Golden Mean OCR

8/14/2019 Golden Mean OCR

1/23

SOME OBSERVATIONS ON THE GEOMETRIC CAUSE OF THE GOLDEN MEAN

RELATIONSHIPS IN NATURE

A RESPONSE TO KEPLER

BY MICHAEL CARR

August 14, 1993

PREFATORY NOTE: For quite some time now I have beenexamining geometrical aspects of solid state fusion, theMoon-Hecht nuclear model and the Platonic solids. In the courseof this work I stumbled across an apparently "unrelated" ideawhich may help to demystify the cause of the Golden Mean ratioin Nature. Although I have briefly discussed this with variouspeople, it really requires a more thoroughly developed

presentation in written form to facilitate discussion. I hopeyou will find this useful and stimulating.Comments, criticisms and other ideas will be greatly

appreciated.


2/23

SOLID" OR LATTICE GEOMETRIES

Even though we have spent years trying to develop theoutlook that action (whether cycloidal, conical, circular etc.)should form the basis of measure, it remains the case that whenwe look at "solid matter" we tend to axiomatically superimpose aCartesian coordinate structure upon it. For example, the numberpi is an artifact of the process of measuring circular action ina totally arbitrary superimposed and unrelated metric.

Even in crystallography there are artifacts of thisproblem. For example the very common structure which is closestpacked around a central sphere or atom which results in acubeoctahedral configuration is labeled the "face centeredcubic arrangement;" this name is completely misleading as to thecause, nature and significance of this structure. I wasrecently shocked to see an exhibit in the Houston Museum ofNatural History purporting to show that an octahedral crystal ofa mineral is actually built of tiny cubes which form stepped orsawtooth edges which "are too small for the naked eye tosee"!!! Octahedral minerals are no more composed of tiny cubes

than are human beings.Let's digress for a moment to do a little geometricalexperimentation; we'll return to our main theme in a moment.

Suppose that you were to attempt to create a crystal orlattice of any uniform substance; how would you begin? Forpurposes of modelling we could use uniform spheres such asstyrofoam balls or marbles, although to make our structuresclearly intelligible I have found the use of small sticks(toothpicks bound together with dried peas or mini marshmallows,or cotton swabs glued together) to make our structures mostclearly visible. Of course most minerals aren't made of uniformcomponents and mixing marbles and baseballs does have an effecton what kind of model structures you can create. Similarly, the

number of chemical bonds which an element can make or thepresence of external fields of various types can effect the typeof structures it can compose, but let us see if we can make anygeneral observations beginning with uniform components having nospecial bonding predispositions.

Two marbles together create the minimum relationship (one)which can be conveniently symbolized by a cotton swab. By addingone more marble we create two more relationships or two morecotton swabs and create the triangle. This is the minimum twodimensional construction: the two legs of each angle arestabilized by the third leg to create two dimensional tensionand compression strength independent of materials or glue used.The triangle gives you minimum internal dimensions. If we addsides to transform the triangle into a quadrilateral or pentagonetc., we lose compression strength even though we increasepotential internal area. We use the term "potential internalarea" because for polygons beyond the triangle there must be


3/23

FIGURE 1Beyond the triangle only external action, bracingor "forces" create any regularity to "planefigure" constructions.

FIGURE 2

CONSTRUCTED TETRAHEDRA


4/23

FIGURE 3

CLOSEST PACKING AROUND ONE SPHERE

Notice that the constructed cubeoctahedron ischaracterized by four triangulated hexagonsemanating out from the nucleus. When expandedfurther outward in each dimension, each vertexbecomes the center of a new cubeoctahedron. Alsonotice that the half octahedra which togetherwith the tetrahedra compose the cubeoctahedronare combined with other half octahedra so thatthe expanded structure which Buckminster Fuller

labeled the "Octet Truss" is actually composed ofoctahedra and tetrahedra.


5/23

some external "force" or other bracing structure to maintainmaximum internal area associated with for example a regularpentagon. Blowing up a balloon inside a cotton swab pentagonwill give us a regular pentagon; without the application ofthis internal pressure which puts the cotton swab pentagon intouniform tension, it has no intrinsic regularity or uniformity(see Figure 1).

Let us return to our marble model: If we place a fourthmarble atop our triangular base, we create the three dimensionalpacking minimum: the tetrahedron. This is as close as closestpacking ever gets; this is the "hardest" any structure ever getsin terms of compression, independent of materials used (seeFigure 2).

