Conway polyhedron notation

# Conway polyhedron notation

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Conway polyhedron notation is used to describe polyhedra
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

based on a seed polyhedron modified by various operation
Operation (mathematics)
The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....

s.

The seed polyhedra are the Platonic solid
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

s, represented by their first letter of their name (T,O,C,I,D); the prism
Prism (geometry)
In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...

s (Pn), antiprism
Antiprism
In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...

s (An) and pyramid
Pyramid (geometry)
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base....

s (Yn). Any convex polyhedron can serve as a seed, as long as the operations can be executed on it.

John Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...

extended the idea of using operators, like truncation
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.- Uniform truncation :...

defined by Kepler, to build related polyhedra of the same symmetry. His descriptive operators can generate all the Archimedean solid
Archimedean solid
In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...

s and Catalan solid
Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....

s from regular seeds. Applied in a series, these operators allow many higher order polyhedra to be generated.

## Operations on polyhedra

Elements are given from the seed (v,e,f) to the new forms, assuming seed is a convex polyhedron: (a topological sphere, Euler characteristic
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...

=2)
OperatorNameAlternate
construction
verticesedgesfacesDescription
Seed v e f Seed form
rReflect
(Hart)
|v e f Mirror image for chiral forms
ddual
Dual polyhedron
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

f e v dual of the seed polyhedron - each vertex creates a new face
aambo e 2e 2+e The edges are new vertices, while old vertices disappear. (rectify
Rectification (geometry)
In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points...

)
jjoin da e+2 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.
ttruncate
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.- Uniform truncation :...

dkd 2e 3e e+2 truncate all vertices.
-- -- dk 2e 3e e+2 Dual of kis, (bitruncation
Bitruncation
In geometry, a bitruncation is an operation on regular polytopes. It represents a truncation beyond rectification. The original edges are lost completely and the original faces remain as smaller copies of themselves....

)
---- kd e+2 3e 2e Kis of dual
kkis
Kleetope
In geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope is another polyhedron or polytope formed by replacing each facet of with a shallow pyramid...

dtd e+2 3e 2e raises a pyramid on each face.
c chamfer
Chamfer
A chamfer is a beveled edge connecting two surfaces. If the surfaces are at right angles, the chamfer will typically be symmetrical at 45 degrees. A fillet is the rounding off of an interior corner. A rounding of an exterior corner is called a "round" or a "radius"."Chamfer" is a term commonly...

e+v 4e 2e+f New hexagonal faces are added in place of edges.
- - dc 2e+f 4e e+v
eexpand
Expansion (geometry)
In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements...

aa 2e 4e 2e+2 Each vertex creates a new face and each edge creates a new quadrilateral. (cantellate
Cantellation (geometry)
In geometry, a cantellation is an operation in any dimension that cuts a regular polytope at its edges and vertices, creating a new facet in place of each edge and vertex. The operation also applies to regular tilings and honeycombs...

)
oortho de 2e+2 4e 2e Each n-gon faces are divided into n quadrilaterals.
ppropellor
(Hart)
|v+2e 4e e+f A face rotation that creates quadrilaterals at vertices (self-dual)
- - dp e+f 4e v+2e
ssnub
Snub (geometry)
In geometry, an alternation is an operation on a polyhedron or tiling that removes alternate vertices. Only even-sided polyhedra can be alternated, for example the zonohedra. Every 2n-sided face becomes n-sided...

dg 2e 5e 3e+2 "expand and twist" - each vertex creates a new face and each edge creates two new triangles
ggyro ds 3e+2 5e 2e Each n-gon face is divided into n pentagons.
bbevel ta 4e 6e 2e+2 New faces are added in place of edges and vertices, Omnitruncation (Known as cantitruncation in higher polytopes).
mmeta db & kj 2e+2 6e 4e n-gon faces are divided into 2n triangles

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:
• the dual of a tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

is dT;
• the truncation
Truncation
In mathematics and computer science, truncation is the term for limiting the number of digits right of the decimal point, by discarding the least significant ones.For example, consider the real numbersThe result would be:- Truncation and floor function :...

of a cube is t3C or tC;
• the truncation
Truncation
In mathematics and computer science, truncation is the term for limiting the number of digits right of the decimal point, by discarding the least significant ones.For example, consider the real numbersThe result would be:- Truncation and floor function :...

of a Cuboctahedron
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...

is t4aC or taC.

All operations are symmetry-preserving except twisting ones like s and g which lose reflection symmetry.

## Examples

The cube
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...

can generate all the convex Octahedral symmetry
Octahedral symmetry
150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...

uniform polyhedra
Uniform polyhedron
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...

