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User:Tomruen/Conway polyhedron notation

< User:Tomruen
See also: User:Tomruen/Semioperators
This example chart shows how 11 new forms can be derived from the cube using 3 operations. The new polyhedra are shown as maps on the surface of the cube so the topological changes are more apparent. Vertices are marked in all forms with circles.
This chart adds 3 more operations: George Hart's p=propellor operator that adds quadrilaterals, g=gyro operation that creates pentagons, and a c=Chamfer operation that replaces edges with hexagons

In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.

Conway and Hart extended the idea of using operators, like truncation defined by Kepler, to build related polyhedra of the same symmetry. The basic descriptive operators can generate all the Archimedean solids and Catalan solids from regular seeds. For example, tC represents a truncated cube, and taC, parsed as t(aC), is a truncated cuboctahedron. The simplest operator dual swaps vertex and face elements, like a dual cube is an octahedron: dC=O. Applied in a series, these operators allow many higher order polyhedra to be generated. A resulting polyhedron will have a fixed topology (vertices, edges, faces), while exact geometry is not constrained: it can be thought of as one of many embeddings of a polyhedral graph on the sphere.

The seed polyhedra are the Platonic solids, represented by the first letter of their name (T,O,C,I,D); the prisms (Pn) for n-gonal forms, antiprisms (An), cupolae (Un), anticupolae (Vn) and pyramids (Yn). Any polyhedron can serve as a seed, as long as the operations can be executed on it. For example regular-faced Johnson solids can be referenced as Jn, for n=1..92.

In general, it is difficult to predict the resulting appearance of the composite of two or more operations from a given seed polyhedron. For instance ambo applied twice becomes the same as the expand operation: aa=e, while a truncation after ambo produces bevel: ta=b. There has been no general theory describing what polyhedra can be generated in by any set of operators. Instead all results have been discovered empirically.

Operations on polyhedra

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Elements are given from the seed (v,e,f) to the new forms, assuming seed is a convex polyhedron: (a topological sphere, Euler characteristic = 2) An example image is given for each operation, based on a cubic seed. The basic operations are sufficient to generate the reflective uniform polyhedra and their duals. Some basic operations can be made as composites of others. One way to classify operations is by the ratio of the number of edges after the operation to the number before: for a large class of operations, this is an integer value that does not vary depending on the seed.[1]

Special forms

The kis operator has a variation, kn, which only adds pyramids to n-sided faces.
The truncate operator has a variation, tn, which only truncates order-n vertices.

The operators are applied like functions from right to left. For example, a cuboctahedron is an ambo cube, i.e. t(C) = aC, and a truncated cuboctahedron is t(a(C)) = t(aC) = taC.

Chirality operator

  • r – "reflect" – makes the mirror image of the seed; it has no effect unless the seed was made with s or g. Alternately an overline can be used for picking the other chiral form, like s = rs.

The operations are visualized here on cube seed examples, drawn on the surface of the cube, with blue faces that cross original edges, and pink faces that center at original vertices.

Basic operations
Operator Example Name Alternate
construction		 vertices edges faces Description
  Seed rr
dd		 v e f Seed form
r reflect v e f Mirror image for chiral forms
d   dual f e v dual of the seed polyhedron - each vertex creates a new face
a   ambo dj
djd		 e 2e f+v New vertices are added mid-edges, while old vertices are removed.
Also called rectification, or the medial graph in graph theory
This creates valence 4 vertices.
j   join da
dad		 v+f 2e e The seed is augmented with pyramids at a height high enough so that 2 coplanar triangles from 2 different pyramids share an edge.
This creates quadrilateral faces.
k
kn		   kis nd = dz
dtd		 v+f 3e 2e raises a pyramid on each face.
Akisation. Also called Kleetope, cumulation,[2] accretion, or pyramid-augmentation.
t
tn		   truncate dn = zd
dkd		 2e 3e v+f truncate all vertices.
conjugate kis
n   needle kd = dt
dzd		 v+f 3e 2e Dual of truncation, triangulate with 2 triangles across every edge. This bisect faces across all vertices and edges, while removing original edges.
This transforms geodesic polyhedron (a,b) into (a+2b,a-b), for a>b.
It transforms (a,0) into (a,a), and (a,a) into '(3a,0), and (2,1) into (4,1), etc.
z   zip dk = td
dnd		 2e 3e v+f Dual of kis or truncation of the dual. This create new edges perpendicular to original edges, a truncation beyond "ambo" with new edges "zipped" between original faces. It is also called bitruncation.
This transforms Goldberg polyhedron G(a,b) into G(a+2b,a-b), for a>b.
It transforms Goldberg G(a,0) into G(a,a), and G(a,a) into G(3a,0), and G(2,1) into G(4,1), etc.
e   expand aa
dod = do		 2e 4e v+e+f Each vertex creates a new face and each edge creates a new quadrilateral. (cantellate)
o   ortho daa
ded = de		 v+e+f 4e 2e Each n-gon faces are divided into n quadrilaterals.
g
rg=g		   gyro dsd = ds v+2e+f 5e 2e Each n-gon face is divided into n pentagons.
s
rs=s		   snub dgd = dg 2e 5e v+2e+f "expand and twist" – each vertex creates a new face and each edge creates two new triangles
b   bevel dkda = ta
dmd = dm		 4e 6e v+e+f New faces are added in place of edges and vertices. (cantitruncation)
m   meta
medial 		kda = kj
dbd = db		 v+e+f 6e 4e Triangulate with added vertices on edge and face centers.

