the regular polytopes

The regular polytopes are the analogues, in dimension higher than three, of the regular polyhedra in dimension three and of the regular polygons in dimension two. A regular polyhedron is an assembling of regular polygons, the faces, neighboring faces having an edge in common. Likewise the regular 4D-polytopes, described by Schläfli, are assemblies of regular polyhedra, the cells, neighboring cells having a face in common.

Here are the characteristics of the six convex regular 4D-polytopes (there are also ten stellated).
With at least three cells around each edge there are only six possibilities: 3, 4 or 5 tetrahedra, 3 cubes, 3 octahedra and 3 dodecahedra.
They verify the Euler formula found by Schläfli:  V + F = E + C

The simplex and the 24-cell are their own duals. The hypercube (also called tesseract) and the 16-cell (cross polytope) are dual, the 120-cell and the 600-cell too.
name
simplex (5-cell)
hypercube (8-cell)
cross polytope (16-cell)
24-cell
120-cell
600-cell 		cells
5 = d+1  tetrahedra
8 = 2d  cubes
16 = 2d  tetrahedra
24  octahedra
120  dodecahedra
600  tetrahedra 		faces
10
24
32
96
720
1200 		edges
10
32
24
96
1200
720 		vertices
5
16
8
24
600
120

But how to represent a polytope?  In the same manner as we do for a polyhedron: by projection on an hyperplane, what reduces the dimension by one unit; we get a representation which don't preserves all the regularity (cf perspective).
Here are representations (Schlegel diagrams) and nets of the four simplest 4D-polytopes. Don't forget that in dimension four they are regular, in particular all their edges have same length and those of the hypercube are mutually perpendicular. For the last two it's much more complicated.


The 120-cell - equivalent of the regular dodecahedron in 3D - is limited by 120 cells (regular dodecahedra), four per vertex, three per edge and two per face. This structure is an approximation of an hypersphere of 4D (just like the 12 faces of the regular dodecahedron build an approximation of a sphere in 3D).
The puzzle on the right is a an exploded view of a projection in 3D of the 120-cell; it is made up of 1+12+20+12=45 dodecahedra: only the first - at the center - is regular, the others are more or less flattened (distortions due to the projection from 4D on a 3D-hyperplane) and laid-out in three successive layers.
The projections of the 30 cells orthogonal to the projection's hyperplane are the hexagonal faces (flattened dodecahedra) of the chamfered dodecahedron which contains the assembling of the 45 pieces.

More precisely, the blue dodecahedron is the projection of the nearest cell (a pole of the hypersphere); the three layers (green, brown and orange dodecahedra) correspond to cells situated on a same parallel of the hypersphere; the dodecahedra flattened into grey hexagons correspond to the cells situated on the equator.
Two cells symmetric with respect to the equator have same projection.

Want to build your own (paper models or 3D printing)?

Remarks:
A 4D-parallel (4D-circle) is a sphere in 3D, thus the spherical layers in the projection.
How can a dodecahedron be flattened into a hexagon? To visualize it rotate the small dodecahedron with the mouse to superimpose two opposite edges at the center of the diagram: 4 faces are reduced to segments, the 8 others are superimposed by pairs.

Thanks to Arnaud Chéritat who kindly sent me the data for this animation
and to Nicolas Hannachi who transformed them with Mathematica to make them usable with LiveGraphics3D.

Here are videos (found on YouTube) to better understand the structure of the hypercube and the 120-cell:
    • the tesseract  (hypercube) by Vladimir Panfilov (1'38 - 4.3 Mb)  -  construct the hypercube  (Delta Simplex - 5'12 - 16 Mb)
    • the 120-cell  (hyperdodecahedron) by Gian Marco Todesco (2'15 - 9 Mb)  -  construct the 120-cell  (Delta Simplex - 4'20 - 13 Mb)
The 120-cell may be considered as 12 interlinked rings of 10 dodecahedra; here are two animations (rotations in 4D) made by Roice Nelson using "120 Cell Explorer" (see references) which show 6 of these rings, that is 60 of the 120 dodecahedra:
    • the first (0'14 - 0,9 Mo) shows the richness of the 120-cell's symmetries
    • the second (0'15 - 0,8 Mo) presents a different point of view.

There are only three regular convex polytopes - with respectively d+1, 2d and 2d hyperfaces (the three first rows of the table) - in each dimension higher than four; the first, corresponding to the tetrahedron, is self dual, the two others, corresponding to the cube and the octahedron, are dual.
Remark: the characteristics of the nD-hypercube can be deduced from the coefficients of the polynomial  (1+2x)n , for n=4   (1+2x)4 = 1 + 8x + 24x2 + 32x3 + 16x4
In this video (YouTube - 7'20) Oliver Knill presents the six 4-polytopes and the three 5-polytopes.

references: •  To understand and visualize the 4th dimension view the video-animations proposed by dimensions-math (2 hours of beautiful mathematics!); there you'll also learn - with an elementary proof - what a stereographic projection is.
•  Two other approaches: 4D Visualization  and La quatrième dimension  (Micmaths on YouTube, in French), a set of four videos (définition - représentation - curiosités - hypercube) by Mickaël Launay
•  The 4D-polytopes on Wikipedia (French version)
•  To play with 4D-polytopes - at least with their projections in 3D - you need Stella4D, the outstanding program by Robert Webb.
•  The Tesseract  by Alex Bogomolny (with interactive applets)
•  Le 120  (hyperdodecahedron) by Arnaud Chéritat (in French) - build your own (paper models or 3D printing)
•  "120 Cell Explorer" allows you to discover and handel the 120-cell
•  Die Platonischen Polychora  by Marco Möller (in German)
•  Regular 4d Polytope Foldouts  by Andrew Weimholt
•  Images for Mathematics: 4D geometry (with animations)
•  Barn Raisings of Four-Dimensional Polytope Projections  by George W. Hart
•  FLATLAND : geometric fiction from Edwin A. Abbott (1884) which presents a two dimension world (it inspired at least two animation movies)

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French 		convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects october 1999
updated 31-07-2022