Regular Polytopes (Platonic solids) in 4D

Regular polytopes or platonic solids are convex solids (closed) where all the building blocks (vertices, edges, faces, hyperfaces) have the same characteristics. That is, vertices have the same number of neighbours, edges are all the same length, polygons are all the same shape and area, and hyperfaces have the same volume.

In 2 dimensions there an infinity of regular polytopes. For example, a circle can be approximated by polygons with 2 edges (line), 3 edges (triangle), 4 edges (square), 5 edges, 6 edges ..... indeed any number of edges. In 2 dimensions, every regular polytope is its own dual.

In 3 dimensions there are just 5, these are typically known as the platonic solids. In 3 dimensions, the tetrahedron is self-dual. The dual of the cube is the octahedron. And the dual of the dodecahedron is the icosahedron.

Regular polytopes in 4 Dimensions

In 4 dimensions there are 6 regular polytopes, this is the highest number that exist in any dimension greater than 2. They are listed and described in order of increasing cell numbers below. Each regular polytope is supplied as data in at least two versions, the first is a simple ascii format listing vertices, edges, and faces. The second is as an OFF file ready for GeomView.

Simplex

• Also known as pentatope or pentachoron
  
• Self-dual
  
• 4D equivalent of the tetrahedron
  
• 5 tetrahedral cells, 10 triangular faces, 10 edges, 5 vertices
  
• 3 tetrahedra meet at an edge
  
• It can be intuitively formed by choosing a point in one higher dimension equidistant to all the vertices in the current dimension and connecting this new point to all the current vertices.
  The sequence is: Point - Line - Triangle - Tetrahedron - Simplex
  
• Vertices, edges, and faces
  
• OFF version
  

Hypercube

• Also known as the Tesseract
  
• Dual with cross polytope
  
• 4D equivalent of the cube
  
• 8 cubic cells, 24 square faces, 32 edges, 16 vertices
  
• 3 cubes meet at an edge
  
• It can be intuitively formed by moving the profile in the current dimension in a direction perpendicular to that dimension by an edge length.
  The sequence is: Point - Line - Square - Cube - Hypercube
  
• Vertices, edges, and faces
  
• OFF version
  

Cross Polytope

• Also known as 16 cell or hexadecachoron
  
• Dual with hypercube
  
• 4D equivalent of the octahedron
  
• 16 tetrahedral cells, 32 triangular faces, 24 edges, 8 vertices
  
• 4 tetrahedra meet at an edge
  
• The generation sequence is: Point - Line - Square - Octahedron - Crosspolytope
  
• Vertices, edges, and faces
  
• OFF version
  

24 cell

• Also known as the icositetrachoron
  
• Self-dual
  
• No equivalent in other dimensions
  
• 24 octahedral cells, 96 triangular faces, 96 edges, 24 vertices
  
• 3 octahedra meet at an edge
  
• Vertices, edges, and faces
  
• OFF version
  

120 Cell

• Also known as the hecatonicosachoron
  
• Dual with 600 cell
  
• 4D equivalent of the dodecahedron
  
• 120 dodecahedral cells, 720 five sided faces, 1200 edges, 600 vertices
  
• 3 dodecahedra meeting per edge
  
• Vertices, edges, and faces
  
• OFF version
  

600 Cell

• Also known as the hexacosichoron
  
• Dual with 120 cell
  
• 4D equivalent to the icosahedron
  
• 600 tetrahedral cells, 1200 triangular faces, 720 edges, 120 vertices
  
• 5 tetrahedra meeting per edge
  
• Vertices, edges, and faces
  
• OFF version
  

Higher Dimensions

In higher dimensions (5, 6, 7 ....) there are only 3 regular polytopes in any particular dimensions! These 3 regular polytopes are the equivalent of the tetrahedron, cube, and octahedron in 3 dimensions, they are normally called the n-simplex, n-cube, and n-crosspolytope respectively where n stands for the dimension.

• Also known as the hypertetrahedron
  
• Self-dual
  
• n + 1 cells each of which is an (n-1) simplex
  
• n + 1 vertices
  
• n (n + 1) / 2 edges
  

n-cube

• Also known as the hypercube
  
• Dual with n-crosspolytope
  
• 2 n cells each of which is an (n-1) cube
  
• 2 ⁿ vertices
  
• n 2 ^{n - 1} edges
  
• 2 n (n - 1), (n-2) squares
  

n-crosspolytope

• Also known as the hyperoctahedron
  
• Dual with n-cube
  
• 2 ⁿ cells each of which is an (n-1) simplex
  
• 2 n vertices
  
• 2 n (n - 1) edges
  
• n 2 ^{n - 1}, (n-2) crosspolytopes
  

Definitions

Dual
The `dual' of a regular polytope is another polytope, also regular, having one vertex in the center of each cell of the polytope we started with. The dual of the dual of a regular polytope is the one we started with (only smaller).

Self-dual
It is possible for a regular polytope to be it's own dual, for example, all the regular polytopes in 2 dimensions. In 3 dimensions the tetrahedron is self-dual. Obviously for a polytope to be self-dual it must have the same number of cells as vertices.

Polyhedral formula
Number of vertices - number of edges + number of faces - number of cells = 0

Vertex/Face ascii format
These files are formatted as follows, hopefully the description below along with an example from above will give all the information needed to translate the geometry into other formats. For more information see VEF format

• The first line contains the number of vertices
  
• The next lines (one for each vertex) contain 4 numbers, each consists of the 4 floating point numbers being the w,x,y,z coordinate of that vertex. The line number (counting from 0) is the vertex ID.
  
• The next line contains the number of edges
  
• The subsequent lines describe each edge, each consists of two vertex IDs making up that edge.
  
• The next line contain the number of faces
  
• The subsequent lines describe each face. The first number of each line is the number of vertices in the face. The rest of the line contains a list of vertex IDs making up that face.

References

Experiments in 4 Dimensions Heiserman, David L.
TAB Books Inc

Introduction to Geometry Coxeter, H.S.M.
John Wiley & Sons Inc

4 Dimensional Space Eckhart, L.
Indiana University Press