. The first row generates the Archimedean solid
Archimedean solid
In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...

s and the second row the Catalan solid
Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....

s, the second row forms being duals of the first. Comparing each new polyhedron
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

with the cube, each operation can be visually understood. (Two polyhedron forms don't have single operator names given by Conway.)
Cube
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...

"seed"
ambo
(rectify)
Rectification (geometry)
In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points...

truncate
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.- Uniform truncation :...

bitruncate
Bitruncation
In geometry, a bitruncation is an operation on regular polytopes. It represents a truncation beyond rectification. The original edges are lost completely and the original faces remain as smaller copies of themselves....

expand
Expansion (geometry)
In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements...

(cantellate)
Cantellation (geometry)
In geometry, a cantellation is an operation in any dimension that cuts a regular polytope at its edges and vertices, creating a new facet in place of each edge and vertex. The operation also applies to regular tilings and honeycombs...

bevel
(omnitruncate)
Omnitruncation (geometry)
In geometry, an omnitruncation is an operation applied to a regular polytope in a Wythoff construction that creates a maximum number of facets...

snub
Snub (geometry)
In geometry, an alternation is an operation on a polyhedron or tiling that removes alternate vertices. Only even-sided polyhedra can be alternated, for example the zonohedra. Every 2n-sided face becomes n-sided...

C
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...

aC = djC
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...

tC = dkdC
Truncated cube
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices....

tdC = dkC
Truncated octahedron
In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....

eC = aaC = doC
Rhombicuboctahedron
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles...

bC = taC = dmC = dkjC
Truncated cuboctahedron
In geometry, the truncated cuboctahedron is an Archimedean solid. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges...

sC = dgC
Snub cube
In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid.The snub cube has 38 faces, 6 of which are squares and the other 32 are equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron, that is, it has two distinct forms, which are mirror images of each...

dual
Dual polyhedron
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

join kis
(vertex-bisect)
ortho
(edge-bisect)
meta
(full-bisect)
gyro

dC
Octahedron
In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

jC = daC
Rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.-Properties:...

kdC = dtC
Triakis octahedron
In geometry, a triakis octahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube.It can be seen as an octahedron with triangular pyramids added to each face; that is, it is the Kleetope of the octahedron. It is also sometimes called a trisoctahedron, or, more...

kC = dtdC
Tetrakis hexahedron
In geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid. It can be seen as a cube with square pyramids covering each square face; that is, it is the Kleetope of the cube....

oC = deC = daaC
Deltoidal icositetrahedron
In geometry, a deltoidal icositetrahedron is a Catalan solid which looks a bit like an overinflated cube. Its dual polyhedron is the rhombicuboctahedron....

mC = dbC = kjC
Disdyakis dodecahedron
In geometry, a disdyakis dodecahedron, or hexakis octahedron, is a Catalan solid and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons...

gC = dsC
Pentagonal icositetrahedron
In geometry, a pentagonal icositetrahedron is a Catalan solid which is the dual of the snub cube. It has two distinct forms, which are mirror images of each other....

## Generating regular seeds

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 = aY3 (Rectified tetrahedron)
• C = daY3 (dual to rectified tetrahedron)
• I = sY3 (snub tetrahedron)
• D = dsY3 (dual to snub tetrahedron)
• Triangular antiprism
Antiprism
In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...

: A3
(An octahedron is a special antiprism)
• O = A3
• C = dA3
• Square prism
Prism (geometry)
In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...

: P4
(A cube is a special prism)
• C = P4
• Pentagonal antiprism
Pentagonal antiprism
In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for a total of 12 faces...

: A5
• I = k5A5 (A special gyroelongated dipyramid)
• D = t5dA5 (A special truncated trapezohedron
Truncated trapezohedron
An n-agonal truncated trapezohedron is a polyhedron formed by a n-agonal trapezohedron with n-agonal pyramids truncated from its two polar axis vertices....

)

## Extensions to Conway's symbols

The above operations allow all of the semiregular polyhedron
Semiregular polyhedron
The term semiregular polyhedron is used variously by different authors.In its original definition, it is a polyhedron with regular faces and a symmetry group which is transitive on its vertices, which is more commonly referred to today as a uniform polyhedron...

s and Catalan solid
Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....

s to be generated from regular polyhedron
Regular polyhedron
A regular polyhedron is a polyhedron whose faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e. it is transitive on its flags...

s. Combined many higher operations can be made, but many interesting higher order polyhedra require new operators to be constructed.

For example, geometric artist George W. Hart
George W. Hart
George William Hart is a geometer who expresses himself both artistically and academically. He is also a research professor in the department of computer science at the State University of New York in Stony Brook, New York....

created an operation he called a propellor, and another reflect to create mirror images of the rotated forms.
• p - "propellor" (A rotation operator that creates quadrilateral
In Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...

s at the vertices). This operation is self-dual: dpX=pdX.
• r - "reflect" - makes the mirror image of the seed; it has no effect unless the seed was made with s or p.

