User:Tomruen/Conway polyhedron notation

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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.

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.

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

• Triangular pyramid: Y3 (A tetrahedron is a special pyramid)
  • T = Y3
  • O = aT (ambo tetrahedron)
  • C = jT (join tetrahedron)
  • I = sT (snub tetrahedron)
  • D = gT (gyro tetrahedron)
• Triangular antiprism: A3 (An octahedron is a special antiprism)
  • O = A3
  • C = dA3
• Square prism: P4 (A cube is a special prism)
  • C = P4
• Pentagonal antiprism: A5
  • I = k5A5 (A special gyroelongated dipyramid)
  • D = t5dA5 (A special truncated trapezohedron)

The regular Euclidean tilings can also be used as seeds:

• Q = Quadrille = Square tiling
• H = Hextille = Hexagonal tiling = dΔ
• Δ = Deltille = Triangular tiling = dH

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

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 Example: A Euclidean hexagonal tiling seed (H)

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.

Derived operations
Operator	Example	Name	Alternate construction	vertices	edges	faces	Description
	Seed	v	e	f	Seed form
at		akd	3e	6e	v+2e+f	ambo of truncate
jk		dak	v+2e+f	6e	3e	join of kis. Similar to ortho except new quad faces are inserted in place of the original edges
ak		dajd	3e	6e	v+2e+f	ambo of kis. Similar to expand, except new vertices are added on the original edges, forming two triangles.
jt		dakd = dat	v+2e+f	6e	3e	join of truncate. dual of ambo of truncate
tj		dka	4e	6e	v+e+f	truncate join
ka		v+e+f	6e	4e	kis ambo
ea or ae		aaa	4e	8e	v+3e+f	expanded ambo, triple ambo
oa or je		daaa = jjj	v+3e+f	8e	4e	ortho of ambo, triple join
kt		kdkd dtkd nn	v+e+f	9e	7e	kis truncate, triangulate, dividing edges into 3, and adding new vertices in the center of original faces. It transforms geodesic polyhedron (a,b) into (3a,3b).
tk		dkdk dktd zz	v+e+f	9e	7e	truncate kis, expand with hexagons around each edge It transforms Goldberg polyhedron G(a,b) into G(3a,3b).
nk		kdk = dtk = ktd	7e	9e	v+e+f	needled kis
tn		dkdkd = dkt = tkd	7e	9e	v+e+f	truncate needle
tt		dkkd	7e	9e	v+e+f	double-truncate
kk		dttd	v+2e+f	9e	6e	double-kis
nt		kkd = dtt	v+e+f	9e	7e	needle truncate
tz		dkk = ttd	6e	9e	v+2e+f	truncate zip
ke		kaa	v+3e+f	12e	8e	Kis expand
to		dkaa	8e	12e	v+3e+f	truncate ortho
ek		aak	6e	12e	v+5e+f	expand kis
ok		daak = dek	v+5e+f	12e	6e	ortho kis
et		aadkd	6e	12e	v+5e+f	expanded truncate
ot		daadkd = det	v+5e+f	12e	6e	ortho truncate
te or ba		dkdaa	8e	12e	v+3e+f	truncate expand
ko or ma		kdaa = dte ma = mj	v+3e+f	12e	8e	kis ortho
ab or am		aka = ata	6e	12e	v+5e+f	ambo bevel
jb or jm		daka = data	v+5e+f	12e	6e	joined bevel
ee		aaaa	v+7e+f	16e	8e	double-expand
oo		daaaa = dee	8e	16e	v+7e+f	double-ortho

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

Chiral derived operations
Operator	Example	Name	Construction	vertices	edges	faces	Description
	Seed	v	e	f	seed form
ag		as djsd = djs	v+4e+f	10e	5e	ambo gyro
jg		dag = js dasd = das	5e	10e	v+4e+f	joined gyro
ga		gj dsjd = dsj	v+5e+f	10e	4e	gyro ambo
sa		dga = sj dgjd = dgj	4e	10e	v+5e+f	snub ambo
kg		dtsd = dts	v+4e+f	15e	10e	kis gyro
ts	dkgd = dkg	10e	15e	v+4e+f	truncated snub
gk	dstd	v+8e+f	15e	6e	gyro kis
st	dgkd	6e	15e	v+8e+f	snub truncation
sk	dgtd	v+8e+f	15e	6e	snub kis
gt	dskd	6e	15e	v+8e+f	gyro truncation
ks		kdg dtgd = dtg	v+4e+f	15e	10e	kis snub
tg	dkdg dksd	10e	15e	v+4e+f	truncated gyro
eg	es aag	v+9e+f	20e	10e	expanded gyro
og	os daagd = daag	10e	20e	v+9e+f	expanded snub
ge	go gaa	v+11e+f	20e	8e	gyro expand
se	so dgaad = dgaa	8e	20e	v+11e+f	snub expand
gg		gs dssd = dss	v+14e+f	25e	10e	double-gyro
ss	sg dggd = dgg	10e	25e	v+14e+f	double-snub

