170 1 2

7 5 266 1 4

2 8 10 392 1 3

9 1 5

10 7 2 1 10

11

12 12 1 7

13

14 14

15

16 16

17

3 18 27 4182

19

20 24

21

22 26

23

24 34

25

26 28

27

28 47

29

30 38

31

4 32 56 22816

110

TABLE 3 THE FIRST 24 ATOMIHEDRA shells Ahn, n = 2(k^2), k = 1, 2, 3,…

Ah

k n total circuits link volume

1 2 1 •6 1

2 8 10 392

3 18 27 4182

4 32 56 22816

5 50 81 85750

6 72 186 253896

7 98 176 636902

8 128 292 1414272

9 162

10 200

11 242

12 288

13 338

14 392

15' 450

16 512

17 578

18 648

19 722

20 800

21 882

22 968

23 1058

24 1152

111

ABLE 4 Ph CIRCUITS, AND THEIR FLEX CYCLES

Ph n = CIRCUIT SPACE NO. OF NO. OF TOTAL FLEXIBLE

LENGTH TWIST FORMS CYCLES # OF POS. IN 3D?

8n n (PL-8)/8 =F =C =F*C

8 1 0 3 1 3 Y

16 2 1 4 16 64 Y

24 3 2 5 6 30 Y

32 4 3 7 32 224 N

40 5 4 N

TABLE 5 Ph CIRCUITS THAT LINK TOGETHER

# LOOPS an Eh CIRCUIT

THAT THAT APPROX -

8n n INTERLOCK IMATES

8 1 0

16 2 2 2

24 3 2

32 4 4 8

40 5 2 2

48 6 2 6

TABLE 6 EDGES, FACES, VERTICES OF SMALL Eh CIRCUITS

CIRCUIT NO. NO. NO.

LENGTH OF OF OF

N EDGES FACES VERTICES e=f+v-2 ?

2 52 24 30 Y Unity

6 90 40 52 Y Ec6

6 94 40 58 N Ah2

6 104 46 58 N Nh2

8 96 44 52 N 8M’

112

A brief history (from my web output on puzzleatomic, 2013)

https://www.puzzleatomic.com/ATOMIC%20pg1.htm This is the link to refer to for the

page numbers cited in this history area.

The Atomihedron is an outgrowth of experiments with flexgons and flexahedrons.

Martin Gardner wrote a famous column in Scientific American on mathematical

recreations. In the May, 1958 issue he described hexaflexagons. These devices are made

of twisted loops in the shape of a hexagon consisting of hinged equilateral triangle strips.

These Hexaflexagons have the very exotic almost magical ability to turn inside out in a

regular manner that seems organic and hypnotic.

Hexaflexagons were invented in 1939 by Arthur H. Stone. Every so often someone

devises new versions of these devices. My experiments began with linear hinged strips

of right triangles about a year before I became aware of the hexaflexagons. These could

be wound up in helix fashion. Once introduced to these amusing devices I was hooked

and began experimenting with solid flexagons I called flexahedrons. Several different

mathematical articles were published on these devices such as rings of regular tetrahedra,

hybid flexahedrons, hexaflexatetrahedron, etc.

It was found that you could not make structures that always flexed inside out when you

increased the number of links and increased the twist as much as possible. You had to

have almost maximum twist or the structure would have no organised system to it. The

twist removes most of the freedom and restricts it to organized movement only . This

being the case, and with the current research at the time showing how DNA is twisted I

theorized that twist is a very important self-organizing principle.

I decided to experiment with a tetrahedron that fills space, is symmetrical and has two 90

degree solid angles and four 60 degree solid angles. Around 1964 I began

experimenting with twisted loops. This involved a lot of head scratching to attempt to

find the best possible configuration and correct twist. After many models the Eh or

Electrihedron, a loop of 24 tetrahedra hinged together into a loop with two twists, was the

only one that seemed to satisfy my requirements.

Further experimentation with the Eh gave me the idea that the Eh loops could be linked to

each other. Of course, when linked then the flexing inside out is no longer possible

except as a kind of time slicing system. From that time, about 1964 and on I have

experimented with these linked Eh circuits at odd times. In the early 70's I discovered the

Atomihedron structure which can consist of many Eh in linked and interwoven circuits as

the order n, of Ahn increases.

