List of regular polytopes
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This page lists the regular polytopes in Euclidean, spherical and hyperbolic spaces.
The Schläfli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each.
The regular polytopes are grouped by dimension and subgrouped by convex, nonconvex and infinite forms. Nonconvex forms use the same vertices as the convex forms, but have intersecting facets. Infinite forms tessellate a one lower dimensional Euclidean space.
Infinite forms can be extended to tessellate a hyperbolic space. Hyperbolic space is like normal space at a small scale, but parallel lines diverge at a distance. This allows vertex figures to have negative angle defects, like making a vertex with 7 equilateral triangles and allowing it to lie flat. It can't be done in a regular plane, but can be at the right scale of a hyperbolic plane.
Contents |
Regular polytope summary count by dimension
| Dimension | Convex | Nonconvex | Convex Euclidean tessellations | Convex hyperbolic tessellations |
|---|---|---|---|---|
| 2 | ∞ polygons | ∞ star polygons | 1 | 1 |
| 3 | 5 Platonic solids | 4 Kepler-Poinsot solids | 3 tilings | ∞ |
| 4 | 6 convex polychora | 10 nonconvex polychora | 1 honeycomb | 4 |
| 5 | 3 convex 5-polytopes | 0 nonconvex 5-polytopes | 3 tessellations | 5 |
| 6+ | 3 | 0 | 1 | 0 |
Two-dimensional regular polytopes
The two dimensional polytopes are called polygons. Regular polygons are equilateral and cyclic.
Usually a regular polygon is considered convex, but star polygons, like the pentagram, can also be considered regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than once to complete.
Star polygons should be called nonconvex rather than concave because the intersecting edges do not generate new vertices and all the vertices exist on a circle boundary.
Convex forms (2D)
The Schläfli symbol {p} represents a regular p-agon:
The infinite set of convex regular polygons are:
| Name | Schläfli Symbol {p} |
|---|---|
| equilateral triangle | {3} |
| square | {4} |
| pentagon | {5} |
| hexagon | {6} |
| heptagon | {7} |
| octagon | {8} |
| enneagon | {9} |
| decagon | {10} |
| Hendecagon | {11} |
| Dodecagon | {12} |
| ...n-agon | {n} |
Nonconvex forms (2D)
There exist also non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers: star polygons.
In general, for any natural number n, there are n-pointed non-convex regular polygonal stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m}={n/(n-m)}) and m and n are coprime.
| Name | Schläfli Symbol {n/m} |
|---|---|
| pentagram | {5/2} |
| heptagrams | {7/2}, {7/3} |
| octagram | {8/3} |
| enneagrams | {9/2}, {9/4} |
| decagram | {10/3} |
| hendecagrams | {11/2} {11/3}, {11/4}, {11/5} |
| dodecagram | {12/5} |
| ...n-agrams | {n/m} |
| Image:Green pentagram.svg {5/2} | Image:Obtuse heptagram.ant.png {7/2} | Image:Acute heptagram.ant.png {7/3} | Image:Star polygon(8-3).png {8/3} |
Three-dimensional regular polytopes
In three dimensions, the regular polytopes are called polyhedra:
A regular polyhedron with Schläfli symbol {p,q} has a regular face type {p}, and regular vertex figure {q}.
A polyhedral vertex figure is an imaginary polygon can can be seen by connecting a polygon by the neighboring vertices to a given vertex. For regular polyhedra, this vertex figures is always a regular (and planar) polygon.
Existence of a regular polyhedron {p,q} is constrained by an inequality, related to the vertex figure's angle defect:
- 1/p + 1/q > 1/2 : Polyhedron (existing in Euclidean 3-space)
- 1/p + 1/q = 1/2 : Euclidean plane tiling
- 1/p + 1/q < 1/2 : Hyperbolic plane tiling
By enumerating the permutations, we find 6 convex forms, 10 nonconvex forms and 3 plane tilings, all with polygons {p} and {q} limited to: {3}, {4}, {5}, {5/2}, and {6}.
Beyond Euclidean space, there's an infinite set of regular hyperbolic tilings.