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 operations
Operator	Example	Name	Alternate construction	vertices	edges	faces	Description
	Seed	v	e	f	Seed form
c		chamfer	dud	v + 2e	4e	f + e	An edge-truncation. New hexagonal faces are added in place of edges. Goldberg (0,2)
-		-	dc	f + e	4e	v + 2e	Dual of chamfer
u		subdivide	dcd	v+e	4e	f+2e	Ambo while retaining original vertices Similar to Loop subdivision surface for triangle faces
-		cd	f+2e	4e	v+e	dual to subdivision
l ln		loft	v+2e	5e	f+2e	An augmentation of each face by prism, adding a smaller copy of each face with trapezoids between the inner and outer ones.
	dl dln	f+2e	5e	v+2e	Dual to loft
	ld lnd	f+2e	5e	v+2e	loft of dual
	dld dlnd	v+2e	5e	f+2e	Conjugate to loft
	dL0	f+3e	6e	v+2e	Dual to joined-lace
	L0d	f+2e	6e	v+3e	Joined-lace of dual
	dL0d	v+3e	6e	f+2e	Conjugate joined-lace
q		quinto	v+3e	6e	f+2e	ortho followed by truncation of vertices centered on original faces. This create 2 new pentagons for every original edge.
-		dq	f+2e	6e	v+3e	dual of quinto
	qd	v+2e	6e	f+3e	quinto of dual
-		dqd	f+3e	6e	v+2e	conjugate of quinto
L0		joined-lace	v+2e	6e	f+3e	Similar to lace, except new with quad faces across original edges
L Ln		Lace	v+2e	7e	f+4e	An augmentation of each face by an antiprism, adding a twist smaller copy of each face, and triangles between. An index can be added to limit the operation to faces of that many sides.
	dL dLn	f+4e	7e	v+2e	Dual of laced
	Ld Ldn	f+2e	7e	v+4e	Lace of dual
	dLd dLnd	v+4e	7e	f+2e	Dual of lace of dual
K Kn		staKe	v+2e+f	7e	4e	Subdivide faces with central quads, and triangles. An index can be added to limit the operation to faces of that many sides.
	dK dKn	4e	7e	v+2e+f	Dual of stake
	Kd	v+2e+f	7e	4e	Stake of dual
	dKd	4e	7e	v+2e+f	Conjugate of stake
M3		edge-medial-3	v+2e+f	7e	4e	Similar to m3 except no diagonal edges added, creating quad faces there
	dM3	4e	7e	v+2e+f	dual of edge-medial-3
	M3d	v+2e+f	7e	4e	edge-medial-3 of dual
	dM3d	4e	7e	v+2e+f	Conjugate of edge-medial-3
M0		joined-medial	v+2e+f	8e	5e	Like medial, but new rhombic faces in place of original edges.
	dM0	v+2e+f	8e	5e	dual of joined-medial
	M0d	v+2e+f	8e	5e	joined-medial of dual
	dM0d	5e	8e	v+2e+f	Conjugate of joined-medial
m3		medial-3	v+2e+f	9e	7e	triangulate with 2 vertices added on edge and face centers.
b3		bevel-3	dm3	7e	9e	v+2e+f	dual to medial-3
	m3d	7e	9e	v+2e+f	medial-3 of dual
	dm3d	v+2e+f	9e	7e	conjugate of medial-3
o3		ortho-3	de3	v+4e	9e	f+4e	ortho operator with 3 edge divisions
e3		expand-3	do3	f+4e	9e	v+4e	expand operator with 3 edge divisions
X		cross	v+f+3e	10e	6e	Combination of kis and subdivide operation. Original edges are divided in half, with triangle and quad faces.
	dX	6e	10e	v+f+3e	Dual to cross
	Xd	6e	10e	v+f+3e	Cross of dual
	dXd	v+f+3e	10e	6e	Conjugate of cross
m4		medial-4	v+3e+f	12e	8e	triangulate with 3 vertices added on edge and face centers.
u5		subdivide-5	v+8e	25e	f+16e	Subdivide edges into 5th This operator divides edges and faces so there are 6 triangles around each new vertex.

