1. SUMMARY I – Platonic solids II – A few definitions
III – Regular convex 4D polytopes

2. I – Introduction: regular 3D polyhedra (the platonic solids)
Regular polyhedron = a solid figure whose faces are equal regular polygons and whose vertices are « all the same »
i. e. a figure such that: every face is a p-gone
and every vertex is surrounded by q faces
Conversely, the numbers p et q uniquely define the polyhedron.
It is written {p q} : Schläfli symbol

3. Sum of angles around a vertex < 360°
p ≥ 3
q ≥ 3

4. Tetrahedron ({3 3}) 4 vertices 6 edges 4 faces Triangular pyramid
Octahedron ({3 4})
6 vertices
12 edges
8 faces
Square bipyramid
Triangular antiprism
Icosahedron ({3 5})
12 vertices
30 edges
20 faces
Pentagonal antiprism with two pentagonal pyramids attached

5. Cube ({4 3})
8 vertices
12 edges
6 faces
Square prism
Dodecahedron ({5 3})
20 vertices
30 edges
12 faces

6. Duality : vertices faces
edges edges
faces vertices
(hence dual's dual = itself)
regular regular
polyhedron polyhedron
{p q} {q p}

8. The tetrahedron is a « half-cube » != +
The tetrahedron is a « half-cube » ! + =
Stella octangula

10. II - Definitions
Polytope = generalisation of a polyhedron in n dimensions.
0-dimensional: point
1-dimensional: segment → bounded by two points
2-dimensional: polygon → bounded by several edges
3-dimensional: polyhedron → bounded by several faces
4-dimensional: polychoron → bounded by several cells

11. Symmetry = isometry of space that leaves the polytope invariant:
Regular polytope
Symmetry = isometry of space that leaves the polytope invariant:
Regular polytope:
- All elements of dimension n-1 are regular and equal.
- For any two vertices A and B, one can find a symmetry that sends A to B.

12. Essential property: it is inscriptible in a sphere
Projection of a polytope onto the sphere :

13. ? ? ?

14. Small stellated dodecahedron, {5/2 5}

15. Cuboctahedron

16. Section of the « Great prismosaurus »

17. Regular tiling of the plane, {6 3}

18. Regular tiling of the hyperbolic plane, {3 7}

19. Regular non planar octodecagone

20. Regular compound of five tetrahedra

21. III – (Convex) regular polychora
Polychoron = polytope en 4D
(point, segment, polygone, polyhedron, polychoron, ...)
Thus it is a 4D object bounded by vertices, edges, faces (which are polygons) and cells (which are polyhedra)
Regular polychoron: regular and equal cells, « identical » vertices -> hence « identical » edges and faces
i. e.
r polyhedra {p q} around every edge
-> written {p q r}, with p ≥ 3, q ≥ 3, r ≥ 3
Sum of angles around every edge is < 360°

22. Duality : vertices cells
edges faces
faces edges
cells vertices
A dual's dual is still itself
regular regular
polychoron polychoron
{p q r} {r q p}

23. For example, for a tetrahedron:
We want to know the possible values of r for every regular polyhedron. To do this, we need to calculate the angle between two of its faces.
For example, for a tetrahedron:
3 2 1
3 2
α=2arcsin ≈70°32′
and similarly for the other polyhedra.

24. Tetrahedron: α ≈ 70°32' → r = 3, 4 ou 5
Cube : α = 90° → r = 3
Octahedron: α ≈ 109°28' → r = 3
Dodecahed.: α ≈ 116°34' → r = 3
Icosahedron: α ≈ 138°19' → impossible

25. {3 3 3} : 5-cell, or hypertetrahedron= +
Tetrahedron + =
Triangle

26. + = 4+1 = 5 vertices 6+4 = 10 edges 4+6 = 10 triangular faces
1+4 = 5 tetrahedrical cells
Auto-dual
5-cell

27. {4 3 3} : Tesseract, or hypercube+ =
Cube = +
Square

28. + = 2x8 = 16 vertices 2x12 + 8 = 32 edges 2x6 + 12 = 24 square faces
2x1 + 6 = 8 cubical cells
Tesseract

29. = =

32. {3 3 4} : 16-cell, or hyperoctahedron= + +
Octahedron = + +
Square

33. + + = 6 +2x1 = 8 vertices 12+2x6 = 24 edges
8 +2x12 = 32 triangular faces
2x8 = 16 tetrahedrical cells
Dual of the tesseract
16-cell

34. =

35. The 16-cell is thus a « half-tesseract » !!!= = +
16-cell
The 16-cell is thus a « half-tesseract » !!!
16-cell

36. {3 4 3} : 24-cell
(1)
(2)
(3)
(1), (2) and (3) are 16-cells

37. (1)+(2) (1)+(3) (2)+(3) (1)+(2), (2)+(3) and (1)+(3) are tesseracts
(1), (2) and (3) play symmetrical roles

38. + = (1)+(2)+(3) : regular? Edges ? Faces? Cells ? 3D equivalent:
Rhombic dodecahedron

39. Face of a rhombic dodecahedron:
Cell of our polychoron:
Edges: edges of the tesseract
+ edges that connect it
to the 16-cell
NB : every edge connects (1) and (2), (1) and (3) or (2) and (3)

40. 3x8 = 24 vertices 32+8x8 = 96 edges 8x12 = 96 triangular faces
24-cell
3x8 = 24 vertices
32+8x8 = 96 edges
8x12 = 96 triangular faces
24 octahedrical cells
Auto-dual

41. {3 3 5} : 600-cell, or hypericosahedron
20 tetrahedra arranged around every vertex, like an icosahedron:
120 vertices
720 edges
1200 triangular faces
600 tetrahedrical cells

44. {5 3 3} : 120-cell, or hyperdodecahedron
600 vertices
1200 edges
720 pentagonal faces
120 dodecahedrical cells
Dual of the 600-cell

45. References Coxeter H. S. M., Regular polytopes
Abbott, Edwin Abbott, Flatland: A Romance of Many Dimensions
Uniform Polytopes in Four Dimensions (George Olshevsky): members.aol.com/Polycell/uniform.html
Lo Jacomo François, Visualiser la quatrième dimension
Wolfram Mathworld encyclopedia: mathworld.wolfram.com
Wikipedia : en.wikipedia.org

46. Acknowledgements
I'd like to thank Mr. Yves Duval, the organiser of the Mathematical Seminar of Louis-le-Grand Students, who encouraged and helped me prepare this talk.