This page lists the regular polytopes in Euclidean space. A dodecahedron, one of the five Platonic solids. ... In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ...

The Schläfli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each. In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ...

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. In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. ... In geometry, the defect of a vertex of a polyhedron is the amount by which the sum of the angles at the vertex falls short of a full circle. ...

Regular Polytope summary count by dimension

This page lists the regular polytopes in Euclidean space. ... This page lists the regular polytopes in Euclidean space. ... This page lists the regular polytopes in Euclidean space. ... This page lists the regular polytopes in Euclidean space. ... This page lists the regular polytopes in Euclidean space. ... This page lists the regular polytopes in Euclidean space. ... This page lists the regular polytopes in Euclidean space. ... This page lists the regular polytopes in Euclidean space. ... This page lists the regular polytopes in Euclidean space. ... This page lists the regular polytopes in Euclidean space. ... This page lists the regular polytopes in Euclidean space. ...

Two dimensional regular polytopes

The two dimensional polytopes are called polygons. Regular polygons are equilateral and cyclic. Look up Polygon in Wiktionary, the free dictionary For other use please see Polygon (disambiguation) A polygon (literally many angle, see Wiktionary for the etymology) is a closed planar path composed of a finite number of sequential line segments. ... In geometry, an equilateral polygon has all sides of the same length. ... In geometry, a circumcircle of a given two-dimensional geometric shape is the smallest circle which contains the shape completely within it. ...

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. In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ... In geometry, a star polygon is a complex, regular polygon, so named for its starlike appearance, created by extending lines in a regular pattern from one vertex of a simple, regular, n-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. ... A pentagram, pentacle, pentalpha, or pentangle A pentagram is a five-pointed star drawn with five straight strokes. ...

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: In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ... A polygon (from the Greek poly, for many, and gonos, for angle) is a closed planar path composed of a finite number of sequential straight line segments. ...

{3}

{4}

{5}

{6}

{7}

{8}

{9}

{10}

In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... In plane geometry, a square is a polygon with four equal sides and equal angles. ... In geometry, a pentagon is any five-sided polygon. ... A regular hexagon A hexagon is a polygon with six edges and six vertices. ... A heptagon is a plane figure with seven sides and seven angles. ... One of the 8 semi-regular tessellations: octagons and squares An octagon is a polygon that has eight sides. ... In geometry, an enneagon or nonagon is a nine-sided polygon. ... An image of a Regular Decagon In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular decagon, having all sides of equal length and all angles equal to 144°. Its Schläfli symbol is {10}. The area of a regular decagon... Look up Polygon in Wiktionary, the free dictionary For other use please see Polygon (disambiguation) A polygon (literally many angle, see Wiktionary for the etymology) is a closed planar path composed of a finite number of sequential line segments. ... File links The following pages link to this file: Triangle Categories: GFDL images ... http://fr. ... Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ... an heptagon This image is ineligible for copyright and therefore in the public domain, because it consists entirely of information that is common property and contains no original authorship. ... Image File history File links Octagon. ... Image File history File links Nonagon. ... Image File history File links Decagon. ...

Nonconvex forms (2D)

There exist also non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers: star polygons. In geometry, a star polygon is a complex, regular polygon, so named for its starlike appearance, created by extending lines in a regular pattern from one vertex of a simple, regular, n-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. ...

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. Coprime - Wikipedia /**/ @import /skins-1. ...

{5/2}

{7/2}

{7/3}

{8/3}

In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... A pentagram, pentacle, pentalpha, or pentangle A pentagram is a five-pointed star drawn with five straight strokes. ... Acute heptagram Obtuse heptagram Acute and obtuse heptagrams inscribed within a heptagon. ... In geometry, a star polygon is a complex, equilateral equiangular polygon, so named for its starlike appearance, created by connecting one vertex of a simple, regular, n-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. ... The Enneagram Figure The Enneagram (or Enneagon) is a nine-pointed diametric figure which is used to indicate the dynamic ways that aspects of things and processes are connected and change. ... The decagram (symbol dag, sometimes dcg) is an SI unit of mass. ... A hendeceagram is a star polygon that has eleven points. ... In geometry, a star polygon is a complex, regular polygon, so named for its starlike appearance, created by extending lines in a regular pattern from one vertex of a simple, regular, n-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. ... Wikipedia does not have an article with this exact name. ... Image File history File links File links The following pages link to this file: Heptagram ... Image File history File links File links The following pages link to this file: Heptagram ... Image File history File links Sketch of construction of a {8/3} star polygon, sketched by me. ...