If we keep adding layers to our tetrahedron in militarycannonball fashion, we create a structure or crystal in whicheach marble not on an edge or face is surrounded by twelve othermarbles or vertexes in cubeoctahedral array. We notice thatvertexes are arrayed in four hexagonal planes around the centralmarble or vertex (see Figure 3). We notice that our structureis actually composed of tetrahedra and octahedra. We also

notice that this structure contains what I call a structuralcube as opposed to a Platonic cube. (If one were to build asguare and then a cube using the same methods described above,we would discover that no matter how many layers of cubes we addto one cube, a cube or layers of cubes are inherently unstableand twist and bend in all directions [see Figure 4]. That iswhy architects have to put diagonal stabilizers in buildings. Astable cube is actually double braced with two tetrahedra: eachtetrahedron makes one diagonal across each face [see Figure 5].If all vertexes are joined, what I call a structural cube iscreated. In the tetrahedral structure we created, find anoctahedron and remove everything around it except the adjoiningtetrahedron on each face; that's a structural cube [see Figure

6]).So what is the significance of all this? Well, we have

discovered that the tetrahedron, octahedron and cube are allcontained within a closest packing geometry; each ischaracterized by hexagonal geometry. Each is in a sense atransformation of the other (see Figures 7 and 8). Each ischaracterized by immediate or entropie communication ordissipation of "energy" or other impinging "forces" to twelveeguidistant neighbors in four hexagonal planar patterns. Thereis no more efficient way to dissipate "energy" or to communicateaction in a solid than through a hexagonal geometry. As Keplerrefers to this in hypothesizing the cause of the hexagonal

geometry of the snowflake in his On the Six Cornered Snowflake"...the formative faculty chooses six-corneredness, not underduress of any material or spatial necessity, but solely becauseit is allured by this aptness whereby the hexagon elsewhere canform a plane without remainder and more than any other shapewith the same capacity resembles a circle."


6/23

FIGURE 4

The twelve stick cube model depends on outside orinside "forces" or bracing to maintain anysemblance of regularity; otherwise you will endup with a near planar hexagon as the cubecollapses upon itself.

FIGURE 5

TETRAHEDRAL BRACING FOR A STABLE CUBE


7/23

FIGURE 6

THE STRUCTURAL CUBE WITHIN CLOSEST PACKING

FIGURE 7

The tetrahedron, octahedron, structural cube andcubeoctahedron are each found as cuts withinclosest packing geometry.


8/23

While Golden Mean characteristics exist in the geometry ofthe marble, sphere or atom because of their curvature, hexagonalgeometry predominates in the realm of crystals and compounds.As we shall see shortly, the break between the geometry of thetetrahedron, octahedron and cube versus that of the icosahedronand dodecahedron is as definite as the asteroid belt whichKepler hypothesized to exist between the inner and outerplanets.

The point to be made here is not that all inorganicmaterials exist only in a hexagonal geometry or that living ororganic substances only have pentagonal characteristics; ratherwe seek the cause for the very great correlation, noticed by theancient Greeks as well as Kepler, between the Golden Mean andthe presence of life. Nor do I mean to imply that all inorganicsubstances form structures which can be easily compared to aclosest packed lattice: shortly we shall examine some inorganicsubstances with pentagonal surface geometries.

Think instead about what separates the living from thenon-living. The drop of water becoming a snowflake is readilydrained of its heat along hexagonal axes which readily transmit

energy to approaching cold air; crystals and metals, because oftheir hexagonal geometries are excellent conductors of energy invarious forms and at various electromagnetic wavelengths. Onthe other hand a plant is able to take impinging light energyand "twist it back upon itself" to conserve it or store it in"storage rings" (through the excitation of electrons to higherorbitais) for use in the building of the plant. Living beingsdo not just transmit energy but process or "curve into loops" atleast parts of the energy impinging upon them to be used forvarious purposes.

Living beings must take stored energy and curve it anddirect it within themselves to accomplish tasks necessary forsurvival, growth and reproduction. To facilitate this process,

living beings must build surfaces and channels wich must reflectthe necessity of "curving" or conserving energy. Of coursethere is a great difference between curvature of action which isa continuous process and "curvature" of construction which is adiscrete process; nevertheless, the two are related and bothexhibit the Golden Mean ratio. Pasteur put a great deal ofground-breaking effort into investigating the effects of thiscurvature of constructed organic substances on the refraction oflight; he also hypothesized the connection between curvature andlife. This leads us across the great divide.


9/23

GOLDEN MEAN OR SURFACE GEOMETRIES

Crossing the great Keplerian divide brings us to the realmof the non-nuclear Platonic solids or surface solids: theicosahedron and dodecahedron.

Using the same construction methods as above,let's create atriangulated hexagon. Notice that it can be bent in all sortsof waysit has no special curvature (see Figure 9).