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.

Chiral extended operations
Operator	Example	Name	Alternate construction	vertices	edges	faces	Description
	"Seed"	v	e	f	Seed form
p rp=p		propellor	v + 2e	5e	f + 2e	A gyro followed by an ambo of vertices centered at original faces
-		-	dp = pd	f + 2e	5e	v + 2e	Same vertices as gyro, except new faces at original vertices
-		4e	7e	v+2e+f	Like snub, except pentagons are around original faces rather than triangles
-		-	-	v+2e+f	7e	4e
w=w2=w2,1 rw=w		whirl	v+4e	7e	f+2e	gyro followed by truncation of vertices centered at original faces. This create 2 new hexagons for every original edge, Goldberg (2,1) The derived operator wrw transforms G(a,b) into G(7a,7b).
v rv=v		volute	dwd	f+2e	7e	v+4e	dual of whirl, a snub followed by kis on original faces. The derived operator vrv transforms geodesic (a,b) into (7a,7b).
g3 rg3=g3		gyro-3	v+6e	11e	f+4e	Gyro operation create 3 pentagons along each original edge
s3 rs3=s3		snub-3	dg3d = dg3	f+4e	11e	v+6e	Dual of gyro-3, snub operation which divides edges into 4 middle triangles with triangles at the original vertices
w3,1 rw3,1=w3,1		whirl-3,1	v+8e	13e	f+4e	create 4 new hexagons for every original edge, Goldberg (3,1)
w3=w3,2 rw3=w3		whirl-3,2	v+12e	19e	f+6e	create 12 new hexagons for every original edge, Goldberg (3,2)

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

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

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

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 w_a by w_b gives (3ab − 2(a + b) + 1,a + b − 1), and w_a by reverse w_b 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).

Whirl-n operators
Name	Seed	Whirl	Whirl-3	Whirl-4	Whirl-5	Whirl-6	Whirl-7	Whirl-8	Whirl-9	Whirl-10	Whirl-11	Whirl-12	Whirl-13	Whirl-14	Whirl-15	Whirl-16	Whirl-17	Whirl-18	Whirl-19	Whirl-20	Whirl-n
Operator (Composite)	–	w=w2	w3	w4	w5	w6 wrw3,1	w7	w8 w3,1w3,1	w9 ww5,1	w10	w11	w12	w13 ww7,2	w14	w15	w16 ww9,2	w17 w3w6,1	w18	w19 w3,1w7,3	w20 ww11,3	wn
Goldberg	(1,0)	(2,1)	(3,2)	(4,3)	(5,4)	(6,5)	(7,6)	(8,7)	(9,8)	(10,9)	(11,10)	(12,11)	(13,12)	(14,13)	(15,14)	(16,15)	(17,16)	(18,17)	(19,18)	(20,19)	(n,n − 1)
T composite	1	7	19	37	61	91 ×ばつ13	127	169 ×ばつ13	217 ×ばつ31	271	331	397	469 ×ばつ67	547	631	721 ×ばつ103	817 ×ばつ43	919	1027 ×ばつ79	1141 ×ばつ163	3n(n − 1) + 1
Example
Vertices	v	v + 4e	v + 12e	v + 24e	v + 40e	v + 60e	v + 84e	v + 112e	v + 144e	v + 180e	v + 220e	v + 264e	v + 312e	v + 364e	v + 420e	v + 480e	v + 544e	v + 612e	v + 684e	v + 760e	v + 2n(n − 1)e
Edges	e	7e	19e	37e	61e	91e	127e	169e	217e	271e	331e	397e	469e	547e	631e	721e	817e	919e	1027e	1141e	e + 3n(n − 1)e
Faces	f	f + 2e	f + 6e	f + 12e	f + 20e	f + 30e	f + 42e	f + 56e	f + 72e	f + 90e	f + 110e	f + 132e	f + 156e	f + 182e	f + 210e	f + 240e	f + 272e	f + 306e	f + 342e	f + 380e	f + n(n − 1)e
wnwn	(1,0)	(5,3)	(16,5)	(33,7)	(56,9)	(85,11)	(120,13)	(161,15)	(208,17)	(261,19)	(320,21)	(385,23)	(456,25)	(533,27)	(616,29)	(705,31)	(800,33)	(901,35)	(1008,37)	(1121,39)	((n – 1)(3n − 1),2n − 1)
wnrwn	(1,0)	(7,0)	(19,0)	(37,0)	(61,0)	(91,0)	(127,0)	(169,0)	(217,0)	(271,0)	(331,0)	(397,0)	(469,0)	(547,0)	(631,0)	(721,0)	(817,0)	(919,0)	(1027,0)	(1141,0)	(1+3n(n − 1),0)
wnz	(1,1)	(4,1)	(7,1)	(10,1)	(13,1)	(16,1)	(19,1)	(22,1)	(25,1)	(28,1)	(31,1)	(34,1)	(37,1)	(40,1)	(43,1)	(46,1)	(49,1)	(52,1)	(55,1)	(58,1)	(3n − 2,1)