Ahn is the Edual of the Nhn, Neutrohedron which reverses the twist of the Atomihdron.

Very recently I have found that these Eduals always fill space in combination with

Eduals. This is a unique mathematical property but it has not yet been fully proven.

113

So far this system has not attracted a lot of attention since the mathematics of it has not

been fully developed. As far as the names used, ie. Electrihedron, Atomihedron,

Neutrohedron, Protohedron, these are fanciful and speculative since no direct connection

with quantum theory has been proven. However, I would point out that the system does

many things that no other system does. It self-organizes on many different levels. That

alone should make it worth serious study. All this self-organization comes about from

ultimate simplicity, a single link, the Eh. This link can be seen as a loop of 24-unit I, j, k

vectors.

This research started in the early 60's and continues in the present year 2013, representing

over 50 years, on and off tackling this structure.

The Atomihedron is quite involved. As it goes to infinity the circuits become very

complicated. Clusters of circuits form at rational fraction distances of the order n, from

the edge to the center, with the biggest cluster in the center.

In the mid 80's I submitted a paper to The American Mathematical Monthly. They

wanted to publish it but only if I could add more mathematics to the description. At the

time I could not find a period of time where my attention would be solely turned to the

Atomihedron, so I never resubmitted the paper. Two other papers were published, both

only minor portions of the EH system. One in "The Mathematics Teacher" in October,

1968 titled "Can Space Be Overtwisted?", the other in a student mathematics journal

called "The Pentagon" Spring, 1972 titled "How a Flexible Tetrahedral Ring Became a

Sphinxx". This paper showed the order 2 Atomihedron and the order 2 Neutrohedron as

well as the Eh2 identity circuit for Eduality and the Eh6 circuits. Higher order

Atomihedra could have been illustrated but space was limited to 8 pages. These papers

may eventually be included on this web site. I also self published two booklets, one

called "A Philosophy of Twist" in the 60's and one titled "A Theory of Replication" in the

80's.

I made puzzles of the 6-link version of wood and plastic in the 80's. The plastic ones

were made with a steel mold I welded together, and an injection plastic molding machine

I made of steel pieces of scrap and a Texas Instruments PLC controller, which now, for

many years sits unused. A friend ran the machine producing hundreds of the Eh links. At

present an injection mold has been made costing me several thousand dollars. It is a two

cavity and produces two different Eh links each with a different gap allowing the links to

snap together on a very satisfying way to produce all the known Eh circuits. These links

are quite precise and fun to work with. Then about 2016 I purchased two 3D printers for

my friend Troy Black and had him print Eh links for me. These are great since they can

be scaled to a small size making it possible to produce a much larger Ah, Nh, models to

experiment with.

Anyone wishing to work on this theory is welcome. I believe a program could reveal if it

has more properties that emulate atomic particles. I can provide information if interested.

114

This system has a beautiful built in self organization property. Every time I come back to

it and resume experimenting new discoveries come to light. It is very non intuitive. The

way the links combine is always a mind game to figure out, even for figures that I have

built many times. I have added solution videos to Puzzleatomic.com.

Twist calc discussion

A specific Eh circuit has several different kinds of twist. These Eh circuits are

geometrically simple and precise and therefore each kind of twist can be exactly

calculated. The previously developed total twist for Ah and Nh ignores unit twist plus

strucural twist =utt. However any attempt to relate twist to energy in a specific circuit

will need to sum these constant twists with the total circuit twist in a given circuit. Thus

the total twist when including utt would be an addition to the polynomials already given.

The topological twist has both a global and a local metric. A short segment of Eh links

has a local twist. The entire closed circuit has a local twist confined to its own ribbon

twist and a global twist depending on how many entwinned circuits are occupied that link

with it. An unoccupied circuit leaves holes and does not add linking twist. The

entwinning and up down meandering add linking and writhing twist respectively. This

complicates the calculations but by breaking them down to specific types of twist the

exact twist can be determined. A computer program is the best way to do this but this

discussion here showing how to write the program is needed.