Generating regular seeds

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All of the five regular polyhedra can be generated from prismatic generators with zero to two operators:

The regular Euclidean tilings can also be used as seeds:

Examples

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The cube can generate all the convex uniform polyhedra with octahedral symmetry. The first row generates the Archimedean solids and the second row the Catalan solids, the second row forms being duals of the first. Comparing each new polyhedron with the cube, each operation can be visually understood.

Cube
"seed"		 ambo truncate zip expand bevel snub
 
C
dO
      		 
aC
aO
      		 
tC
zO
      		 
zC = dkC
tO
      		 
aaC = eC
eO
      		 
bC = taC
taO
      		 
sC
sO
     
dual join needle kis ortho medial gyro
 
dC
O
      		 
jC
jO
      		 
dtC = kdC
kO
      		 
kC
dtO
      		 
oC
oO
      		 
dtaC = mC
mO
      		 
gC
gO
     

The truncated icosahedron, tI or zD, which is Goldberg polyhedron G(2,0), creates more polyhedra which are neither vertex nor face-transitive.

Truncated icosahedron seed
"seed" ambo truncate zip expand bevel snub
 
zD
tI 		 
azI
atI 		 
tzD
ttI 		 
tdzD
tdtI 		 
aazD = ezD
aatI = etI 		 
bzD
btI 		 
szD
stI
dual join needle kis ortho medial gyro
 
dzD
dtI 		 
jzD
jtI 		 
kdzD
kdtI 		 
kzD
ktI 		 
ozD
otI 		 
mzD
mtI 		 
gzD
gtI

Geometric coordinates of derived forms

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In general the seed polyhedron can be considered a tiling of a surface since the operators represent topological operations so the exact geometric positions of the vertices of the derived forms are not defined in general. A convex regular polyhedron seed can be considered a tiling on a sphere, and so the derived polyhedron can equally be assumed to be positioned on the surface of a sphere. Similar a regular tiling on a plane, such as a hexagonal tiling can be a seed tiling for derived tilings. Nonconvex polyhedra can become seeds if a related topological surface is defined to constrain the positions of the vertices. For example, toroidal polyhedra can derive other polyhedra with point on the same torus surface.

Example: A dodecahedron seed as a spherical tiling
 
D 		 
tD 		 
aD 		 
zD = dkD 		 
eD 		 
bD = taD 		 
sD
 
dD 		 
nD = dtD 		 
jD = daD 		 
kD = dtdD 		 
oD = deD 		 
mD = dtaD 		 
gD
Example: A Euclidean hexagonal tiling seed (H)
 
H 		 
tH 		 
aH 		 
tdH = H 		 
eH 		 
bH = taH 		 
sH
 
dH 		 
nH = dtH 		 
jH = daH 		 
dtdH = kH 		 
oH = deH 		 
mH = dtaH 		 
gH = dsH

Derived operations

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Mixing two or more basic operations leads to a wide variety of forms. There are many more derived operations, for example, mixing two ambo, kis, or expand, along with up to 3 interspaced duals. Using alternative operators like join, truncate, ortho, bevel and medial can simply the names and remove the dual operators. The numbers of total edges of a derived operation can be computed as the product of the number of total edges of each individual operator.

Operator(s) d a
j		 k, t
n, z		 e
o		 g
s		 a&k a&e k&k k&e
k&a2		 e&e
edge-multiplier 1 2 3 4 5 6 8 9 12 16
Unique derived operators 8 2 8 10 2

The operations are visualized here on cube seed examples, drawn on the surface of the cube, with blue faces that cross original edges, and pink faces that cross original vertices.

Chiral derived operations

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There are more derived operators mixing at least one gyro with ambo, kis or expand, and up to 3 duals.