## Geometric coordinates of derived forms

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 torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

-shaped polyhedra can derive other polyhedra with point on the same torus surface.
 D tDTruncated dodecahedronIn geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.- Geometric relations :... aDIcosidodecahedronIn geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon... tdDTruncated icosahedronIn geometry, the truncated icosahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids whose faces are two or more types of regular polygons.It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges.... eDRhombicosidodecahedronIn geometry, the rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.... teDTruncated icosidodecahedronIn geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.... sDSnub dodecahedronIn geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.... dDIcosahedronIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids.... dteDDisdyakis triacontahedronIn geometry, a disdyakis triacontahedron, or hexakis icosahedron is a Catalan solid and the dual to the Archimedean truncated icosidodecahedron. As such it is face uniform but with irregular face polygons...

 H tHTruncated hexagonal tilingIn geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons and one triangle on each vertex.... aHTrihexagonal tilingIn geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex... tdH = H eH teH sHSnub hexagonal tilingIn geometry, the Snub hexagonal tiling is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex... dH dtHTriakis triangular tilingIn geometry, the Triakis triangular tiling is a tiling of the Euclidean plane. It is an equilateral triangular tiling with each triangle divided into three obtuse triangles from the center point.... daH dtdH = dH deHDeltoidal trihexagonal tilingIn geometry, the deltoidal trihexagonal tiling is a dual of the semiregular tiling.Conway calls it a tetrille.The edges of this tiling can be formed by the intersection overlay of the regular triangular tiling and a hexagonal tiling.- Dual tiling :... dteH dsHFloret pentagonal tilingIn geometry, the floret pentagonal tiling is a dual semiregular tiling of the Euclidean plane. It is one of 14 known isohedral pentagon tilings. It is given its name because its six pentagonal tiles radiate out from a central point, like petals on a flower...
 TTetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids... tTTruncated tetrahedronIn geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :... aTOctahedronIn geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.... tdTTruncated tetrahedronIn geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :... eTCuboctahedronIn geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,... bTTruncated octahedronIn geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron.... sTIcosahedronIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids.... dTTetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids... dtTTriakis tetrahedronIn geometry, a triakis tetrahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated tetrahedron.It can be seen as a tetrahedron with triangular pyramids added to each face; that is, it is the Kleetope of the tetrahedron... jTCubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... kTTriakis tetrahedronIn geometry, a triakis tetrahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated tetrahedron.It can be seen as a tetrahedron with triangular pyramids added to each face; that is, it is the Kleetope of the tetrahedron... oTRhombic dodecahedronIn geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.-Properties:... mT gT
Example: A hyperbolic heptagonal tiling seed
{7,3}
"seed"
truncate
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.- Uniform truncation :...

ambo
(rectify)
Rectification (geometry)
In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points...

bitruncate
Bitruncation
In geometry, a bitruncation is an operation on regular polytopes. It represents a truncation beyond rectification. The original edges are lost completely and the original faces remain as smaller copies of themselves....

expand
Expansion (geometry)
In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements...

(cantellate)
Cantellation (geometry)
In geometry, a cantellation is an operation in any dimension that cuts a regular polytope at its edges and vertices, creating a new facet in place of each edge and vertex. The operation also applies to regular tilings and honeycombs...

bevel
(omnitruncate)
Omnitruncation (geometry)
In geometry, an omnitruncation is an operation applied to a regular polytope in a Wythoff construction that creates a maximum number of facets...

snub
Snub (geometry)
In geometry, an alternation is an operation on a polyhedron or tiling that removes alternate vertices. Only even-sided polyhedra can be alternated, for example the zonohedra. Every 2n-sided face becomes n-sided...

dual
Dual polyhedron
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

join kis
(vertex-bisect)
ortho
(edge-bisect)
meta
(full-bisect)
gyro

## Other polyhedra

Iterating operators on simple forms can produce progressively larger polyhedra, maintaining the fundamental symmetry of the seed element. The vertices are assumed to be on the same spherical radius. Some generated forms can exist as spherical tilings, but fail to produce polyhedra with planar faces.

### Icosahedral symmetry

Rhombic:
Triangular:
Dual triangular:
Triangular chiral:
Dual triangular chiral:

• Uniform polyhedra
• Computer graphics
Computer graphics
Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....

algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

s:
• Doo–Sabin subdivision surface - expand operator
• Catmull–Clark subdivision surface - ortho operator