A NEW ORDER IN SPACE ASPECTS OF A THREEFOLD ORDERING … · vi : a new order in space ~ platonic...

CH. VI : A NEW ORDER IN SPACE ~ PLATONIC AND ARCHIMEDEAN POLYHEDRA AND TILINGS 41

Chapter VI

A NEW ORDER IN SPACE ~ASPECTS OF A THREEFOLD ORDERING OF

THE FUNDAMENTAL SYMMETRIES OF EMPIRICAL SPACE,AS EVIDENCED IN THE PLATONIC AND ARCHIMEDEAN POLYHEDRA ~

TOGETHER WITH A TWOFOLD EXTENSION OF THE ORDER TO INCLUDE THEREGULAR AND SEMI-REGULAR TILINGS OF THE PLANAR SURFACE

FIRST PUBLISHED IN THE INTERNATIONAL JOURNAL OF SPACE STRUCTURES, VOL. 6 NO.1,UNIVERSITY OF SURREY, UNITED KINGDOM, 1991 (PREPARED IN RESPONSE TO THE EDITOR�’S INVITATION).

~ DEDICATED TO KEITH CRITCHLOW FRCA ~

ABSTRACTThere are generally considered to be five Platonic(Regular) and thirteen Archimedean (Semi-Regular)polyhedra (leaving aside the Kepler-Poinsot polyhedra,and disregarding the regular prisms and antiprisms).1These are important as they represent the most regulararticulation that the space of our experience is capableof. But as far as I am aware, a fully satisfactory patternwhich situates these polyhedra in their properinterrelationship has not previously been advanced.

Keith Critchlow proposes the tetrahedron andtruncated tetrahedron as nuclear or �‘over�’ solids, togetherwith a twofold periodic arrangement of secondary regular�‘parent�’ solids and their truncations for the remainder.2Cundy and Rollett propose a similar arrangement,3 withthe first two solids of tetrahedral symmetry, and the restof octahedral or icosahedral symmetry. Coxeter suggeststhe Platonic solids fall naturally into two classes, the�“crystallographic�” solids (tetrahedron, octahedron andcube), and the �“pentagonal�” polyhedra (icosahedron,dodecahedron and Kepler-Poinsot solids).4 But in myopinion these arrangements, although valuable, do notadequately integrate all the polyhedra into a rigorous andcomprehensive order.

This writer proposes a threefold arrangement, whichdoes appear rigorous and comprehensive. Thus the regularand semi-regular polyhedra are conveniently classifiedinto three parallel sets, according to which one of onlythree spatial patterns of symmetries each exhibits: {2,3,3}-fold TetraTetrahedral, {2,3,4}-fold OctaHexahedral, or{2,3,5}-fold IcosiDodecahedral symmetry. These are thesymmetries of the quasi-regular octahedron,cuboctahedron, and icosidodecahedron respectively. Fourpolyhedra exhibit two of these fundamental patterns, andaccordingly reappear as different elements in either ofthe sets to which they belong. Two polyhedra each appeartwice as different elements in the same set, in accord withtheir alternative orientations. Polyhedra which reappearare given alternative names. There are therefore in total

three parallel sets of eight polyhedra each, making twenty-four polyhedra in all; there being then in effect six regularand eighteen semi-regular polyhedra.

Each element of a set strictly correlates with itscorresponding elements in the other sets. Further,particular patterns - which are termed aspects - arereplicated across each of the sets. Within each set of eight,subsets of elements are discernible according to variousrules: sequential truncation, rotation and displacement,and octaving of faces. The particular aspect whichcharacterises a subset of one set correlates with therespective aspects of the other sets. Properties of elementsand of aspects can then be generalised across classes,and this predictive ability is confirmed in practice. AGestalt pattern is advanced for each set which integratesthe constituent aspects, and this is further developed toinclude the whole three sets.

The classification is also regularly extended torecognise a fourth and fifth class, which together comprisethe three regular and all but one of the eight semi-regulartilings of the planar surface, (the exception 33.42 beingconsidered a degenerate case). These classes are of{2,3,6}-fold and {2,4,4}-fold symmetry, of the quasi-regular TriangularHexagonal Array and SquareSquareArray respectively. Within either two-dimensional classsome elements are repeated with different colourings asdifferent facial arrangements are associated with therespective symmetry pattern of that class, the repeatedelements being given alternative names. Discounting thedegenerate exception, there are then in effect a total offour regular and twelve semi-regular tilings of the plane.But they do not exhaust the thirty-two possible uniformcolourings of regular and semi-regular tilings.

The harmony of interrelationship which characterisesthe entire order reveals the elegant structure of space.This research is from the author�’s forthcoming work onthe Platonic and Archimedean Polyhedra, which will beavailable from the author.

NATURAL HARMONY ~ ESSAYING STRUCTURAL MORPHOLOGY42

THE EIGHTEEN ARCHIMEDEAN or SEMI-REGULAR SOLIDS: SIX TRUNCATIONSOF THE TETRAHEDRON, OF THE OCTAHEDRON, AND OF THE ICOSAHEDRON

Truncated!!!" (Octahedron) Great Rhombic Snub Small Rhombic Truncated!!!#Tetrahedron TetraTetrahedron TetraTetrahedron TetraTetrahedron TetraTetrahedron TetrahedronTruncated (Cuboctahedron) Great Rhombic Snub Small Rhombic Truncated

Octahedron OctaHexahedron OctaHexahedron OctaHexahedron OctaHexahedron HexahedronTruncated Great Rhombic Snub Small Rhombic Truncated

Icosahedron IcosiDodecahedron IcosiDodecahedron IcosiDodecahedron IcosiDodecahedron Dodecahedron

THE ARCHIMEDEAN or SEMI-REGULAR SOLIDS:SIX TRUNCATIONS OF THE OCTAHEDRON AND OF THE ICOSAHEDRON

Truncated (Dymaxion) Truncated Snub Rhombic TruncatedOctahedron Cuboctahedron Cuboctahedron Hexahedron Cuboctahedron HexahedronTruncated Truncated Snub Rhombic Truncated

Icosahedron IcosiDodecahedron IcosiDodecahedron IcosiDodecahedron IcosiDodecahedron Dodecahedron

THE SIX REGULAR PLATONIC �‘PARENT�’ SOLIDS AS THREE POLAR PAIRSI " Tetrahedron # TetrahedronII Octahedron HexahedronIII Icosahedron Dodecahedron

THE NUCLEAR or �‘OVER�’ SOLIDS THE SECONDARY REGULAR �‘PARENT�’ SOLIDSTetrahedron Octahedron Cube or Hexahedron

Truncated Tetrahedron Icosahedron Dodecahedron

HIS PAPER DEVELOPS KEITH CRITCHLOW�’S WORK,Order in Space, and is hence respectfullydescribed as a �“New Order in Space�”. Hegives a progression of the regular polyhedra

as a Regular Tetraktys, which pattern I do not address inthis paper.

