# Regular Polyhedra

This page shows all the regular polyhedra that can be drawn on the genus-0 orientable manifold, the sphere. For the purpose of these pages, a "regular polyhedron" is regarded as a regular map on the sphere.

This definition gives five regular polyhedra, also known as the five platonic solids; and some other less interesting things. There are many better web pages about the five platonic solids, for instance on Wikipedia, on Wolfram, and a page which lets you rotate them.

nameSchläfli
symbol
pictureV
F
E
Eu
dual

Petrie dual

rotational
symmetry
group
Antipodes, etc.Commentsqy
tetrahedron{3,3} 4
4
6
2
self-dual

hemicube

A4Each face is antipodal to a vertex and vice versa; and each edge to an edge.

Cantellation of the tetrahedron yields the octahedron.

These are the five "Platonic solids". 2
cube{4,3} 8
6
12
2
octahedron

{6,3}(2,2}

S4Each face is antipodal to one face, each vertex to one vertex, and each edge to one edge.3
octahedron{3,4} 6
8
12
2
cube

C4{6,4}

S4Each face is antipodal to one face, each vertex to one vertex, and each edge to one edge.3
dodecahedron{5,3} 20
12
30
2
icosahedron

C6{10,3}5

A5Each face is antipodal to one face, each vertex to one vertex, and each edge to one edge.6
icosahedron{3,5} 12
20
30
2
dodecahedron

C14{10,5}

A5Each face is antipodal to one face, each vertex to one vertex, and each edge to one edge.6
The things below are not normally considered to be polyhedra. However they fit my definition.
n-hosohedron
(heptahosohedron, etc.)
{2,n} 2
n
n
2
di-n-hedron

n=3: S1:{6,3}(1,1)
n=4: S1:{4,4}(1,1)
n=5: S2:{10,5}
n=6: S2:{6,6}
n=7: S3:{14,7}
n=8: S3:{8,8}2

D2n n odd:   Each face is antipodal to one edge and vice versa; each vertex is antipodal to the other vertex.

n even:   Each face is antipodal to one face, each edge to one edge, and each vertex to the other vertex. The image uses 7 as an example value for n.
1
di-n-hedron
(diheptagon, etc.)
{n,2} n
2
n
2
n-hosohedron

n odd: C1:{2n,2}
n even: self-Petrie dual

D2n n odd:   Each vertex is antipodal to one edge and vice versa; each face is antipodal to the other face.

n even:   Each vertex is antipodal to one vertex, each edge to one edge, and each face to the other face. The image uses 7 as an example value for n.
1
2-hosohedron,
digonal dihedron
{2,2} 2
2
2
2
self-dual

self-Petrie dual

C2×C2 Each face is antipodal to the other face, each vertex to the other vertex, and each edge to the other edge.

Cantellation of the 2-hosohedron yields the 4-hosohedron.  1
monodigon
{2,1} 2
1
1
2
dimonogon

self-Petrie dual

C2 Each vertex is antipodal to the other vertex; the face is antipodal to the edge and vice versa.   ½
dimonogon
{1,2} 1
2
1
2
monodigon

hemi-digonal hosohedron

C2 Each face is antipodal to the other face; the vertex is antipodal to the edge and vice versa.   ½
edgeless map
{0,0} 1
1
0
2
self-dual

self-Petrie dual

C1 The face is antipodal to the vertex and vice versa. There is no edge.

Our definition of a regular map requires that each face must have the topology of a disk. So this is a valid regular map; but in all other manifolds there must be at least one edge.  0

Our definition of a regular map requires each face to have the topology of a disk. This cannot be achieved by embedding the empty set in any compact manifold, so every regular map must have at least one vertex.

Index to other pages on regular maps;
indexes to those on S0 C1 S1 S2 C4 C5 S3 C6 S4.
Some pages on groups Copyright N.S.Wedd 2009,2010