Conway polyhedron notation

Conway polyhedron notation is a textual notation for a certain class of convex polyhedra. Introduced by Conway, the applications of the notation to date have been mainly artistic, with the sculptor George Hart helping to popularize it.

The notation begins with "seed" polyhedron and and modifies it with a series of "operators" to obtain the desired polyhedron. These operators include standard ones such duals, truncation, and rectification, but also some more unusual ones. The notation itself is a string of mostly letters, representing the seed polyhedron as an uppercase letter on the right-hand side of the string, and operators as lowercase letters that are applied from right to left. For example, gtC is a cube (C) subjected to truncation (t) and then a "gyro" operation (g).

The notation can be straightforwardly generalized for nonconvex polyhedra by e.g. starting with a nonconvex seed. It has not been applied to dimensions other than 3.

Description[edit | edit source]

Commonly-used seed polyhedra
Polyhedron	Tetrahedron	Octahedron	Cube	Icosahedron	Dodecahedron	$n$ -gonal prism	$n$ -gonal antiprism	$n$ -gonal pyramid
Polyhedron
Seed letter	T	O	C	I	D	P $n$	A $n$	Y $n$

Commonly-used operators in Conway notation (Pink and blue regions are cut away; pink is related to original vertices and blue is related to original edges)
Operator name	Conway notation	Notes
(Seed) (Example: cube)	C	= dd
dual	dC	Faces replaced with vertices. Vertices replaced with faces.
truncate	tC	Cuts off each vertex, with the cuts independent of one another. A number after the t can mean "only apply this operator to vertices surrounded by this many edges."
ambo	aC	Rectification. Cuts off each vertex, to the extent that the vertices of the cuts just touch each other.
expand; explode	eC	Cantellation, which is the 3D form of expansion. = aa, = do
bevel	bC	The 3D form of omnitruncation. = ta, = dm
reflect	rC	Produces the mirror image of a chiral polyhedron. Does not change achiral polyhedra.
kis	kC	Puts a vertex in the middle of each face, and connects it with edges to the existing vertices. This adds a pyramid to each face. This operator creates the tetrakis hexahedron from the cube. A number after the k can mean "only apply this operator to faces with this many edges."
join	jC	Creates quadrilateral faces. The operator that creates the rhombic dodecahedron from the cube (or octahedron). = da
ortho	oC	Basically turns each $n$ -gon face into $n$ quadrilaterals. The operator that creates the deltoidal icositetrahedron from the cube (or octahedron). = daa, = de
meta	mC	The operator that creates the disdyakis dodecahedron from the cube (or octahedron). = kj, = db
snub	sC	Snubbing. Produces a chiral polyhedron. = dg
gyro	gC	Basically turns each $n$ -gon face into $n$ pentagons. Chiral. The operator that creates the pentagonal icositetrahedron from the cube (or octahedron). = ds

The Platonic solids can be represented as Conway operators applied to prisms, antiprisms, and pyramids.

T =	Y3
O =	aY3	A3	dP4
C =	jY3	dA3	P4
I =	sY3			k5A5
D =	gY3				t5dA5

The Archimedean and Catalan solids can also be represented as Conway operators applied to Platonic solids. (Applying these to the tetrahedron might result in a Platonic solid.)

truncate	rectify	truncate dual	cantellate	omnitrunate	snub	Catalan solid corresponding to one of these
t	a	td	e; aa	b; ta	s	d- followed by this operator(s)
kis of dual	"rhombic" solid	kis	"deltoidal" solid	disdyakis	"pentagonal" solid	Archimedean solid corresponding to one of these
kd; dt	j; da	k; dtd	o; daa	m; dta	g; ds	d- followed by this operator(s)

The wide variety of possibilities by which new faces, edges, and vertices can be created and by which old ones can be removed may give rise to many different operators from different sources. Some are not default or standardized, and might have conflicting notation or might not be found in all implementations of the Conway notation.

Examples of less-official operators in Conway notation
Operator name	Image	Notes
(Seed)
zip		Bitruncation. The operator that creates the truncated octahedron from the cube. = dk, = td
needle		The operator that creates the triakis octahedron from the cube. = kd, = dt
chamfer		The operator that creates the chamfered cube from the cube.
propellor
whirl
loft		Equivalent to augmenting the faces with prisms.
lace		Equivalent to augmenting the faces with antiprisms.
quinto