Critchlow then gives the following classification,5

arranging the Archimedean Solids from left to right as aseries of increasing truncations of the parent Octahedronor Icosahedron, that is as two classes of symmetry:

I amend this by distinguishing three classes, whichotherwise following Critchlow�’s schema would comprise

successive truncations of the parent Tetrahedron,Octahedron, or Icosahedron:

But I develop this further, as the simple linear sequence isunsatisfactory. (For example, if the polyhedra are arrangedin reverse as successive truncations of the Tetrahedron,Hexahedron, and Dodecahedron, the relative order of therhombic and snub polyhedra changes). Coxeter properlyplaces importance on the octahedron, cuboctahedron, andicosidodecahedron as Quasi-Regular solids.6 I term these

solids the TetraTetrahedron, OctaHexahedron, andIcosiDodecahedron, and recognise their significance ascentral elements of their respective classes. I have soughtto use a consistent terminology, which necessitates longand rather clumsy terms. Further subsets are identified.The rich patterns are made apparent by contemplation ofthe accompanying illustrations.

THREE FUNDAMENTAL SYSTEMS OF POLYHEDRAL SYMMETRYAND TWO FUNDAMENTAL SYSTEMS OF POLYGONAL TILING SYMMETRYThe three classes of polyhedra and two classes ofpolygonal tiling are given by the integral three-dimensional patterns of the three Quasi-Regularpolyhedra, and the integral two-dimensional patterns ofwhat I postulate to be the two Quasi-Regular tilings. Theseare shown in Table 1 opposite.

The important quasi-regular elements from which thesesymmetry patterns are derived, shown in Fig 1, are termedthe central or neutral elements of first degree. Kepler gavewhat is thought to be the first systematic treatment of theregular and semi-regular tilings of the plane, andconsidered them as analogues of the regular and semi-regular polyhedra.7


Class Class Class Class

I II III IVthe {2,3,3}-fold the {2,3,4}-fold the {2,3,5}-fold the {2,3,6}-foldsymmetry of the symmetry of the symmetry of the symmetry of theTetraTetrahedron OctaHexahedron IcosiDodecahedron Triangular

Hexagonal Array(3-D) (3-D) (3-D) (2-D)

Class

Vthe {2,4,4}-foldsymmetry of the

SquareSquare Array(Chess Board)

(2-D)

Figure 1 : Quasi-Regular elements from which the fundamental symmetry classes are derived.

Table 1: Fundamental Symmetry Patterns

In a similar spirit, I have extended the polyhedral orderto include the two latter classes, which together generatethe three regular and seven of the eight semi-regular tilingsof the planar surface.8 I omit an eighth semi-regular tiling33.42 as in my opinion it is degenerate.

The orderly progression of elements through the threepolyhedral classes thus continues into a fourth polygonalclass, which develops {2,3,6}-fold symmetry on the plane.The second polyhedral class is developed orthogonally tothat fourfold sequence to yield the fifth polygonal class,which is of {2,4,4}-fold symmetry on the plane. Thereare appropriately three classes for three-dimensionalspace, and two classes for two-dimensional space.

In both classes IV and V, all axes are parallel and areperpendicular to the plane, rather than coincident at thecentre of the polyhedra. It will therefore be appreciatedthat Classes I, II and III refer to the symmetries of regularthree-dimensional polyaxial centralised solids of finite

extent; whilst Classes IV and V refer to the symmetriesof regular two-dimensional polyaxial decentralised tilingsof infinite extent, (but which could be considered ascentralised forms where the Centre is at infinity).Elsewhere I discuss three-dimensional polyaxialcentralised zonahedral clusters,9 which contrast with thedecentralised space filling patterns described byCritchlow,10 and in a separate work I fully document onesuch centralised expansion.11

Within the two-dimensional classes, elements reappearas for the polyhedral classes, ascribing different facialarrangements to the patterns of positive, neutral andnegative symmetry axes. These correspond to differentuniform colourings of the same semi-regular tiling. Ingeneral, aspects can be generalised from the first threeclasses to the other, with common-sense being exercisedin the change in dimension.

V{2,4,4}

SquareSquare Array

II{2,3,4}

OctaHexahedron

I{2,3,3}

TetraTetrahedron

III{2,3,5}

IcosiDodecahedron

IV{2,3,6}

TriHexagonal Array


Figure 2 : Symmetry systems of positive, neutral and negative axes of the five classes.Axes through the vertices of the elements shown passthrough the centres of the polyhedra in Classes I-III,and are orthogonal to the page in Classes IV and V.

SYSTEMS OF POSITIVE, NEUTRALAND NEGATIVE AXES OF EACH CLASSSystems of positive and negative axes are dual for each ofthe five classes. Class V is akin to Class I, in that its polesare self-dual.

The positive axes of each symmetry system are givenby the centres of one of the two sets of faces of the centralelements of first degree, as shown in the left column ofFig 2.

Positive Axes(Male)

Neutral Axes(Neuter)

Negative Axes(Female)

I

II

III

IV

V



I II III IVcentres of the centres of the centres of the centres of the

triangular faces of the square faces of the pentagonal faces of the hexagonal faces of theTetraTetrahedron OctaHexahedron IcosiDodecahedron TriHexagonal Array

(other tetrahedral set only)

Class

Vcentres of the

square faces of theSquareSquare Array

(other square set only)


I II III IVvertices of the vertices of the vertices of the vertices of the

TetraTetrahedron OctaHexahedron IcosiDodecahedron TriHexagonal Array

Class

Vvertices of the

SquareSquare Array


I II III IVcentres of the centres of the centres of the centres of the

triangular faces of the triangular faces of the triangular faces of the triangular faces of theTetraTetrahedron OctaHexahedron IcosiDodecahedron TriHexagonal Array

(one tetrahedral set only)

Class

Vcentres of the

square faces of theSquareSquare Array(one square set only)

Table 2: Positive Axes

In the polyhedral classes, these accord with the axesof the centres of the faces of the Male Tetrahedron,Octahedron, and Icosahedron; and of the vertices of theFemale Tetrahedron, Hexahedron, and Dodecahedron.