Three dimensional regular polytopes

In three dimensions, the regular polytopes are called polyhedra: This article is about the geometric shape. ...

A regular polyhedron with Schläfli symbol {p,q} has a regular face type {p}, and regular vertex figure {q}. In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ...

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 contrained by an inequality, related to the vertex figure's angle defect: In geometry, the defect of a vertex of a polyhedron is the amount by which the sum of the angles at the vertex falls short of a full circle. ...

• 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}. // Mathematics In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ...

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: This article is about the geometric shape. ... In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with all its faces being congruent regular polygons, and the same number of faces meeting at each of its vertices. ...

{3,3}

{4,3}

{3,4}

{5,3}

{3,5}

In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ... The symmetry group of an object (e. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... A cube (or regular hexahedron) is a three-dimensional Platonic solid composed of six square faces, with three meeting at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... An icosahedron [ˌaıkəsəhiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant. ... Download high resolution version (643x607, 26 KB)Tetrahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (742x826, 50 KB)Hexahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (862x862, 41 KB)Octahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (847x829, 63 KB)Dodecahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (819x791, 71 KB)Icosahedron, made by me using POV-Ray, see image:poly. ...

Nonconvex forms (3D)

The nonconvex regular polyhedra are call the Kepler-Poinsot solids and there are 4 of them, based on the vertices of the dodecahedron {5,3} and icosahedron {3,5}: A Kepler solid (also called Kepler-Poinsot solid) is a regular non-convex polyhedron, all the faces of which are identical regular polygons and which has the same number of faces meeting at all its vertices (compare to Platonic solids). ... A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... An icosahedron [ˌaıkəsəhiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant. ...

Infinite forms (3D)

Tessellations of the plane are called tilings. There are 3 regular tilings: In geometry, a tiling (also called tessellation, mosaic or dissection) of a given shape S consists of a collection of other shapes which precisely cover S. Often the shape S to be tiled is the Euclidean plane, but other shapes and three-dimensional objects are considered as well. ...

{4,4}

{3,6}

{6,3}

In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ... The symmetry group of an object (e. ... In geometry, the Square tiling is a regular tiling of the Euclidean plane. ... In geometry, the triangular tiling is a regular tiling of the Euclidean plane. ... In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. ... Image File history File links Tile4444bc. ... Image File history File links Tile333333bc. ... Image File history File links Tile666bc. ...

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. In geometry, a star polygon is a complex, regular polygon, so named for its starlike appearance, created by extending lines in a regular pattern from one vertex of a simple, regular, n-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. ...

Hyperbolic infinite forms (3D)

Tessellations of hyperbolic 2-space can be called hyperbolic tilings. In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. ...

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:

In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ... The symmetry group of an object (e. ...

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}. In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora) (from Greek poly meaning many and choros meaning room or space), 4-polytope, or polyhedroid. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ...

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 a regular polychora, this edge figure will always be a regular polygon.

The existence of a regular polychoron {p,q,r} is contrained by the existence of the regular polyhedra {p,q}, {q,r}.

It 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. In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ...

Convex Forms (4D)

The 6 convex polychora are as follows: In mathematics, a convex regular 4-polytope (or polychoron) is 4-dimensional polytope which is both a regular and convex. ...