An operation u_n 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.

Subdivisions on a cube example
Operator	u1	u2 = u	u3 = nn	u4 = uu	u5	u6 = unn	u7 =vrv	u8 = u3	u9 =n4
Example
Conway	C	uC	nnC	uuC	u5C	unnC	vrvC	u3C	n4C
Vertices	v	v+e	v+e+f	v+4e	v+8e	v+11e+f	v+16e	v+21e	v+26e+f
Edges	e	4e	9e	16e	25e	36e	49e	64e	81e
Faces	f	f+2e	7e	f+8e	f+16e	24e	f+32e	f+42e	54e
Full triangulation
Operator	u1k	u2k =uk	u3k =nnk	u4k =uuk	u5k	u6k =unnk	u7k =vrvk	u8k =u3k	u9k =n4k
Example
Conway	kC	ukC	nnkC	uukC	u5kC	unnkC	vrvkC	u3kC	n4kC
Goldberg dual	{3,n+}1,1	{3,n+}2,2	{3,n+}3,3	{3,n+}4,4	{3,n+}5,5	{3,n+}6,6	{3,n+}7,7	{3,n+}8,8	{3,n+}9,9

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.

For Goldberg duals, an operator u_k 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 I: Subdivision operations on an icosahedron as geodesic polyhedra
(m,0)	(1,0)	(2,0)	(3,0)	(4,0)	(5,0)	(6,0)	(7,0)	(8,0)	(9,0)	(10,0)	(11,0)	(12,0)	(13,0)	(14,0)	(15,0)	(16,0)
T	1	4	9	16	25	36	49	64	81	100	121	144	169	196	225	256
Operation Composite	u1	u2=u =dcd	u3=nn =kt	u4 =u22 =dccd	u5	u6=u2u3 =dctkd	u7 =vv =dwrwd	u8=u23 =dcccd	u9=u32 =ktkt	u10=u2u5	u11	u12=u22u3 =dccdkt	u13 v3,1v3,1	u14=u2u7 =uvv =dcwrwd	u15= u3u5 =u5nn	u16=u24 =dccccd
Face triangle
Icosahedron Conway Geodesic	I {3,5+}1,0	uI=k5aI {3,5+}2,0	nnI=ktI {3,5+}3,0	u2I {3,5+}4,0	{3,5+}5,0	unnI {3,5+}6,0	vrvI {3,5+}7,0	u3I {3,5+}8,0	nn2I {3,5+}9,0	{3,5+}10,0	{3,5+}11,0	u2nnI {3,5+}12,0	{3,5+}13,0	uvrvI {3,5+}14,0	{3,5+}15,0	u4I {3,5+}16,0
Dual operator	c	zz =tk	cc	c5	czz =ctk	ww =wrw	ccc	z2 =tktk	cc5	c11	cczz =cctk	w3,1w3,1	cww =cwrw	c5zz	cccc
Dodecahedron Conway Goldberg	D {5+,3}1,0	cD {5+,3}2,0	zzD {5+,3}3,0	ccD {5+,3}4,0	c3D {5+,3}5,0	czzD {5+,3}6,0	wrwD {5+,3}7,0	cccD {5+,3}8,0	zz2D {5+,3}9,0	cc5D {5+,3}10,0	c11D {5+,3}11,0	cczzD {5+,3}12,0	w3,1rw3,1D {5+,3}13,0	cwrwD {5+,3}14,0	c5zzD {5+,3}15,0	ccccD G{5+,3}16,0