List of twists in an Eh circuit:

unit linking twist= ult= +1 per link (see page 11)

unit structural twist= ust = -1/4 per link (see page 11). It could also be seen as -1/2 twist

per link since an adjacent link is in the same orientation when it is in a straight line such

as xx. Thus as the new link moves forward along itself it must rotate 180 degrees to put it

back in the same orientation, but moved forward one link.

unit total twist = utt = ult+ ust = 0.75 per link=+0.75L where L is the total circuit length.

If the twist of ust is -1/2 then utt = 1/2.

The below twists are negative for AHn and positive for interior NHn circuits.

circuit twist = ct = 1/2 times no. of changes of xy or traverse direction in the circuit.

Circuit twist is linear when seeking total twist in AHn or NHn, see polynomial for it in

calcs.

linking twist = lt = Sum((xt>=3)-1)/2 (see page 6)

Where each linear x traverse of links is 3 or longer subtract 1 and add them all up then

divide by 2 to get the total linking number. Each unit of linking number is equivalent to 1

twist(from knot theory).

Linking twist is linear when seeking total twist in AHn or NHn, see polynomial for it in

115

calcs.

writhe twist = wt = (Sums(MinMax(abs)Sum of z-4))/4 (see page 6 Puzzleatomic

Atomihedron writeup. It is way to crammed with information!! But may help here.)

Writhe twist is not linear and therefore it must be

calculated separately per n if seeking Ahn or Nhn total twist. A polynomial for its

average estimate is possible.

circuit total twist = ctt = ct+lt+wt+utt.

the above should be enough info on twists to begin progrmming

Each of these twists is affected if some circuits are missing in an entangled group.

More notes from 2013

This is an attempt to show that the twist energy of these central circuits matches the

electron energy as far as the order of filling orbitals of the elements is concerned but has

not been finished or mathematically correct/proven, the reasoning it exists if there is any

physical connection is sound.

According to knot theory a pair of linked loops has two crossing in a plane projection

therefore it has a linking number of 1 (divide crossing number by 2). This is exactly

equivalent to a twist of -1, 0, or +1, or anywhere in between such as +0.333, depending

on the projection of the linked loops you set up. To calculate the twist we could include

the unit twist caused by each Eh linking to its adjacent neighbor. As stated before this is

equal to 1 per Eh so is equal to the number of Eh in any given circuit. (I have since

changed this to zero for a straight line of links, type L connections. But it becomes

negative in Ah circuits and positive in Nh circuits and is just the circuit twist 5, 2020)

Since the circuit, linking and writhe twist or clw, in Ah is negative we subract these from

the unit twist, utw to get the twist energy of a given circuit.

However as some orbitals are not occupied the clw would have a smaller l than if the

entire meta tetrahedron was complete. At present we will just assume that all orbitals are

filled to calculate the linking twist in a given circuit and see how this matches actual

orbital filling.

More notes from 2013 (speculative)

clw = circuit + linking + writhe twist

Circuit twist is the twist of a closed circuit. Linking twist in a Ah or Nh is the twist added

by one closed circuit linking to another closed circuit. Writhe twist occurs because the

circuits can wind around opposite to normal twist. See the knot definition of it.

utw = unit twist (eh to eh linking, utw can be ignored for now)

stw = s orbital twist stw … ptw, p orbital, etc.

s orbital twist stw = 2n+2 - clw where 2n+2=utw or unit twist or circuit length

c=1, l=(n-2)/2, w=0. so stw = 2n+2 -(1+(n-2)/2+0)

stw = 3n/2+2

116

or stw=2n+1 minus lk=n/2-1for all crossing lks

p orbital twist ptw. Let 6(n-5) +12 =6n+10 = utw

c=3, l=6(n-2)/2, w=3 so ptw= 6n+10-(3+3n-2+3)

ptw= 3n+6

for 1 loop

p length surface =12 blength= 16 al=6(n-2)

utw=6n+16

6n+16 -(3+3n-2+3)= 3n+12 =

stw's modified for links not crossed since not yet filled.

n=2, and 1stw=5

n=8 and 2stw=14 2ptw=30

n=18 and 3stw=33 3ptw=60

n=32 and 4stw=62 4ptw=102

n=50 and 5stw=104 5ptw=156