Now, remove one side of the triangulated hexagon to create atriangulated pentagon or end of an icosahedron (see Figure 10).Now we have created "curvature"a surface with an inside and anoutside: we have created a structure which does not communicateaction in a plane but deflects action by the minimum uniformlyconstructible deflection from the plane. This minimumdeflection is the basis of the Golden Mean or Golden Sectionratio across the inside of the triangulated pentagon.

Now take the same materials and build a row of equilateraltriangles around the perimeter of the triangulated pentagon sothat six cotton swabs converge at each vertex or corner of thetriangulated pentagon. Now connect the extended triangular

"teeth" with cotton swabs between the vertexes of the triangular"teeth" so that we have created in effect an expandedtriangulated pentagon. Note that the Golden Mean ratio stillholds for the expanded pentagon. But also note that around eachof the edge vertexes or corners of the original triangulatedpentagon we have six equilateral triangles. In other words wehave grown the triangulated pentagon through hexagonalextension. By iteration of this process you can create gianttriangulated pentagons that are composed of umpteen hexagonsexcept for one triangulated pentagon which determines thecurvature and defines the process of expansion (see Figure 11).

Similarly if you build an icosahedron using cotton swabs andtake off the top triangulated pentagon, you can extend the

midsection of the icosahedron through hexagonal extension tocreate a tubular structure which preserves the pentagonal crosssection of the tube even though the tube is entirely composed ofhexagonal arrays around each vertex except at the twoicosahedral end caps (see Figure 12). Of course you may alsoevenly grow an icosahedron through hexagonal extension to nearlyinfinite, nearly spherical dimensions (see Figure 13). Similarstructures can be made from triangulated dodecahedra. Noticethe the pentagons determining the curvature of the Radiolarianskeleton in Figure 14. Though this example is not triangulatedand suffers some deformation, its structural principle is clear.

It is interesting to note that the two most critical

substances necessary for life on Earth are known tospontaneously form these structures. Buckyballs and Buckytubes


10/23

FIGURE 8THE OCTAHEDRON, TETRAHEDRON AND STRUCTURAL CUBE

IN CLOSEST PACKING OR OCTET TRUSS CONFIGURATION

FIGURE 9THE FLEXIBLE TRIANGULATED HEXAGON

10


11/23

FIGURE 10

THE TRIANGULATED PENTAGON AS THE CAUSE OFCURVATURE IN SURFACE CONSTRUCTION

11


12/23

of carbon have been discussed at great length in 21st Centurymagazine and the New Federalist newspaper, but it may not be sowell known that water forms similar structures around foreigncompounds in hydrates (see the table by Irving Klotz in Figure15).

12


13/23

FIGURE 11EXTENSION OF PENTAGONAL CURVATURE THROUGH

HEXAGONAL EXTENSION

13


14/23

FIGURE 11 (continued)

FIGURE 12TUBULAR HEXAGONAL EXTENSION OF AN ICOSAHEDRON

PRESERVING PENTAGONAL CROSS SECTION

14


15/23

FIGURE 12 (continued)

FIGURE 13EXPANSION OF ICOSAHEDRON THROUGH HEXAGONAL

EXTENSION

15


16/23

FIGURE 14

RADIOLARIAN SKELETON

(from Symmetry by Herman Weyl)

Faces

Vertices

Edges

Volumeenclosed

12

20

30

160 A3

14

24

36

230A3

15

26

39

260 A3

16

28

42

290 A3

Simple polyhedra found in crystalline clathrate hydrates ofClass I and Class II molecules.

FIGURE 15

HYDRATE POLYHEDRA

(from "Polyhedral Clathrate Hydrates" by IrvingKlotz delivered to the Frozen Cell Symposium,1969)

16


17/23

FIGURE 16VIEWS OF A CRUDE SNAIL SHELL MODEL BUILT USING A

COMBINATION OF PENTAGONAL CURVATURE AND HEXAGONAL

EXTENSION

17


18/23

FIGURE 17CREATION OF THE PENTAGON AND GOLDEN MEAN THROUGH

SIXFOLD CIRCULAR ACTION

FIGURE 18SIX SPHERES SURROUNDING ONE IN A PLANE

18


19/23

FIGURE 19THE TRIANGULATED PENTAGONTHE MINIMUM, WHOLE

UNIT CONSTRUCTED DEFLECTION FROM THE PLANE

19


20/23

CONCLUSION

Just as the Golden Mean is characteristic of theastronomical and atomic realms because of their curvature, Ihypothesize that living beings are characterized by surfaces ofmany types, sizes, purposes etc., but that it is this geometryof pentagonal "curvature" determining hexagonal extension ofcurved surfaces within living beings which is the reason thatliving beings are characterized by Golden Mean ratios.