Convex forms (3D)
The convex regular polyhedra are called the 5 Platonic solids:
| Name | Schläfli Symbol {p,q} | Faces {p} | Edges | Vertices {q} | χ | Symmetry | dual |
|---|---|---|---|---|---|---|---|
| Tetrahedron | {3,3} | 4 {3} | 6 | 4 {3} | 2 | Td | Self-dual |
| Cube (hexahedron) | {4,3} | 6 {4} | 12 | 8 {3} | 2 | Oh | Octahedron |
| Octahedron | {3,4} | 8 {3} | 12 | 6 {4} | 2 | Oh | Cube |
| Dodecahedron | {5,3} | 12 {5} | 30 | 20 {3} | 2 | Ih | Icosahedron |
| Icosahedron | {3,5} | 20 {3} | 30 | 12 {5} | 2 | Ih | Dodecahedron |
| Image:Tetrahedron.jpg {3,3} | Image:Hexahedron.jpg {4,3} | Image:Octahedron.jpg {3,4} | Image:Dodecahedron.jpg {5,3} | Image:Icosahedron.jpg {3,5} |
Nonconvex forms (3D)
The nonconvex regular polyhedra are call the Kepler-Poinsot solids and there are four of them, based on the vertices of the dodecahedron {5,3} and icosahedron {3,5}:
| Name | Schläfli Symbol {p,q} | Faces {p} | Edges | Vertices {q} | χ | Symmetry | Dual |
|---|---|---|---|---|---|---|---|
| Small stellated dodecahedron | {5/2,5} | 12 {5/2} | 30 | 12 {5} | -6 | Ih | Great dodecahedron |
| Great dodecahedron | {5,5/2} | 12 {5} | 30 | 12 {5/2} | -6 | Ih | Small stellated dodecahedron |
| Great stellated dodecahedron | {5/2,3} | 12 {5/2} | 30 | 20 {3} | 2 | Ih | Great icosahedron |
| Great icosahedron | {3,5/2} | 20 {3} | 30 | 12 {5/2} | 2 | Ih | Great stellated dodecahedron |
Infinite forms (3D)
Tessellations of the plane are called tilings. There are three regular tilings:
| Name | Schläfli Symbol {p,q} | Face type {p} | Vertex figure {q} | χ | Symmetry | Dual |
|---|---|---|---|---|---|---|
| Square tiling | {4,4} | {4} | {4} | 0 | p4m | Self-dual |
| Triangular tiling | {3,6} | {3} | {6} | 0 | p6m | Hexagonal tiling |
| Hexagonal tiling | {6,3} | {6} | {3} | 0 | p6m | Triangular tiling |
| Image:Tile4444bc.gif {4,4} | Image:Tile333333bc.gif {3,6} | Image:Tile666bc.gif {6,3} |
Euclidean star-tilings
There are no plane tilings of star polygons. There are many enumerations that fit in the plane (1/p + 1/q = 1/2), like {8/3,8}, {10/3,5}, {5/2,10}, {12/5,12}, etc, but none repeat periodically.
Hyperbolic infinite forms (3D)
Tessellations of hyperbolic 2-space can be called hyperbolic tilings.
There are infinitely many regular hyperbolic tilings. As stated above, every positive integer pairs {p,q} such that 1/p + 1/q < 1/2 is a hyperbolic tiling.
A sampling:
| Name | Schläfli Symbol {p,q} | Face type {p} | Vertex figure {q} | χ | Symmetry | Dual |
|---|---|---|---|---|---|---|
| Order-5 square tiling | {4,5} | {4} | {5} | - | ? | {5,4} |
| Order-4 pentagonal tiling | {5,4} | {5} | {4} | - | ? | {4,5} |
| Order-7 triangular tiling | {3,7} | {3} | {7} | - | ? | {7,3} |
| Order-3 heptagonal tiling | {7,3} | {7} | {3} | - | ? | {3,7} |
| Order-6 square tiling | {4,6} | {4} | {6} | - | ? | {6,4} |
| Order-4 hexagonal tiling | {6,4} | {6} | {4} | - | ? | {4,6} |
| Order-5 pentagonal tiling | {5,5} | {5} | {5} | - | ? | Self-dual |
| Order-8 triangular tiling | {3,8} | {3} | {8} | - | ? | {8,3} |
| Order-3 octagonal tiling | {8,3} | {8} | {3} | - | ? | {3,8} |
| Order-7 square tiling | {4,7} | {4} | {7} | - | ? | {7,4} |
| Order-4 heptagonal tiling | {7,4} | {7} | {4} | - | ? | {4,7} |
| Order-6 pentagonal tiling | {5,6} | {5} | {6} | - | ? | {6,5} |
| Order-5 hexagonal tiling | {6,5} | {6} | {5} | - | ? | {5,6} |
| Order-9 triangle tiling | {3,9} | {3} | {9} | - | ? | {9,3} |
| Order-3 enneagonal tiling | {9,3} | {9} | {3} | - | ? | {3,9} |
| Order-8 square tiling | {4,8} | {4} | {8} | - | ? | {8,4} |
| Order-4 octagonal tiling | {8,4} | {8} | {4} | - | ? | {4,8} |
| Order-7 pentagonal tiling | {5,7} | {5} | {7} | - | ? | {7,5} |
| Order-5 heptagonal tiling | {7,5} | {7} | {5} | - | ? | {5,7} |
| Order-6 hexagonal tiling | {6,6} | {6} | {6} | - | ? | Self-dual |
There's a number of different ways to display the hyperbolic plane, including the Poincaré disc model below which maps the plane into a circle, as shown below. It should be recognized that all of the polygon faces in the tilings below are equal-sized and only appear to get smaller near the edges due to the projection applied, very similar to the effect of a camera fisheye lens.