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 II: Orthogonal subdivision operations
(m,m)	(1,1)	(2,2)	(3,3)	(4,4)	(5,5)	(6,6)	(7,7)	(8,8)	(9,9)	(10,10)	(11,11)	(12,12)	(13,13)	(14,14)	(15,15)	(16,16)
T= m2×ばつ3	3 ×ばつ3	12 ×ばつ3	27 ×ばつ3	48 ×ばつ3	75 ×ばつ3	108 ×ばつ3	147 ×ばつ3	192 ×ばつ3	243 ×ばつ3	300 ×ばつ3	363 ×ばつ3	432 ×ばつ3	507 ×ばつ3	588 ×ばつ3	675 ×ばつ3	768 ×ばつ3
Operation	u1n n =kd	u2n =un =dct	u3n =n3 =ktkd	u4n =u22n =dcct	u5n	u6n =u2=u3n =dctkt	u7n =vvn =dwrwt	u8n =u23n =dccct	u9n =u32n =ktktkd	u10n =u2u5n	u11n	u12n =u22u3n =dcctkt	u13n	u14n =u2u7n =dcwrwt	u15n =u3u5n	u16n =u24n =dcccct
Face triangle
Icosahedron Conway Geodesic	nI {3,5+}1,1	unI {3,5+}2,2	nnnI {3,5+}3,3	u2nI {3,5+}4,4	{3,5+}5,5	unnnI {3,5+}6,6	vrvnI {3,5+}7,7	u3nI {3,5+}8,8	n5I {3,5+}9,9	{3,5+}10,10	{3,5+}11,11	u2n3I {3,5+}12,12	{3,5+}13,13	dcwrwdnI {3,5+}14,14	{3,5+}15,15	u4nI {3,5+}16,16
Dual operator	z =dk	cz =cdk	z3 =tkdk	c2z =ccdk	c5z	cz3 =ctkdk	wwz =wrwdk	c3z =cccdk	z5 =tktkdk	cc5z	c11z	c2z3 =c2tkdk	c13z	cwwz =cwrwdk	c3c5z	c4z =ccccdk
Dodecahedron Conway Goldberg	zD {5+,3}1,1	czD {5+,3}2,2	zzzD {5+,3}3,3	cczD {5+,3}4,4	{5+,3}5,5	czzzD {5+,3}6,6	wrwzD {5+,3}7,7	c3zD {5+,3}8,8	z5D {5+,3}9,9	{5+,3}10,10	G{5+,3}11,11	cczzzD {5+,3}12,12	{5+,3}13,13	cwrwzD G{5+,3}14,14	{5+,3}15,15	cccczD {5+,3}16,16

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.

Class III: Unequal subdivision operations
Operation Composite	v2,1 =v	v3,1	v3,2=v3	v4,1 =vn	v4,2 =vu	v5,1	v4,3=v4	v5,2 =v3n	v6,1	v6,2 =v3,1u	v5,3 =vv	v7,1 =v3n	v5,4=v5	v6,3 =vnn	v7,2
T	7	13	19	21 ×ばつ3	28 ×ばつ4	31	37	39 ×ばつ3	43	52 ×ばつ4	49 ×ばつ7	57 ×ばつ3	61	63 ×ばつ7	67
Face triangle
Icosahedron Conway Geodesic	vI {3,5+}2,1	v3,1I {3,5+}3,1	v3I {3,5+}3,2	vnI {3,5+}4,1	vuI {3,5+}4,2	{3,5+}5,1	v4I {3,5+}4,3	v3nI {3,5+}5,2	{3,5+}6,1	v3,1uI {3,5+}6,2	vvI {3,5+}5,3	v3nI {3,5+}7,1	v5I {3,5+}5,4	vnnI {3,5+}6,3	v7,2I {3,5+}7,2
Operator	w	w3,1	w3	wz	wc	w5,1	w4	w3,1z	w6,1	w3,1c	ww	w3z	w5	wzz	w7,2
Dodecahedron Conway	wD {5+,3}2,1	w3,1D {5+,3}3,1	w3D {5+,3}3,2	wzD {5+,3}4,1	wcD {5+,3}4,2	w5,1D {5+,3}5,1	w4D {5+,3}4,3	w3zD {5+,3}5,2	{5+,3}6,1	w3,1cD {5+,3}6,2	wwD {5+,3}5,3	w3zD {5+,3}7,1	w5D {5+,3}5,4	wzzD {5+,3}6,3	w7,2D {5+,3}7,2