As examples I have built crude snail shell models using thisprinciple of pentagonal construction for curvature, andhexagonal construction for extension (see Figure 16).

Note that the same hexagonal extension which in thecrystalline structures allowed for maximum entropy in fourseparate planes outward from a central sphere in closest packedarrangement, in the case of surfaces preserves the curvaturecreated by the triangulated pentagon even as it "smoothes" thecurvature to be more spherical.

In Kepler's words: "Of the two regular [Platonic] solids,the dodecahedron and the icosahedron, the former is made up

precisely of pentagons, the latter of triangles but trianglesthat meet five at a point. Both of these solids, and indeed thestructure of the pentagon itself, cannot be formed without thedivine proportion as modern geometers call it." In a sense thisis true, but this approach tends to identify the effect as thecause and thereby creates unnecessary mystification andconfusion. Of course there are ways to create a pentagon whichignore its real derivation as a mere projection onto the planeof a sixfold equal division of circular action on the surface ofa sphere (see Figure 17) or more relevant to this caseatriangulated pentagon. This also causes mystification.

Let's do one last quick experiment. Take a marble andsurround it with one layer of other marbles on a plane surface.

Exactly and precisely six similar marbles will surround it (seeFigure 18). Remove one marble encircling the center marble.Now constrict the remaining five encircling marbles until eachis touching the center marble and two other encircling marbles.What happens? As we did before with cotton swabs we havedeflected our geometry by the minimum whole unit difference fromthe plane and have created the maximum "curvature" inconstructive terms (see Figure 19). The Golden Mean or DivineProportion in constructed surfaces and in living beings is theeffectnot the cause of this mini-max process.

Of course all of this seems to be so ridiculously simplethat one is tempted to say that it's too simple. Yet every

other treatment of the Golden Mean which I've seen does notaccount for the necessity of the Golden Mean as a commoncharacteristic of living organisms except perhaps in the case ofmollusks. If one starts from the standpoint of taking analgebraic ratio and then creating "Golden" triangles,

20


21/23

rectangles, pentagons or even spirals within the Cartesiancoordinate system, it is possible to make interestingconstructions; however, the central question is leftunanswered. Nature does not go around calculating square rootsor pi. These irrational numbers are the result of oursuperimposition over nature of an arbitrary rectilinearCartesian coordinate system, or in other words the attempt tosquare the circle or to understand the living in terms of thenon-living. The development of supercomputers and partialdifferential equations has allowed for the solution toinfinitesimal degrees of accuracy of extremely complex functionsas in computational fluid dynamics; however, this is equivalentto putting the square pegs in the round holesit can be done,but it lacks the simplicity of truth.

21


22/23

FIGURE 20

PALLADIUM-DEUTERIUM LOADING CONFIGURATION

In a one to one loading configuration, deuteriumoccupies the center of every octahedron withinthe closest packed palladium lattice. As thismodel demonstrates, the constructedcubeoctahedron and the structural cube areinversions of each other in which each vertex ofthe cubeoctahedron occupies the center of eachoctahedron (complete or partial) within theconstructed cube and conversely each vertex of

the constructed cube occupies the center of eachpartial octahedron within the cubeoctahedron.

What you have is palladium in close packed oroctet truss arrangement with respect to eachother palladium atom and deuterium in closepacked or octet truss arrangement with respect toeach other deuteron. It is almost as if thepalladium is a matrix for creating deuterium orhydrogen metal.

Note that each deuteron-deuteron relationshipcrosses each palladium-palladium relationship atright angles. Is this what allows the Coulombrepulsion of deuterons to be screened out?

Also note that after this one to one loadingis achieved, the next largest spaces which couldbe occupied by deuterons are the interiors of thetetrahedra within the palladium octet truss. Are

22


23/23

these tetrahedra the crucibles of solid statefusion? As Dr. Takahashi asked, is it possibleto drive two deuterons within the interior of oneof the tetrahedra to facilitate a fusionreaction?

Or, does the fact that each deuteron issurrounded equidistantly by twelve otherdeuterons facilitate some other coherent wavephenomenon as in a laser or in Dr. Preparata'stheory of superradiance? By sealing one or moreexterior surfaces of this deuterium lattice, isit possible to use the entropie character of theoctet truss or lattice in a negentropic way toconcentrate energy or action onto surface"anvils" to fuse two deuterons?

While the answers to these questions isunknown, there is little doubt in my mind thatthe geometry of both the palladium lattice andthe deuterium lattice are of primary importance.

Golden Mean OCR

Documents

Transcript of Golden Mean OCR