In Class IV, these accord with the axes of the centresof the faces of the Triangular Array, and of the vertices ofthe Hexagonal array; and form a hexagonal grid.

In Class V, these accord with the axes of the centres ofthe faces of the Male Square Array, and of the vertices ofthe Female Square Array; and form a square grid.

The neutral axes of each symmetry system are givenby the vertices of the central elements of first degree, asshown in the middle column of Fig 2 opposite.

Table 3: Neutral Axes

These correspond to the axes of the mid-edges of themale and female polar elements, i.e. Male and FemaleTetrahedra, Octahedron and Hexahedron, Icosahedron andDodecahedron, Triangular and Hexagonal Array, and Maleand Female Square Arrays. The neutral axes of Class IV

form a trihexagonal grid; those of Class V form a squaregrid on the 2 diagonal.

The negative axes of each system are given by thecentres of the other of their two sets of faces of the centralelements:

Table 4: Negative Axes


These accord with the axes of the vertices of the MaleTetrahedron, Octahedron, Icosahedron, Triangular Array,and Male Square Array as shown in the right column ofFig 2; and with the axes of the centres of the faces of theFemale Tetrahedron, Hexahedron, Dodecahedron,Hexagonal Array, and Female Square Array. The negativeaxes of Class IV form a triangular grid, while those ofClass V form a square grid.

In the first three classes, with certain regular exceptions,each face of every regular or semi-regular polyhedra isorthogonal to and centred about its respective fundamentalaxis, that is the specific axis of the particular symmetrypattern with which it is associated. The exceptions arethe paired neutral faces of the three Snub Polyhedra, therethen being two adjoining neutral faces for each neutralaxis instead of one, with their common edge beingorthogonal to and centred about the axis.12 Similarly inthe last two classes, with certain regular exceptions, eachface of every regular or semi-regular tiling is centred aboutits respective fundamental axis. The exceptions are thepaired neutral faces of the two Skew Polygonal Tilings,there being two adjoining neutral faces for each neutralaxis instead of one, with their common edge being bisectedby the axis.

Note that the positive and negative axes of Class I takentogether are the positive axes of Class II, which are inturn a subset of the positive axes of Class III. The neutralaxes of Class I are the negative axes of Class II and asubset of the neutral axes of Class III. Positive, neutraland negative axes of Class V are square grids, the neutralaxis being rotated by /4 to the other two and scaled.

The shaded illustrations of polyhedra and tilingsconsistently depict male faces as dark, neutral faces asmedium, and female faces as light shading. Within eachillustration, the circumradii of polyhedra depicted areconstant. It is of course only possible to illustrate arepresentative portion of each element in Classes IV andV, these elements being of infinite extent. All three-dimensional views are either projections of constantrotation of the regulating cube,13 or lie along female axes.14

Classes IV and V also are shown centred about a femaleaxis. It is generally more convenient to depict Class Velements in line with elements of the other four classes,but their proper relationship is orthogonal to that axis fromthe corresponding Class II element.

Figure 3 : Ad Triangulum, Ad Quadratum, and Ad Pentangulum proportional harmonies used in traditional architecture(after Ghyka). Left: Dome of Milan, Caesar Caesariano, 1521; centre: Cuboctahedron and Byzantine Cupolas;

right: Two Gothic Standard Plans (Moessel).


POSITIVE NEUTRAL NEGATIVE SYMMETRY KEY PROPORTION NUMBER POLYGONI 3 2 3 {2,3,3} 3 Ad Triangulum ( 3) 3 TriangleII 3 2 4 {2,3,4} 4 Ad Quadratum ( 2) 4 SquareIII 3 2 5 {2,3,5} 5 Ad Pentangulum (ø) 5 PentagonIV 3 2 6 {2,3,6} 6 Ad Triangulum ( 3) 6 HexagonV 4 2 4 {2,4,4} 4 Ad Quadratum ( 2) 4 Square

THE THREE PROPORTIONAL SYSTEMSINHERENT IN THE SYMMETRY AXESThe KAIROS School teaches three traditional proportionalsystems which have been used in sacred architecturethroughout history.15 Fig 3, after Ghyka,16 shows examplesof each of these. They are:

- Ad Triangulum, which exploits 3 ratios derivedfrom the equilateral triangle;

- Ad Quadratum, which exploits 2 ratios derivedfrom the square;17 and

- Ad Pentangulum, which exploits ø = ( 5+1)/2(Golden Section) ratios, derived from the regularpentagon, and from the 5 diagonal of the doublesquare.

Positive axes of Classes I-IV develop threefoldsymmetry, with Class V of fourfold symmetry.

Neutral axes of all five classes develop twofoldsymmetry, including the neutral square faces of therhombic solids.

Negative axes develop threefold, fourfold, fivefold,sixfold, and fourfold symmetry respectively. In the caseof the snub and skew elements, the symmetries arerotational only but not in general reflective.18 It will thenbe seen that the three polyhedral classes exhibit AdTriangulum, Ad Quadratum and Ad Pentangulumproportional harmonies respectively, interwoven with AdTriangulum. Class IV is of Ad Triangulum harmony, andClass V of Ad Quadratum. A limited Ad Quadratum isdeveloped throughout all classes about the neutral axes.

THE POLAR ELEMENTS AND SHORTHORIZONTAL SEQUENCES OF FIRST DEGREEThe five Platonic polyhedra are reorganised as six polarsolids comprising three male-female duals.19 The threePlatonic tilings are reorganised as four polar tilingscomprising two male-female duals, of triangular andhexagonal grid, and square and square grid respectively.Fig 4 shows how these polarise about the central neutralelement as one pair for each set.

Thus the three neutral polyhedra (tetratetrahedron,octahexahedron and icosidodecahedron) and two neutraltilings (triangularhexagonal and squaresquare arrays) arerecognised as mediating states of their respectivepolarities. Quasi-regular neutral central and regular polarelements of each set are the main elements of theirrespective horizontal sequences.