More class III: Unequal subdivision operations
Operation Composite	v8,1	v6,4 =v3u	v7,3	v8,2 =wcz	v6,5=v6 =vrv3,1	v9,1 =vv3,1	v7,4	v8,3	v9,2	v7,5	v10,1 =v4n	v8,4 =vuu	v9,3 =v3,1nn	v7,6=v7	v8,6 v4u
T	73	76 ×ばつ4	79	84 ×ばつ3	91 ×ばつ7		93	97	103	109	111 ×ばつ3	112 ×ばつ4	117 ×ばつ9	127	148 ×ばつ4
Face triangle
Icosahedron Conway Geodesic	v8,1I {3,5+}8,1	v3uI {3,5+}6,4	v7,3I {3,5+}7,3	vunI {3,5+}8,2	vv3,1I {3,5+}6,5	vrv3,1I {3,5+}9,1	v7,4I {3,5+}7,4	v8,3I {3,5+}8,3	v9,2I {3,5+}9,2	v7,5I {3,5+}7,5	v4nI {3,5+}10,1	vuuI {3,5+}8,4	v3,1nnI {3,5+}9,3	v7I {3,5+}7,6	v4uI {3,5+}8,6
Operator	w8,1	wrw3,1	w7,3	w3,1c	wcz	w3,1w	w7,4	w8,3	w9,2	w7,5	w4z	wcc	w3,1zz	w7	w4c
Dodecahedron Conway	w8,1D {5+,3}8,1	w3cD {5+,3}6,4	w7,3D {5+,3}7,3	wczD {5+,3}8,2	ww3,1D {5+,3}6,5	wrw3,1D {5+,3}9,1	w7,4D {5+,3}7,4	w8,3D {5+,3}8,3	w9,2D {5+,3}9,2	w7,5D {5+,3}7,5	w4zD {5+,3}10,1	wccD {5+,3}8,4	w3,1nnD {5+,3}9,3	w7D {5+,3}7,6	w4cD {5+,3}8,6

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

• t6dtT
  t6dtT
• atT
  atT
• tatT
  tatT
• stT
  stT
• XT (10e)
  XT (10e)
• dXT (10e)
  dXT (10e)
• m3T
  m3T
• b3T
  b3T

• daC (2e)
  daC (2e)
• cC (4e) *
  cC (4e) *
• dcC (4e) *
  dcC (4e) *
• cO (4e)
  cO (4e)
• akC (6e) *
  akC (6e) *
• dakC (6e)
  dakC (6e)
• m3C (6e)
  m3C (6e)
• m3O (6e)
  m3O (6e)
• b3C (6e)
  b3C (6e)
• b3O (6e)
  b3O (6e)
• atC (6e)
  atC (6e)
• qC (6e)
  qC (6e)
• edaC (8e)
  edaC (8e)
• dktO=tkC (9e)
  dktO=tkC (9e)
• taaC (12e) *
  taaC (12e) *
• XO (10e)
  XO (10e)
• XC (10e)
  XC (10e)
• dXO (10e)
  dXO (10e)
• dXC (10e)
  dXC (10e)
• cdkC (12e)
  cdkC (12e)
• ccC (16e)
  ccC (16e)
• tkdkC (18e)
  tkdkC (18e)
• tatO (18e)
  tatO (18e)
• tatC (18e)
  tatC (18e)
• l6l8taC (22e)
  l6l8taC (22e)
• ccdkC (48e)
  ccdkC (48e)
• wrwC (49e)
  wrwC (49e)
• cccC (64e)
  cccC (64e)
• tktkC (81e)
  tktkC (81e)

Chiral

• wC (7e)
  wC (7e)
• saC (10e)
  saC (10e)
• gaC (10e)
  gaC (10e)
• saC (10e)
  saC (10e)
• stO (15e)
  stO (15e)
• stC (15e)
  stC (15e)