Operator(s) d a k e g a&g k&g e&g g&g
edge-multiplier 1 2 3 4 5 10 15 20 25
Unique derived operators 4 8 4 2

Extended operators

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These extended operators can't be created in general from the basic operations above. Some can be created in special cases with k and t operators only applied to specific sided faces and vertices. For example, a chamfered cube, cC, can be constructed as t4daC, as a rhombic dodecahedron, daC or jC, with its valence-4 vertices truncated. A lofted cube, lC is the same as t4kC. And a quinto-dodecahedron, qD can be constructed as t5daaD or t5deD or t5oD, a deltoidal hexecontahedron, deD or oD, with its valence-5 vertices truncated.

Some further extended operators suggest a sequence and are given a following integer for higher order forms. For example, ortho divides a square face into 4 squares, and a o3 can divide into 9 squares. o3 is a unique construction while o4 can be derived as oo, ortho applied twice. The loft operator can include an index, similar to kis, to limit the effect to faces with that number of sides.

The chamfer operation creates Goldberg polyhedra G(2,0), with new hexagons between original faces. Sequential chamfers create G(2n,0).

Extended chiral operators

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These operators can't be created in general from the basic operations above. Geometric artist George W. Hart created an operation he called a propellor.

  • p – "propellor" (A rotation operator that creates quadrilaterals at the vertices). This operation is self-dual: dpX=pdX.

Operations that preserve original edges

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These augmentation operations retain original edges, and allowing the operator to apply to any independent subset of faces. Conway notation supports an optional index to these operators to specific how many sides affected faces will have.

Operator kis loft lace stake kis-kis
Example kC lC LC KC kkC
Edges 3e 5e 6e 7e 9e
Image
on cube		          
Augmentation Pyramid Prism Antiprism

Subdivision

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A subdivision operation divides original edges into n new edges and face interiors into smaller triangles or other polygons.

Square subdivision

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The ortho operator can be applied in series for powers of two quad divisions. Other divisions can be produced by the product of factorized divisions. The propellor operator applied in sequence, in reverse chiral directions produces a 5-ortho division. If the seed polyhedron has nonquadrilaeral faces, they will be retained as smaller copies for odd-ortho operators.

Examples on a cube
Ortho o2=o o3 o4=o2 o5
=prp		 o6=oo3 o7 o8=o3 o9=o32 o10=oo5
=oprp
Example                    
Vertices v v+e+f v+4e v+7e+f v+12e v+17e+f v+24e v+31e+f v+40e v+63e+f
Edges e 4e 9e 16e 25e 36e 49e 64e 81e 128e
Faces f 2e f+4e 8e f+12e 18e f+24e 32e f+40e 64e
Expand
(dual)		 e2=e e3 e4=e2 e5
=dprp		 e6=ee3 e7 e8=e3 e9=e32 e10=ee5
=doprp
Example        

Chiral hexagonal subdivision

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A whirl operation creates Goldberg polyhedra, G(2,1) with new hexagonal faces around each original vertex. Two sequential whirls create G(3,5). In general, a whirl can transform a G(a,b) into G(a + 3b,2a − b) for a>b and the same chiral direction. If chiral directions are reversed, G(a,b) becomes G(2a + 3b,a − 2b) if a ≥ 2b, and G(3a + b,2b − a) if a < 2b. Higher n-whirls can be defined as G(n,n − 1), and m,n-whirl G(m,n).

Whirl-n operators generate Goldberg polyhedra (n,n − 1) and can be defined by dividing a seed polyhedron's edges into 2n − 1 subedges as rings around brick pattern hexagons. Some can also be generated by composite operators with smaller Whirl-m,n operators.

The product of whirl-n and its reverse generates a (3n2 − 3n + 1,0) Goldberg polyhedron. wrw generates (7,0) w3rw3 generates (19,0), w4rw4 generates (37,0), w5rw5 generates (61,0), and w6rw6 generates (91,0). The product of two whirl-n is ((n − 1)(3n − 1),2n − 1) or (3n2 − 4n + 1,2n − 1). The product of wa by wb gives (3ab − 2(a + b) + 1,a + b − 1), and wa by reverse wb is (3ab − a − 2b + 1,a − b) for a ≥ b.

The product of two identical whirl-n operators generates Goldberg ((n − 1)(3n − 1),2n − 1). The product of a k-whirl and zip is (3k − 2,1).

Triangulated subdivision

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Triangular subdivisions u1 to u6 on a square face, repeat their structure in intervals of 3 with new layers of triangles

An operation un divides faces into triangles with n-divisions along each edge, called an n-frequency subdivision in Buckminster Fuller's geodesic polyhedra.[3]

Conway polyhedron operators can construct many of these subdivisions.

If the original faces are all triangles, the new polyhedra will also have all triangular faces, and create triangular tilings within each original face. If the original polyhedra has higher polygons, all new faces won't necessarily be triangles. In such cases a polyhedron can first be kised, with new vertices inserted in the center of each face.