(Regular) (Quasi-Regular) (Regular)MALE POLAR �—�—�—�—�—�—�— NEUTRAL CENTRAL �—�—�—�—�—�—�— FEMALE POLAR

I ! Tetrahedron �—�—�—�—�—�—�— TetraTetrahedron �—�—�—�—�—�—�— " TetrahedronII Octahedron �—�—�—�—�—�—�— OctaHexahedron �—�—�—�—�—�—�— HexahedronIII Icosahedron �—�—�—�—�—�—�— IcosiDodecahedron �—�—�—�—�—�—�— DodecahedronIV Triangular Array �—�—�—�—�—�—�— TriHexagonal Array �—�—�—�—�—�—�— Hexagonal ArrayV ! Square Array �—�—�—�—�—�—�— SquareSquare Array �—�—�—�—�—�—�— " Square Array

THE INTERMEDIARY ELEMENTS AND LONGHORIZONTAL SEQUENCES OF FIRST DEGREEThe long horizontal sequences shown in Figs 5 and 12are completed by the inclusion of two intermediaryelements for each set. The octahedron-cube andicosahedron-dodecahedron truncation sequences shownare well-known, and are illustrated by Holden.20 Theintermediary solids are the truncations of their respectivepolar solids, as the intermediary tilings are the truncationsof their respective polar tilings.

The intermediary elements of first degree of Class IVare the hexagonal-hexagonal grid which is developed asa truncation of the male polar triangular grid, and thetriangular-dodecagonal grid which is developed as atruncation of the female polar hexagonal grid. Theintermediary elements of first degree of Class V are theoctagonal-square grid developed as a truncation of themale polar square grid, and the square-octagonal gridwhich is developed as a truncation of the female polarsquare grid.


Figure 4 : Regular polyhedra and tilings polarised in duals about the quasi-regular polyhedra and tilings.

MALE ! MALE NEUTRAL FEMALE FEMALE "POLAR INTERMEDIARY CENTRAL INTERMEDIARY POLARMale ! Truncated Male (Octahedron) Truncated Female Female "

I Tetrahedron Tetrahedron TetraTetrahedron Tetrahedron TetrahedronTruncated (Cuboctahedron) Truncated (Cube)

II Octahedron OctaHexahedron Hexahedron Hexahedron HexahedronTruncated Truncated (Pentagonal-)

III Icosahedron Icosahedron IcosiDodecahedron Dodecahedron DodecahedronTruncated Truncated

IV Triangular Array Triangular Array TriHexagonal Array Hexagonal Array Hexagonal ArrayMale ! Truncated Male Truncated Female Female "

V Square Array Square Array SquareSquare Array Square Array Square Array

Male Pole Neutral Centre Female Pole

I

II

III

IV

V


MalePolar

MaleIntermediary

NeutralCentre

FemaleIntermediary

FemalePolar

I

II

III

Figure 5 : Long Horizontal Sequences of Classes I-III showing the regular and semi-regular polyhedra of First Degree.See also Fig 12.

THE NEUTRAL CENTRALELEMENTS OF SECOND DEGREETo each of the three central elements of first degree, thereexist corresponding central elements of second and of thirddegree. They are shown as the second row of Figs 6 and13. Each second degree central solid is the small rhombicsolid of the respective solid of first degree: the SmallRhombic TetraTetrahedron, the Small RhombicOctaHexahedron, and the Small RhombicIcosiDodecahedron. These solids are developed from theirfirst degree solids with all faces displacing regularlyoutwards along their axes, (or with faces contractingsymmetrically), while:

- rotating the positive faces about their positive axesby half their central face angle, (triangular faces ofClasses I, II, and II by /3);

- rotating the negative faces about their negative axesby half their central angle, (triangular faces ofClass I by /3, square faces of Class II by /4,pentagonal faces of Class III by /5); and

- developing square faces on the neutral axesto complete the solids.

The second degree central tilings are the small rhombictiling of the respective tiling of first degree: the triangular-square-hexagonal grid, and the square-square-square grid,or the Small Rhombic TriHexagonal Array and the SmallRhombic SquareSquare Array. These tilings are shown inthe second row of Fig 13, and are developed from theirfirst degree tilings with all axes expanding regularly fromone another, (or with faces contracting symmetrically),while:

- rotating the positive faces about their positive axesby half their central face angle,(triangular faces of Class IV by /3 andsquare faces of Class V by /4);

- rotating the negative faces about their negative axesby half their central angle,(hexagonal faces of Class IV by /6 andsquare faces of Class V by /4); and

- developing square faces on the neutral axesto complete the tilings.


~ TetraTetrahedron(Class I)

~ OctaHexahedron(Class II)

~ IcosiDodecahedron(Class III)

Great Rhombic ~(Third degree

Central)

I{2,3,3}

II{2,3,4}

III{2,3,5}

Small Rhombic ~(Second degree

Central)

~(First degree

Central)

Snub ~(Second degreeIntermediary)

Figure 6 : Vertical Sequences of Classes I-III showing the Neutral Central polyhedra of First, Second and Third degreetogether with the Intermediary polyhedra of Second degree. See also Fig 13.

THE INTERMEDIARYELEMENTS OF SECOND DEGREEPart way through the rotational displacements to obtainthe central elements of second degree, the Snub or Skewelements are formed as intermediary elements of seconddegree. These are shown as the third row of Figs 6 and13. For convenience, I illustrate only right-hand forms,which represent both enantiomorphs for each case. Ofparticular interest is the recognition of the true SnubTetraTetrahedron of Class I as the regular Icosahedron

(of Class III), considered in its alternative mode.21 ThisSnub TetraTetrahedron of Class I is strictly paralleled bythe Snub OctaHexahedron of Class II and the SnubIcosiDodecahedron of Class III. In Class IV and V, thesnub intermediary elements of second degree are givenby the skew triangular-triangular-hexagonal grid and theskew square-diamond-square grid, or Skew TriHexagonaland Skew SquareSquare Arrays.


CLASS I II III IV VSymmetry {2,3,3} {2,3,4} {2,3,5} {2,3,6} {2,4,4}

Central Element Great Rhombic Great Rhombic Great Rhombic Great Rhombic Great Rhombicof 3rd Degree TetraTetrahedron OctaHexahedron IcosiDodecahedron TriHexagon Array SquareSquareArray(Inter. Elementof 3rd Degree) �—�— �—�— �—�— �—�— �—�—

Central Element Small Rhombic Small Rhombic Small Rhombic Small Rhombic Small Rhombicof 2nd Degree TetraTetrahedron OctaHexahedron IcosiDodecahedron TriHexagon Array SquareSquareArrayInter. Element Snub Snub Snub Skew Skewof 2nd Degree TetraTetrahedron OctaHexahedron IcosiDodecahedron TriHexagon Array SquareSquareArray

Central Element (Octahedron) (Cuboctahedron)of 1st Degree TetraTetrahedron OctaHexahedron IcosiDodecahedron TriHexagon Array SquareSquareArray

These rotational displacement sequences are the preciseequivalents of R. Buckminster Fuller�’s jitterbug systems.He gives the (Class I) symmetrical contraction of thevector equilibrium (small rhombic tetra tetrahedron)through the icosahedron (snub tetra tetrahedron) to theoctahedron (tetra tetrahedron).22 As far as I am aware,neither Fuller nor his followers discovered the equivalentClass II and III jitterbugs,23 i.e. the symmetricalcontraction from small rhombic octahexahedron throughsnub octahexahedron to octahexahedron, and that fromsmall rhombic icosidodecahedron to snubicosidodecahedron to icosidodecahedron. Rigid maletriangles alternate with rigid female squares and pentagons

respectively to which they are vertex pin-jointed, withthe neutral faces being voids. I have deduced these fromthe order I advance, and the predictability of thesebehaviours across the classes is further confirmation ofits validity.