• kD = daD (2e)
  kD = daD (2e)
• kD (3e) *
  kD (3e) *
• dkD=tI (3e) *
  dkD=tI (3e) *
• cI (4e) *
  cI (4e) *
• t5daD=cD (4e) *
  t5daD=cD (4e) *
• dcI (4e) *
  dcI (4e) *
• dakD (6e) *
  dakD (6e) *
• atD (6e)
  atD (6e)
• atI=akD (6e) *
  atI=akD (6e) *
• qD (6e)
  qD (6e)
• m3D (6e)
  m3D (6e)
• m3I (6e)
  m3I (6e)
• b3D (6e)
  b3D (6e)
• b3I (6e)
  b3I (6e)
• edaD (8e) *
  edaD (8e) *
• tkdD (9e) *
  tkdD (9e) *
• gaD (10e) *
  gaD (10e) *
• XI (10e)
  XI (10e)
• XD (10e)
  XD (10e)
• dXI (10e)
  dXI (10e)
• dXD (10e)
  dXD (10e)
• teD (12e) *
  teD (12e) *
• cdkD (12e)
  cdkD (12e)
• m3aI (12e)
  m3aI (12e)
• tatI = takD (18e)
  tatI = takD (18e)
• tatD (18e)
  tatD (18e)
• atkD (18e)
  atkD (18e)
• m3tD (18e)
  m3tD (18e)
• qtI = t5t6otI (18e)
  qtI = t5t6otI (18e)
• dqtI = k5k6etI (18e)
  dqtI = k5k6etI (18e)
• actI (24e)
  actI (24e)
• kdktI (27e)
  kdktI (27e)
• tktI (27e)
  tktI (27e)
• dctkD (36e)
  dctkD (36e)
• ctkD (36e)
  ctkD (36e)
• k6k5tI
  k6k5tI
• kt5daD
  kt5daD

Chiral

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

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

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 1x1 regular square torus, {4,4}_1,0
• A regular 4x4 square torus, {4,4}4,0
  A regular 4x4 square torus, {4,4}_4,0
• ×ばつ12 projected to torus">×ばつ12 projected to torus
  ×ばつ12 projected to torus
• ×ばつ12 projected to torus">ta×ばつ12 projected to torus
  ta×ばつ12 projected to torus
• ×ばつ8 projected to torus">ac×ばつ8 projected to torus
  ac×ばつ8 projected to torus
• ×ばつ12 projected to torus">×ばつ12 projected to torus
  ×ばつ12 projected to torus
• ×ばつ8 projected to torus">ta×ばつ8 projected to torus
  ta×ばつ8 projected to torus
• ×ばつ12 projected to torus">k×ばつ12 projected to torus
  k×ばつ12 projected to torus

• tQ
  tQ
• cQ
  cQ
• akQ
  akQ

• tH
  tH
• cΔ
  cΔ
• cH
  cH
• ctH
  ctH
• dakH
  dakH
• aaaH
  aaaH
• aaaH, equilateral
  aaaH, equilateral

• Uniform polyhedra
• Computer graphics algorithms:
  • Doo–Sabin subdivision surface – expand operator
  • Catmull–Clark subdivision surface – ortho operator
• Goldberg polyhedron
• Goldberg–Coxeter construction
• Symmetrohedron

• George W. Hart, Sculpture based on Propellorized Polyhedra, Proceedings of MOSAIC 2000, Seattle, WA, August, 2000, pp. 61–70 [2]
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
  • Chapter 21: Naming the Archimedean and Catalan polyhedra and Tilings
• Visualization of Conway Polyhedron Notation Hidetoshi Nonaka, World Academy of Science, Engineering and Technology 50 2009

• George Hart's Conway interpreter: generates polyhedra in VRML, taking Conway notation as input. Also includes helpful explanations of the operations.
  • Polyhedra Names
• Vertex- and edge-truncation of the Platonic and Archimedean solids leading to vertex-transitive polyhedra Livio Zefiro
• polyHédronisme: generates polyhedra in HTML5 canvas, taking Conway notation as input
• Weisstein, Eric W. "Conway Polyhedron Notation". MathWorld .
• John Conway's notation Visualization of Conway Polyhedron Notation
• Weisstein, Eric W. "Truncation". MathWorld . (truncate)
• Weisstein, Eric W. "Rectification". MathWorld . (ambo)
• Weisstein, Eric W. "Cumulation or Apiculation". MathWorld . (kis)
• Conway operators, PolyGloss, Wendy Krieger
• Derived Solids

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