{3,3,3}

{4,3,3}

{3,3,4}

{3,4,3}

{5,3,3}

{3,3,5}

In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ... The pentachoron, also called a pentatope or 4-simplex, is the simplest convex regular polychoron (a type of four-dimensional geometric figure). ... In geometry, the tesseract is the 4-dimensional analog of the cube. ... (Redirected from 16-cell) In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ... (Redirected from 16-cell) In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ... In geometry, the tesseract is the 4-dimensional analog of the cube. ... In geometry, the 24-cell (or icositetrachoron) is the convex regular 4-polytope with Schläfli symbol {3,4,3}. The 24-cell is the unique convex regular 4-polytope without a good 3-dimensional analog. ... In geometry, the 120-cell (or hecatonicosachoron) is the convex regular 4-polytope with Schläfli symbol {5,3,3}. It is sometimes thought of as the 4-dimensional analog of the dodecahedron. ... In mathematics, the 600-cell is the 4-dimensional convex regular polytope with 600 facets. ... In mathematics, the 600-cell is the 4-dimensional convex regular polytope with 600 facets. ... In geometry, the 120-cell (or hecatonicosachoron) is the convex regular 4-polytope with Schläfli symbol {5,3,3}. It is sometimes thought of as the 4-dimensional analog of the dodecahedron. ... Image File history File links Download high resolution version (640x640, 38 KB)Four symmetry views of the 5-cell polytope, edges drawn with an orthographic projection (z-w axes ignored). ... The star-shaped 4D-hypercube. ... Image File history File links Download high resolution version (641x625, 56 KB)Four symmetry views of the 16-cell polytope, edges drawn with an orthographic projection (z-w axes ignored). ... Image File history File links Download high resolution version (675x656, 13 KB)An image of the 24-cell polytope, edges drawn with an orthographic projection (z-w axes ignored). ... Image File history File links Download high resolution version (640x604, 61 KB) Summary A good image of the 120-cell polytope, edges drawn with an orthographic projection (z-w axes ignored). ... Image File history File links Download high resolution version (640x633, 69 KB)An image of the 600-cell polytope, edges drawn with an orthographic projection (z-w axes ignored). ...

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}: In geometry, the 120-cell (or hecatonicosachoron) is the convex regular 4-polytope with Schläfli symbol {5,3,3}. It is sometimes thought of as the 4-dimensional analog of the dodecahedron. ... In mathematics, the 600-cell is the 4-dimensional convex regular polytope with 600 facets. ...

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.

In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ...

Infinite forms (4D)

Tessellations of 3-space are called honeycombs. There is only one regular honeycomb:

{4,3,4}

In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... The cubic honeycomb is the only regular tessellation (or honeycomb) in Euclidean 3-space. ... Image File history File links Download high resolution version (902x902, 1221 KB) Summary View of cubic honeycomb generated by software: [1] Curved Spaces v1. ...

Hyperbolic infinite forms (4D)

Tessellations of hyperbolic 3-space can be called hyperbolic honeycombs. There are 4 regular hyperbolic honeycombs: In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. ...

In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... The hyperbolic small dodecahedral honeycomb is one of four regular tessellation (or honeycomb) in hyperbolic 3-space. ... Image File history File links Download high resolution version (902x902, 1341 KB) Summary View of Hyperbolic small dodecahedral honeycomb generated by software: [1] Curved Spaces v1. ...

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 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

Eumeration of these contraints 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: In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ...

In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. ... In geometry, a measure polytope is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). ... In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ... In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ... In geometry, a measure polytope is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). ...

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:

In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... In geometry, the tesseract is the 4-dimensional analog of the cube. ... (Redirected from 16-cell) In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ... In geometry, the 24-cell (or icositetrachoron) is the convex regular 4-polytope with Schläfli symbol {3,4,3}. The 24-cell is the unique convex regular 4-polytope without a good 3-dimensional analog. ... In geometry, the 24-cell (or icositetrachoron) is the convex regular 4-polytope with Schläfli symbol {3,4,3}. The 24-cell is the unique convex regular 4-polytope without a good 3-dimensional analog. ... (Redirected from 16-cell) In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ...

Hyperbolic infinite forms (5D)

There are five kinds of infinite regular polytopes that can tessellate four dimensional hyperbolic space:

In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ...

Higher dimensional regular polytopes

Convex forms (higher dimension)

In dimensions 5 and higher , there are only three kinds of convex regular polytopes. In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ...

In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. ... In geometry, a measure polytope is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). ... In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ...

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. In geometry, a measure polytope is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). ...

In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... In geometry, a measure polytope is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). ...

Hyperbolic infinite forms (higher dimension)

There are no hyperbolic tessellations in 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