Analogous two-dimensional jitterbug systems alsoexist for Classes IV and V, with rigid male triangles andsquares alternating with female hexagons and squares towhich they are vertex pin-jointed, and with neutral facesbeing voids. In Class IV, the small rhombic trihexsymmetrically contracts through the skew trihex to thetrihex. In Class V, the small rhombic squaresquaresymmetrically contracts through the skew squaresquareto the squaresquare grid.

THE NEUTRAL CENTRALELEMENTS OF THIRD DEGREEJust as the neutral central elements of second degree canbe thought of as higher octaves of the correspondingelements of first degree, as corresponding faces rotate anddisplace from each other, the neutral central elements ofthird degree represent a higher octave again, ascorresponding faces double in frequency and displacefrom each other. This is why the pattern is given verticalarticulation. They are shown as the first row of Figs 6 and13. Within each class, the element of third degree is thegreater element of second degree, and the truncatedelement of first degree. The truncated tetra tetrahedron ortruncated octahedron is known as the Great RhombicTetraTetrahedron, the truncated octahexahedron ortruncated cuboctahedron as the Great RhombicOctaHexahedron, and the truncated icosidodecahedron asthe Great Rhombic IcosiDodecahedron. Referring to the

last two solids, Cundy and Rollett suggest that the �“greatrhombic�” terminology is preferable, as the truncationsrequire an additional distortion to convert rectangles tosquares.24 The syllable �‘rhomb-�’ shows that the neutralsets of square faces lie in the planes of the rhombichexahedron (cube), rhombic dodecahedron, and rhombictriacontahedron respectively, which are the duals of thecentral TetraTetrahedron, OctaHexahedron andIcosiDodecahedron. These three duals are illustrated inFig 14 below. The central elements of third degree ofClasses IV and V are given by the hexagonal-square-dodecagonal grid and the octagonal-square-octagonal grid,or Great Rhombic TriHexagonal and Great RhombicSquareSquare Arrays. Their duals, the diamond and squarearrays, are known as the rhombic trihexagonal andrhombic squaresquare arrays, also shown in Fig 14.

So for the elements of third degree:- Positive faces are formed of double frequency on

the positive axes (hexagonal for Classes I-IV,octagonal for Class V);

- Square Neutral faces are developed on the neutralaxes of all five Great Rhombs; and

- Negative faces are formed of double frequency onthe negative axes (hexagonal for the GreatRhombic TetraTetrahedron, octagonal for the GreatRhombic OctaHexahedron, decagonal for the GreatRhombic IcosiDodecahedron, dodecagonal for the

Great Rhombic TriHexagonal Array, and octagonalfor the Great Rhombic SquareSquare Array).

Clearly, for each class, Neutral Central, Small Rhombicand Great Rhombic elements form a major progression,whilst Neutral Central, Snub/Skew and Small Rhombicform a minor progression. All four are obviously centralelements, rather than polar or intermediary polar.There are no intermediary elements of third degree. Butit will be appreciated that the linear sequential order ofthe vertical axis is less definitive than the horizontal.


I II III IVCENTRE TetraTetrahedron OctaHexahedron IcosiDodecahedron TriHexagon Array

Dual Rhombic Dual Rhombic Dual Rhombic Dual RhombicRHOMBIC TetraTetrahedron OctaHexahedron IcosiDodecahedron TriHexagon Array

DUAL (Rhombic (Rhombic (Rhombic (DiamondHexahedron) Dodecahedron) Triacontahedron) Array)

VCENTRE SquareSquareArray

Dual RhombicRHOMBIC SquareSquareArray

DUAL (SquareArray)

THE META-PATTERN OF EACH SETSo we have discerned a structural pattern of aspects foreach set that accords well with traditional symbolism. Inits short form, this takes the form of an inverted T, shownin Table 6. It comprises a central trunk of the centralelements of first, second, and third degree, with the polarelements of first degree either side. This is an intelligibleand imageable structure, that is readily memorised and isuseful for teaching.

In its expanded form, the structure takes the form ofan �“i�”. Figs 7-11 show this structure for each of the fiveclasses. The intermediary cases are included - there beingno intermediary between the third degree and seconddegree neutral centres. I develop Grünbaum andShephard�’s tiling and colouring symbols for the polyhedraand tilings. The first numerical sequence gives thesequence of frequencies of faces about a vertex,commencing always from a female axis and proceedingclockwise, and differentiating between male, female andneutral faces. The second expression in brackets gives thecolouring of faces in the same sequence, and thereforethe axis type, 1 being male, 2 female, and 3 neutral.Symbolic description & colouring of an element areconsistent throughout the classes.

Although Class IV and V generate all but one of theeleven regular and semi-regular tilings, they by no meansexhaust all thirty-two uniform colourings of these tilings,which Grünbaum and Shephard illustrate in Fig 2.9.2.25

(Note that (36) 121314 appears to be incorrectly shadedin that illustration). Colouring the tilings modifies thesymmetry system in a number of cases. Two colouredtilings although of the correct {2,4,4}-fold symmetry donot appear in this classification. For the remainder, theyeither belong to the degenerate {2,2,2,2}-fold symmetryclass, or to the {3,3,3}-fold symmetry class, both of whichI exclude. Although it might be thought possible to extendthe order to include classes of the remaining {2,4,4}-fold,and {3,3,3}-fold and {2,2,2,2}-fold symmetry elementsto accommodate all of the uniform colourings, theseclasses even if they exist are less regularly defined andincomplete.26

Regular and semi-regular elements that repeat, shownin Table 7, are discerned by disregarding the differencesbetween them in colouring and corresponding ascriptionof faces to symmetry axes. Six polyhedra appear twice inthe first three classes. In Class IV, one tiling appears twice;in Class V, one tiling appears four times, and anotherappears three times.