| Image:Hyperspace tiling 4-5.png {4,5} | Image:Hyperspace tiling 5-4.png {5,4} | Image:Hyperbolic tiling 3-7.png {3,7} | Image:Hyperbolic tiling 7-3.png {7,3} |
Four-dimensional regular polytopes
Regular polychora with Schläfli symbol symbol {p,q,r} have cells of type {p,q}, faces of type {p}, edge figures {r}, and vertex figures {q,r}.
- A polychoral vertex figure is an imaginary polyhedron that can be seen by the arrangement of neighboring vertices around a given vertex. For regular polychora, this vertex figure is a regular polyhedron.
- A polychoral edge figure is an imaginary polygon that can be seen by the arrangement of faces around an edge. For regular polychora, this edge figure will always be a regular polygon.
The existence of a regular polychoron {p,q,r} is constrained by the existence of the regular polyhedra {p,q}, {q,r}.
Each will exist in a space dependent upon this expression:
- sin(π/p) sin(π/r) − cos(π/q)
- > 0 : Hyperspherical surface polychoron (in 4-space)
- = 0 : Euclidean 3-space honeycomb
- < 0 : Hyperbolic 3-space honeycomb
These constraints allow for 21 forms: 6 are convex, 10 are nonconvex, one is a Euclidean 3-space honeycomb, and 4 are hyperbolic honeycombs.
The Euler characteristic χ for polychora is χ = V + F − E − C and is zero for all forms.
Convex forms (4D)
The 6 convex polychora are as follows:
| Name | Schläfli Symbol {p,q,r} | Cells {p,q} | Faces {p} | Edges {r} | Vertices {q,r} | χ | Dual {r,q,p} |
|---|---|---|---|---|---|---|---|
| Pentachoron | {3,3,3} | 5 {3,3} | 10 {3} | 10 {3} | 5 {3,3} | 0 | Self-dual |
| Tesseract | {4,3,3} | 8 {4,3} | 24 {4} | 32 {3} | 16 {3,3} | 0 | 16-cell |
| 16-cell | {3,3,4} | 16 {3,3} | 32 {3} | 24 {4} | 8 {3,4} | 0 | Tesseract |
| 24-cell | {3,4,3} | 24 {3,4} | 96 {3} | 96 {3} | 24 {4,3} | 0 | Self-dual |
| 120-cell | {5,3,3} | 120 {5,3} | 720 {5} | 1200 {3} | 600 {3,3} | 0 | 600-cell |
| 600-cell | {3,3,5} | 600 {3,3} | 1200 {3} | 720 {5} | 120 {3,5} | 0 | 120-cell |
| Image:Cell5-4dpolytope.png {3,3,3} | Image:Hypercube star.png {4,3,3} | Image:Cell16-4dpolytope.png {3,3,4} | Image:Cell24-4dpolytope.png {3,4,3} | Image:Cell120-4dpolytope.gif {5,3,3} | Image:Cell600-4dpolytope.gif {3,3,5} |
Nonconvex forms (4D)
There are ten nonconvex regular polychora and their vertices are based on the convex 120-cell {5,3,3} and 600-cell {3,3,5}:
There are 4 failed potential nonconvex regular polychora permutations: {3,5/2,3}, {4,3,5/2}, {5/2,3,4}, {5/2,3,5/2}. Their cells and vertex figures exist, but they don't repeat periodically on the surface of a hypersphere.