Geodesic polyhedra

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See also: Geodesic polyhedra, Goldberg polyhedra, and Capsid § Triangulation_number

Conway operations can duplicate some of the Goldberg polyhedra and geodesic duals. The number of vertices, edges, and faces of Goldberg polyhedron G(m,n) can be computed from m and n, with T = m2 + mn + n2 = (m + n)2 − mn as the number of new triangles in each subdivided triangle. (m,0) and (m,m) constructions are listed below from Conway operators.

Class I

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For Goldberg duals, an operator uk is defined here as dividing faces with k edge subdivisions, with Conway u = u2, while its conjugate operator, dud is chamfer, c. This operator is used in computer graphics, loop subdivision surface, as recursive iterations of u2, doubling each application. The operator u3 is given a Conway operator nn=kt, and its conjugate operator zz=tk. The product of two whirl operators with reverse chirality, wrw or ww, produces 7 subdivisions as Goldberg polyhedron G(7,0), thus u7=vrv. Higher subdivision and whirl operations in chiral pairs can construct more class I forms. w(3,1)rw(3,1) gives Goldberg G(13,0). w(3,2)rw(3,2) gives G(19,0).

Class II

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Orthogonal subdivision can also be defined, using operator n=kd. The operator transforms geodesic polyhedron (a,b) into (a+2b,a-b), for a>b. It transforms (a,0) into (a,a), and (a,a) into (3a,0). The operator z=dk does the same for the Goldberg polyhedra.

This is also called a Triacon method, dividing into subtriangles along their height, so they require an even number of triangles along each edge.

Class III

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Most geodesic polyhedra and dual Goldberg polyhedra G(n,m) can't be constructed from derived Conway operators. The whirl operation creates Goldberg polyhedra, G(2,1) with new hexagonal faces around each original vertex, and n-whirl genereates G(n,n-1). On icosahedral symmetry forms, t5g is equivalent to whirl in this case. The v=volute operation represents the triangular subdivision dual of whirl. On icosahedral forms it can be made by the derived operator k5s, a pentakis snub.

Two sequential whirls create G(3,5). In general, a whirl can transform a G(a,b) into G(a+3b,2a-b) for a>b and the same chiral direction. If chiral directions are reversed, G(a,b) becomes G(2a+3b,a-2b) if a>=2b, and G(3a+b,2b-a) if a<2b.

Example polyhedra by symmetry

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Iterating operators on simple forms can produce progressively larger polyhedra, maintaining the fundamental symmetry of the seed element.

Tetrahedral symmetry

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Octahedral symmetry

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Chiral

Icosahedral symmetry

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Chiral
  •  
    dsD (5e)
  •  
    sD (5e)
  •  
    wD (7e)
  •  
    k5sD (7e)
  •  
    saD (10e)
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    saD (10e)
  •  
    g3D (11e)
  •  
    s3D (11e)
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    g3I (11e)
  •  
    s3I (11e)
  •  
    stI (15e)
  •  
    stD (15e)
  •  
    wtI (21e)
  •  
    k5k6stI (21e)

Dihedral symmetry

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  •  
    t4daA4=cA4
  •  
    t4daA4=cA4 (side)
  •  
    t4daA4=cA4 (top)
  •  
    tA4
  •  
    tA5
  •  
    eP3 = aaP3
  •  
    eA4 = aaA4

Toroidal symmetry

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Torioidal tilings exist on the flat torus on the surface of a duocylinder in four dimensions but can be projected down to three dimensions as an ordinary torus. These tilings are topologically similar subsets of the Euclidean plane tilings.

  •  
    A 1x1 regular square torus, {4,4}1,0
  •  
    A regular 4x4 square torus, {4,4}4,0
  •  
    tQ24×12 projected to torus
  •  
    taQ24×12 projected to torus
  •  
    actQ24×8 projected to torus
  •  
    tH24×12 projected to torus
  •  
    taH24×8 projected to torus
  •  
    kH24×12 projected to torus

Euclidean square symmetry

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Euclidean triangular symmetry

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See also

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References

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  1. ^ Brinkmann, G.; Goetschalckx, P.; Schein, S. (2017). "Goldberg, Fuller, Caspar, Klug and Coxeter and a general approach to local symmetry-preserving operations". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science. 473 (2206): 20170267. arXiv:1705.02848. doi:10.1098/rspa.2017.0267.
  2. ^ http://mathworld.wolfram.com/Cumulation.html
  3. ^ Anthony Pugh, Polyhedra: a visual approach, (1976), Chapter 6, Geodesic polyhedra, p.63 [1]
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