A META-PATTERN FOR THETHREE CLASSES OF POLYHEDRAThe arrangement for each polyhedral set suggests aconvenient overall pattern of the first three classes insidea cubic matrix. A 3x3x3 cube contains the three short setsas vertical layers. A 5x5x5 cube contains the completesets in the outside and central vertical layers, with twoempty inbetween layers.

With certain regular exceptions, each axis has itscharacteristic frequency of rotational and reflectivesymmetry. Each face of every polyhedron is orthogonalto and centred about its respective fundamental axis, i.e.the specific axis of the particular symmetry pattern of itsclass.

The only exceptions arise in the Snub polyhedra. Herethe symmetries are rotational but not reflective. Positiveand negative faces remain orthogonal to their respectiveaxes, but are displaced along their axes from the centralsolids of first degree and rotated by an irrational measure.But on the neutral axes, pairs of triangular faces form abouta common edge that is perpendicular to and bisected bytheir axis. These neutral faces are not perpendicular tothe neutral axes, nor do the axes pass through themidpoints of the faces.

Table 5: Rhombic Duals


3°2°1°

GREATRHOMB

SMALLRHOMB

MALE NEUTRAL FEMALEPOLE CORE POLE

Table 6: Meta-Pattern of each set

I Male Tetrahedron 33 = 33 Female Tetrahedron I(111) (222)

I Male Truncated Tetrahedron 3.62 = 62.3 Truncated Female Tetrahedron I(211) (221)

I TetraTetrahedron 3.3.3.3 = 34 Octahedron II(2121) (1111)

I Snub TetraTetrahedron 3.3.3.32 = 35 Icosahedron III(23133) (11111)

I Small Rhombic TetraTetrahedron 3.4.3.4 = 4.3.4.3 OctaHexahedron II(2313) (2121)

I Great Rhombic TetraTetrahedron 6.4.6 = 4.62 Truncated Octahedron II(231) (211)

IV Truncated Triangular Array 6.62 = 63 Hexagonal Array IV(2121) (222)

V Male Square Array 44 = 44 Female Square Array V(1111) (2222)

= 4.4.4.4 Central SquareSquare Array V(2121)

= 4.4.4.4 Small Rhombic SquareSquare V(2313)

V Male Truncated Square Array 4.82 = 82.4 Female Truncated Square Array V(211) (221)

= 8.4.8 Great Rhombic SquareSquare V

Table 7: Regular and semi-regular elements that repeat(disregarding alternative symmetry patterns given by alternative colourings of faces)

NATURAL HARMONY ~ ESSAYING STRUCTURAL MORPHOLOGY54Great RhombicNeutral Element

2n.4.2mMETA-PATTERN: {2, m, n} (231)

Third DegreeCentral

(m = 3, n = 3, 4, 5, 6; m = 4, n = 4) See Figs 7-11.

Small RhombicNeutral Element

n.4.m.4(2313)

Second DegreeCentral

SnubNeutral Element

n.3.m.32

(23133)Second DegreeIntermediary

Male Truncate Male Central Truncate Female FemalePolar Element Polar Element Neutral Element Polar Element Polar Element

mn n.2m.2m n.m.n.m 2n.2n.m nm

(1n) (211) (2121) (221) (2m)Male Male First Degree Female FemalePolar Intermediary Central Intermediary Polar

Great RhombicTetraTetrahedron

6.4.6CLASS I: {2, 3, 3} (231)

TruncatedOctahedron

TETRATETRAHEDRAL See Fig. 7.

Small RhombicTetraTetrahedron

3.4.3.4(2313)

Cuboctahedron

SnubTetraTetrahedron

3.3.3.32

(23133)

Icosahedron

Male Truncate Male Central Truncate Female FemaleTetrahedron Tetrahedron TetraTetrahedron Tetrahedron Tetrahedron

33 3.62 3.3.3.3 62.3 33

(111) (211) (2121) (221) (222)Truncated Truncated

Tetrahedron Tetrahedron Octahedron Tetrahedron Tetrahedron


Figure 7 : Regular and semi-regular polyhedra of Class I of TetraTetrahedral symmetry.

NATURAL HARMONY ~ ESSAYING STRUCTURAL MORPHOLOGY56Great Rhombic

OctaHexahedron8.4.6

CLASS II: {2, 3, 4} (231)Truncated

OctaHexahedron

OCTAHEXAHEDRAL See Fig. 8.

Small RhombicOctaHexahedron

4.4.3.4(2313)

�“Square Spin�”

SnubOctaHexahedron

4.3.3.32

(23133)Snub

Cuboctahedron

Male Truncate Male Central Truncate Female FemaleOctahedron Octahedron OctaHexahedron Hexahedron Hexahedron

34 4.62 4.3.4.3 82.3 43

(1111) (211) (2121) (221) (222)Cuboctahedron

�“Mecon�” or �“Dymaxion�” Truncated Cube Cube

Great RhombicIcosiDodecahedron

10.4.6CLASS III: {2, 3, 5} (231)

TruncatedIcosidodecahdrn

ICOSIDODECAHEDRAL See Fig. 9.

Small RhombicIcosiDodecahedron

5.4.3.4(2313)

RhombicDodecahedron

SnubIcosiDodecahedron

5.3.3.32

(23133)Snub

Dodecahedron

Truncate Male Truncate FemaleIcosahedron Icosahedron IcosiDodecahedron Dodecahedron Dodecahedron

35 5.62 5.3.5.3 102.3 53

(11111) (211) (2121) (221) (222)First Pentagonal

Dodecahedron


Figure 8 : Regular and semi-regular polyhedra of Class II of OctaHexahedral symmetry.

Figure 9 : Regular and semi-regular polyhedra of Class III of IcosiDodecahedral Symmetry.

NATURAL HARMONY ~ ESSAYING STRUCTURAL MORPHOLOGY58Great RhombicTriHex Array

12.4.6CLASS IV: {2, 3, 6} (231)

Third DegreeCentral

TRIANGULAR-HEXAGONAL See Fig 10.

Small RhombicTriHex Array

6.4.3.4(2313)


SkewTriHex Array

6.3.3.32


Truncate Truncate FemaleTriangular Array Triangular Array TriHex Array Hexagonal Array Hexagonal Array

36 6.62 6.3.6.3 122.3 63

(111111) (211) (2121) (221) (222)Triangular- Tri-Dodecagonal

Triangular Grid Hexagonal Array Hexagonal Array Intermediary Hexagonal Grid

Great RhombicSqrSquare Array

8.4.8CLASS V: {2, 4, 4} (231)

Third DegreeCentral

SQUARE-SQUARE See Fig. 11.