| Name | Schläfli Symbol {p,q,r} | Cells {p,q} | Faces {p} | Edges {r} | Vertices {q,r} | χ | Dual {r,q,p} |
|---|---|---|---|---|---|---|---|
| Great grand stellated 120-cell | {5/2,3,3} | 120 {5/2,3} | 720 {5/2} | 1200 {3} | 600 {3,3} | 0 | Grand 600-cell |
| Grand 600-cell | {3,3,5/2} | 600 {3,3} | 1200 {3} | 720 {5/2} | 120 {3,5/2} | 0 | Great grand stellated 120-cell |
| Great stellated 120-cell | {5/2,3,5} | 120 {5/2,3} | 720 {5/2} | 720 {5} | 120 {3,5} | 0 | Grand 120-cell |
| Grand 120-cell | {5,3,5/2} | 120 {5,3} | 720 {5} | 720 {5/2} | 120 {3,5/2} | 0 | Great stellated 120-cell |
| Grand stellated 120-cell | {5/2,5,5/2} | 120 {5/2,5} | 720 {5/2} | 720 {5/2} | 120 {5,5/2} | 0 | Self-dual |
| Small stellated 120-cell | {5/2,5,3} | 120 {5/2,5} | 720 {5/2} | 1200 {3} | 120 {5,3} | -480 | Icosahedral 120-cell |
| Icosahedral 120-cell | {3,5,5/2} | 120 {3,5} | 1200 {3} | 720 {5/2} | 120 {5,5/2} | 480 | Small stellated 120-cell |
| Great icosahedral 120-cell | {3,5/2,5} | 120 {3,5/2} | 1200 {3} | 720 {5} | 120 {5/2,5} | 480 | Great grand 120-cell |
| Great grand 120-cell | {5,5/2,3} | 120 {5,5/2} | 720 {5} | 1200 {3} | 120 {5/2,3} | -480 | Great icosahedral 120-cell |
| Great 120-cell | {5,5/2,5} | 120 {5,5/2} | 720 {5} | 720 {5} | 120 {5/2,5} | 0 | Self-dual |
Infinite forms (4D)
Tessellations of 3-space are called honeycombs. There is only one regular honeycomb:
| Name | Schläfli Symbol {p,q,r} | Cell type {p,q} | Face type {p} | Edge figure {r} | Vertex figure {q,r} | χ | Dual |
|---|---|---|---|---|---|---|---|
| Cubic honeycomb | {4,3,4} | {4,3} | {4} | {4} | {3,4} | 0 | Self-dual |
| Image:Cubic honeycomb.png {4,3,4} |
Hyperbolic infinite forms (4D)
Tessellations of hyperbolic 3-space can be called hyperbolic honeycombs. There are 4 regular hyperbolic honeycombs:
| Name | Schläfli Symbol {p,q,r} | Cell type {p,q} | Face type {p} | Edge figure {r} | Vertex figure {q,r} | χ | Dual |
|---|---|---|---|---|---|---|---|
| Icosahedral honeycomb | {3,5,3} | {3,5} | {3} | {3} | {5,3} | 0 | Self-dual |
| Great cubic honeycomb | {4,3,5} | {4,3} | {4} | {5} | {3,5} | 0 | Small dodecahedral honeycomb {5,3,4} |
| Small dodecahedral honeycomb | {5,3,4} | {5,3} | {5} | {4} | {3,4} | 0 | Great cubic honeycomb {4,3,5} |
| Great dodecahedral honeycomb | {5,3,5} | {5,3} | {5} | {5} | {3,5} | 0 | Self-dual |
| Image:Hyperbolic orthogonal dodecahedral honeycomb.png {5,3,4} |
Five-dimensional regular polytopes
In five dimensions, a regular polytope can be named as {p,q,r,s} where {p,q,r} is the hypercell (or tetron) type, {p,q} is the cell type, {p} is the face type, and {s} is the face figure, {r,s} is the edge figure, and {q,r,s} is the vertex figure.
- A 5-polytopal vertex figure is an imaginary polychoron that can be seen by the arrangement of neighboring vertices to each vertex.
- A 5-polytopal edge figure is an imaginary polyhedron that can be seen by the arrangement of faces around each edge.
- A 5-polytopal face figure is an imaginary polygon that can be seen by the arrangement of cells around each face.
A regular polytope {p,q,r,s} exists only if {p,q,r} and {q,r,s} are regular polychora.
The space it fits in is based on the expression:
- (cos2(π/q)/sin2(π/p)) + (cos2(π/r)/sin2(π/s))
- < 1 : Spherical polytope
- = 1 : Euclidean 4-space tessellation
- > 1 : hyperbolic 4-space tessellation
Enumeration of these constraints produce 3 convex polytopes, zero nonconvex polytopes, 3 4-space tessellations, and 5 hyperbolic 4-space tessellations.
