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

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

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

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

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

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

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

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

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.

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

? ? ?

Small stellated dodecahedron, {5/2 5}

Cuboctahedron

Section of the « Great prismosaurus »

Regular tiling of the plane, {6 3}

Regular tiling of the hyperbolic plane, {3 7}

Regular non planar octodecagone

Regular compound of five tetrahedra

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°

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}

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 3 2 1 2 ≈70°32′ and similarly for the other polyhedra.

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

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

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

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

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

= =

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

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

=

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

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

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

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

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)

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

{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

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

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

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.