Small RhombicSqrSquare Array

4.4.4.4(2313)


SkewSqrSquare Array

4.3.4.32


Male Truncate Male SquareSquare Truncate Female FemaleSquare Array Square Array Array Square Array Square Array

44 4.82 4.4.4.4 82.4 44

(1111) (211) (2121) (221) (2222)Octangular- Square-

Square Grid Square Array Chess Board Octangular Array Square Grid


Figure 10 : Regular and semi-regular tilings of Class IV of TriHexagonal Array Symmetry.

Figure 11 : Regular and semi-regular tilings of Class V of SquareSquare Array Symmetry.


Figure 12 : Long Horizontal Sequences of the five classes showing the regular and semi-regular elements of First Degree.See also Fig 5.

THE META-PATTERN OVER ALL FIVE CLASSESReference to Fig 12 confirms that the positive polarelement consists only of positive single-frequency faces,and the negative polar element consists only of negativesingle-frequency faces. The truncated positiveintermediary element consists of double-frequency edge-jointed positive faces which cluster around and isolate thesingle-frequency negative faces, with three faces to avertex. The truncated negative intermediary elementconsists of double-frequency edge-jointed negative faceswhich cluster around and isolate the single-frequencypositive faces, with three faces to a vertex. The neutralcentral element of first degree alternates edge-jointedpositive and negative single-frequency faces, so that a

positive face is only edge-jointed to negative faces andvice versa, with positive faces vertex-jointed to one anotheras are negative faces; there are four faces to a vertex.

Fig 13 shows that part-way in the rotation of positiveand negative faces of the central element of first degreetowards forming the central element of second degree,the intermediary element of second degree occurs in twoenantiomorphs. This snub or skew element alternatesvertex-jointed positive and negative single-frequencyfaces, so that a positive face is only vertex-jointed tonegative faces and vice versa. Paired triangular neutralfaces separate the positive and negative faces, to whichthey are edge-jointed; there are five faces to a vertex.

MalePolar

MaleIntermediary

NeutralCentre

FemaleIntermediary

FemalePolar


Figure 13 : Vertical Sequences of the five classes showing the regular and semi-regular Neutral Central elementsof First, Second and Third Degree, together with the Intermediary elements of Second Degree. See also Fig 6.

The central element of second degree, or Small Rhombof Fig 13, consists of single-frequency positive andnegative faces, separated by square neutral faces to whichthey are edge-jointed, so that a positive face is only vertex-jointed to negative faces and vice versa. Positive andnegative faces of the central element of first degree arerotated by half their central face angle, one set clockwiseand the other counter. There are four faces to a vertex.

The central element of third degree, or Great Rhombof Fig 13, is of double-frequency positive and negativefaces and square neutral faces. Positive faces are edge-jointed to alternating negative and neutral faces; neutralto alternating positive and negative faces; and negativefaces to alternating positive and neutral faces. There arethree faces to a vertex.

These regularities are best appreciated bycontemplation of the illustrations.

FIVE RHOMBIC DUAL ELEMENTS,AND OTHER POLYHEDRAThree rhombic polyhedra, shown in Fig 14, are appendedbecause of their importance as duals of the three keycentral polyhedra of first degree. All of their faces areneutral, and the cube reappears in Class I as the DualRhombic TetraTetrahedron or rhombic hexahedron.Rhombic duals of the key central patterns are also shown

for the two-dimensional classes, with all faces beingneutral. In Class IV this is a regular array of diamondsformed from paired equilateral triangles. In each class,there are two kinds of vertex, with three and three, threeand four, three and five, three and six, and four and fourfaces respectively.

GreatRhomb

SmallRhomb

Snub/Skew

NeutralCentre

I{2,3,3}

TetraTetra

II{2,3,4}

OctaHexa

III{2,3,5}

IcosiDodeca

IV{2,3,6}

TriHex

V{2,4,4}

SquareSquare



I II III IV(TetraTetrahedron) (OctaHexahedron) (IcosiDodecahedron) (TriHexagon Array)

Dual Rhombic Dual Rhombic Dual Rhombic Dual RhombicTetraTetrahedron OctaHexahedron IcosiDodecahedron TriHexagon Array

(Rhombic (Rhombic (Rhombic (DiamondHexahedron) Dodecahedron) Triacontahedron) Array)

Class

V(SquareSquareArray)

Dual RhombicSquareSquareArray

(SquareArray)

Figure 14 : Dual Rhombic elements of the quasi-regular neutral central elements of First Degree of Fig 1.

Table 5: Rhombic Duals

Other polyhedra can be integrated into the order, havingregard to their symmetry patterns. For example, theKepler-Poinsot polyhedra could be integrated as polarsolids of Class III, where edges connect non-adjacentvertices. The joint solids of Stella Octangular(Tetrahedron-Tetrahedron), Octahedron-Cube, and

Icosahedron-Dodecahedron represent a common centralelement of the three polyhedral classes, and equivalenttiling patterns (which are not semi-regular) are found forthe other two classes by overlaying positive, negative andneutral symmetry arrays.

I{2,3,3}

Dual RhombicTetraTetrahedron

III{2,3,5}

Dual RhombicIcosiDodecahedron

IV{2,3,6}

Dual RhombicTriHexagonal Array

V{2,4,4}

Dual RhombicSquareSquare Array

II{2,3,4}

Dual RhombicOctaHexahedron

CONCLUSIONThis paper presents a threefold classification of the regularand semi-regular polyhedra. It then extends that order toinclude a fourth and fifth class which generate the regularand all but one of the semi-regular tilings of the plane.Their development within the overall order advanced isconsistent. Contemplation of Figs 7-11, which show eachof the five classes, and Figs 12-13, which show thehorizontal and vertical axes of the five classes, confirmsthis. Elements are classified according to their symmetrypattern and positioned relative to other elements within

their class. In this process certain elements are repeatedwith different ascription of faces to symmetry axes. Withcertain regular exceptions, all faces are symmetricallydisposed about their respective symmetry axes. Theexceptions are the neutral faces of the snub and skewelements, where it is the paired triangles that aresymmetrically disposed. The poles of each class areregular, and the centres are quasi-regular. Each classculminates in a great rhombic element. The regularity ofthe order allows properties of an element or aspect of


elements of one class to be generalised to the other classes(with common-sense being applied at the change indimension), and this predictive ability is confirmed byexperience and discovery. The order does not account forall thirty-two uniform colourings of the semi-regulartilings of the plane, nor is it intended to.