Convex forms (5D)
There are three kinds of convex regular polytopes in five dimensions:
| Name | Schläfli Symbol {p,q,r,s} | Hypercell type {p,q,r} | Cell type {p,q} | Face type {p} | Face figure {s} | Edge figure {r,s} | Vertex figure {q,r,s} | Dual |
|---|---|---|---|---|---|---|---|---|
| 5-simplex | {3,3,3,3} | {3,3,3} | {3,3} | {3} | {3} | {3,3} | {3,3,3} | Self-dual |
| measure 5-polytope | {4,3,3,3} | {4,3,3} | {4,3} | {4} | {3} | {3,3} | {3,3,3} | cross-5-polytope |
| cross-5-polytope | {3,3,3,4} | {3,3,3} | {3,3} | {3} | {4} | {3,4} | {3,3,4} | measure 5-polytope |
Nonconvex forms (5D)
There are no non-convex regular polytopes in five dimension.
Infinite forms (5D)
There are three kinds of infinite regular polytopes that can tessellate four dimensional space:
| Name | Schläfli Symbol {p,q,r,s} | Hypercell type {p,q,r} | Cell type {p,q} | Face type {p} | Face figure {s} | Edge figure {r,s} | Vertex figure {q,r,s} | Dual |
|---|---|---|---|---|---|---|---|---|
| Tesseract tessellation | {4,3,3,4} | {4,3,3} | {4,3} | {4} | {4} | {3,4} | {3,3,4} | Self-dual |
| 16-cell tessellation | {3,3,4,3} | {3,3,4} | {3,3} | {3} | {3} | {4,3} | {3,4,3} | 24-cell tessellation |
| 24-cell tessellation | {3,4,3,3} | {3,4,3} | {3,4} | {3} | {3} | {3,3} | {4,3,3} | 16-cell tessellation |
Hyperbolic infinite forms (5D)
There are five kinds of infinite regular polytopes that can tessellate four dimensional hyperbolic space:
| Name | Schläfli Symbol {p,q,r,s} | Hypercell type {p,q,r} | Cell type {p,q} | Face type {p} | Face figure {s} | Edge figure {r,s} | Vertex figure {q,r,s} | Dual |
|---|---|---|---|---|---|---|---|---|
| Pentachoron tessellation | {3,3,3,5} | {3,3,3} | {3,3} | {3} | {5} | {3,5} | {3,3,5} | {5,3,3,3} |
| Small 120-cell tessellation | {5,3,3,3} | {5,3,3} | {5,3} | {5} | {3} | {3,3} | {3,3,3} | {3,3,3,5} |
| Great tesseract tessellation | {4,3,3,5} | {4,3,3} | {4,3} | {4} | {5} | {3,5} | {3,3,5} | {5,3,3,4} |
| Great 120-cell tessellation | {5,3,3,4} | {5,3,3} | {5,3} | {5} | {4} | {3,4} | {3,3,4} | {4,3,3,5} |
| Great grand 120-cell tessellation | {5,3,3,5} | {5,3,3} | {5,3} | {5} | {5} | {3,5} | {3,3,5} | Self-dual |
Higher-dimensional regular polytopes
Convex forms (higher dimension)
In dimensions 5 and higher , there are only three kinds of convex regular polytopes.
| Name | Schläfli Symbol {p1,p2,...,pn-1} | Hypercell type | Vertex figure | Dual |
|---|---|---|---|---|
| n-simplex | {3,3,3,...,3} | {3,3,...,3} | {3,3,...,3} | Self-dual |
| measure n-polytope | {4,3,3,...,3} | {4,3,...,3} | {3,3,...,3} | cross-n-polytope |
| cross-n-polytope | {3,...,3,3,4} | {3,...,3,3} | {3,...,3,4} | measure n-polytope |
Nonconvex forms (higher dimension)
There are no non-convex regular polytopes in five dimension or higher.
Infinite forms (higher dimension)
There is only one infinite regular polytope that can tessellate five dimensions or higher, formed by measure polytopes.
| Name | Schläfli Symbol {p1, p2, ..., pn−1} | Hypercell type | Vertex figure | Dual |
|---|---|---|---|---|
| measure polytopes tessellation | {4,3,...,3,4} | {4,3,...,3} | {3,...,3,4} | Self-dual |
Hyperbolic infinite forms (higher dimension)
There are no regular tessellations of hyperbolic 5-space or higher.
External links
- The Platonic Solids
- Kepler-Poinsot Polyhedra
- 4d Polytope Applet
- Regular 4d Polytope Foldouts
- Multidimensional Glossary (Look up Hexacosichoron and Hecatonicosachoron)
- Polytope Viewer