More generally, recognition is made of the orderlyspatial structure underlying the specific expressions ofregularity which constitute these beautiful polyhedra andsurface patterns.

Classification of the regular and semi-regular polyhedrareveals meaningful aspects that illuminate the richharmony of space that manifests these perfect forms.

***

1 The Kepler-Poinsot polyhedra are beyond the scope ofthe present enquiry, but could nevertheless be integratedinto an extended classification according to theirsymmetries. The regular prisms and antiprisms aremonoaxial solids, and thus not relevant to thisclassification which is of polyaxial solids. For somewhatsimilar reasons, I remain to be convinced the tiling of theplane of {2,2,2}-fold symmetry is a true polyaxialdecentralised pattern, and exclude it and its derivativesand colourings from this treatment.

2 Critchlow, K., Order in Space, Thames and Hudson,London, 1969. See especially Appendix 1: A PeriodicArrangement of the Elements of Spatial Order.

3 Cundy, H.M. and Rollett, A.P.,Mathematical Models (Second Edition),Oxford University Press, London, 1961, p. 100.

4 Coxeter, H.S.M., Regular Polytopes,Dover Publications, New York, 1973, p.289.

5 Critchlow, op. cit. See pp. 34-37. 6 Coxeter, Regular Polytopes, op. cit., p.19. 7 Grünbaum, B. and Shephard, G.C.,

Tilings and Patterns,W.H. Freeman and Company, New York, 1987, p.110.

8 See Grünbaum and Shephard, ibid., Fig 2.1.5, p.63, for adiscussion and illustrations of space-filling surfacepatterns. See also Critchlow, op. cit., pp. 60-61.

9 Meurant, R.C., The Aesthetics of the Sacred ~ AHarmonic Geometry of Consciousness and Philosophy ofSacred Architecture, Third Edition. The Opoutere Press,Auckland, 1989. This work also addresses two-dimensional centralised zonagonal clusters, whichcontrast with decentralised regular tilings of the plane.

10 Critchlow, op. cit., see in particular Appendix 2:A Periodic Arrangement of the Multiple Single, Dual,and Triple All-Space Filling Solids Exhibiting aPeriodicity of Eight, Punctuated by a 3,2,3 Symmetry.

11 Meurant, R.C., The Integral Space Habitation ~ Towardsan Architecture of Space.The Opoutere Press, Auckland, 1989.

12 The enantiomorphs of the Snubs and Skewsare not considered as separate elements.

13 From an orthogonal orientation to the camera, these werefirst rotated by 15° clockwise about the vertical axis, andthen 18° downwards about the left-right axis.

14 This necessitates a different orientation of the regulatingcube for each of the three polyhedral classes.

15 See for example Lawlor, R.,Sacred Geometry - Philosophy and Practice,Thames and Hudson, London, 1982.

16 Ghyka, M., The Geometry of Art and Life,Dover Publications, New York, 1977,Plates LXV, XIV, and LIX.

17 See Lund, F.M., Ad Quadratum - A Study of theGeometrical Bases of Classic and Medieval ReligiousArchitecture, B.T. Batsford Ltd., London, 1921. Also seeBrunés, T., The Secrets of Ancient Geometry - and ItsUse, vols. I & II, Rhodos, Copenhagen, 1967; and Watts,D.J. and Watts, C.M., A Roman Apartment Complex,Scientific American, December 1986.

18 Excepting the neutral axes of the snub tetra tetrahedron3.3.3.32 (23133) of I:{2,3,3} (the icosahedron) and theskew squaresquare 4.3.4.32 (23133) of IV: {2, 4, 4}, ifthe positive-negative axis distinction and thus colouringsare ignored.

19 Male-Female polarity is a convenient means ofstructuring complexity, which accords with traditionalsymbolism. The Male and Female Tetrahedra are simplythe two tetrahedra of dual orientation which aredeveloped in a cube. Although alternative orientations ofthe icosahedron and dodecahedron in the cube exist, theyare in contrast not dual to one another (i.e. icosahedronto icosahedron, etc.)

20 Holden, A., Shapes, Space, and Symmetry, ColumbiaUniversity Press, New York, 1971, pp. 40-41.

21 Holden, ibid., p.48, illustrates the left- and right-handedtetrahedral symmetries of the faces of the icosahedron.Both enantiomorphs of the Snub Tetra tetrahedron are ofcourse icosahedra.

22 Fuller, R.B., Synergetics: Explorations in the Geometryof Thinking, Vol.I, MacMillan Publishing Co., New York,1975, Fig 460.08, p.193.

23 For example Edmondson merely repeats Fuller�’s vectorequilibrium contraction, and offers nothing new. SeeEdmondson, A.C., A Fuller Explanation: the SynergeticGeometry of R. Buckminster Fuller, Birkhaüser, Boston,1987, Fig 11-1, p.160.

24 Cundy and Rollett, op. cit., p. 100.25 Grünbaum and Shephard, op. cit.,

Fig 2.9.2, pp.104-6.26 For example, a hybrid Class VI of {3,3,3}-fold symmetry

may be advanced, which is generated orthogonal to thepage from Class I in the metapattern shown above.(Intriguingly, the three extensions of the polyhedralpattern to generate two-dimensional Classes IV, V, andVI then occur along each of the three axes respectively).Using Grünbaum and Shephard�’s symbols, this TriTriclass contains 3.3.3.3.3.3 (212121), 3.6.3.6 (2313), and6.6.6 (231) as central tilings of first, second and thirddegree respectively. With a 3-fold neutral axis, theneutral faces are fully expanded into hexagons. The skewtiling is 3.3.3.33 (231333), containing two-frequencytriangular neutral faces, which occur as contractedhexagons. The vertical sequence is complete. But theremaining elements of the horizontal sequence do notexist. More critically, the positive and negative symmetrysystems are not dual, and the central element should nottherefore be considered quasi-regular. Both poles arevirtual triangular arrays of triangles, which are notpossible without leaving triangular holes; or 36 (111111)and (222222) could be taken as male and female poles,but then the requirements for conservation of spatialarray and orientation would need to be relaxed.Similarly, it is tempting to postulate 3.35 (211111) and(122222) as both male and female intermediaries; butthen the respective polar faces of that intermediary arenot of double frequency, and their spatial array is notconserved. Thus although elements could be foundby relaxing the constraints on the definition,I prefer not to.

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