Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Regular Polytopes Laura Mancinska University of Waterloo, Department of C&O January 23, 2008 Definitions and examples Platonic solids Why only five? How to describe them? Regular polytopes in 4 dimensions Regular polytopes in higher dimensions Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Outline How many regular polytopes are there in n dimensions? Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Outline How many regular polytopes are there in n dimensions? Definitions and examples Platonic solids Why only five? How to describe them? Regular polytopes in 4 dimensions Regular polytopes in higher dimensions Definition n A polytope in R is a finite, convex region enclosed by a finite number of hyperplanes. We denote it by Πn. Examples n = 0, 1, 2, 3, 4. Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Polytope is the general term of the sequence “point, segment, polygon, polyhedron,. ” Examples n = 0, 1, 2, 3, 4. Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Polytope is the general term of the sequence “point, segment, polygon, polyhedron,. ” Definition n A polytope in R is a finite, convex region enclosed by a finite number of hyperplanes. We denote it by Πn. Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Polytope is the general term of the sequence “point, segment, polygon, polyhedron,. ” Definition n A polytope in R is a finite, convex region enclosed by a finite number of hyperplanes. We denote it by Πn. Examples n = 0, 1, 2, 3, 4. We define all Π0 and Π1 to be regular. The regularity of Π2 is understood in the usual sense. Vertex figure at vertex v is a Πn−1 obtained by joining the midpoints of adjacent edges incident to v. Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Definition Regular polytope is a polytope Πn (n ≥ 3) with 1 regular facets 2 regular vertex figures Vertex figure at vertex v is a Πn−1 obtained by joining the midpoints of adjacent edges incident to v. Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Definition Regular polytope is a polytope Πn (n ≥ 3) with 1 regular facets 2 regular vertex figures We define all Π0 and Π1 to be regular. The regularity of Π2 is understood in the usual sense. Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Definition Regular polytope is a polytope Πn (n ≥ 3) with 1 regular facets 2 regular vertex figures We define all Π0 and Π1 to be regular. The regularity of Π2 is understood in the usual sense. Vertex figure at vertex v is a Πn−1 obtained by joining the midpoints of adjacent edges incident to v. Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Definition Regular polytope is a polytope Πn (n ≥ 3) with 1 regular facets 2 regular vertex figures We define all Π0 and Π1 to be regular. The regularity of Π2 is understood in the usual sense. Vertex figure at vertex v is a Πn−1 obtained by joining the midpoints of adjacent edges incident to v. Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Star-polygons 5 7 7 2 2 3 9 = 9 = 9 = 8 9 9 3 2 4 9 = 9 = 9 = Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Kepler-Poinsot solids 5 5 5, 2 3, 2 9 = 9 = 5 5 2 , 5 2 , 3 9 = 9 = Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Two dimensional case In 2 dimensions there is an infinite number of regular polytopes (polygons). 3 4 5 6 8 < 8 < 8 < 8 < 7 8 9 10 8 < 8 < 8 < 8 < 2π π − q < 2π p 2 2 1 − < p q 1 1 1 < + 2 p q Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Necessary condition in 3D Polyhedron {p, q} Faces of polyhedron are polygons {p} Vertex figures are polygons {q}. Note that this means that exactly q faces meet at each vertex. 2 2 1 − < p q 1 1 1 < + 2 p q Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Necessary condition in 3D Polyhedron {p, q} Faces of polyhedron are polygons {p} Vertex figures are polygons {q}. Note that this means that exactly q faces meet at each vertex. 2π π − q < 2π p 1 1 1 < + 2 p q Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Necessary condition in 3D Polyhedron {p, q} Faces of polyhedron are polygons {p} Vertex figures are polygons {q}. Note that this means that exactly q faces meet at each vertex. 2π π − q < 2π p 2 2 1 − < p q Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Necessary condition in 3D Polyhedron {p, q} Faces of polyhedron are polygons {p} Vertex figures are polygons {q}. Note that this means that exactly q faces meet at each vertex. 2π π − q < 2π p 2 2 1 − < p q 1 1 1 < + 2 p q p = 3 p = 4 p = 5 q = 3, 4, 5 q = 3 q = 3 But do the corresponding polyhedrons really exist? Solutions Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Solutions of the inequality Inequality Faces are polygons {p} Exactly q faces meet at each vertex 1 1 1 < + 2 p q q = 3, 4, 5 q = 3 q = 3 But do the corresponding polyhedrons really exist? Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Solutions of the inequality Inequality Faces are polygons {p} Exactly q faces meet at each vertex 1 1 1 < + 2 p q Solutions p = 3 p = 4 p = 5 q = 3 q = 3 But do the corresponding polyhedrons really exist? Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Solutions of the inequality Inequality Faces are polygons {p} Exactly q faces meet at each vertex 1 1 1 < + 2 p q Solutions p = 3 p = 4 p = 5 q = 3, 4, 5 But do the corresponding polyhedrons really exist? Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Solutions of the inequality Inequality Faces are polygons {p} Exactly q faces meet at each vertex 1 1 1 < + 2 p q Solutions p = 3 p = 4 p = 5 q = 3, 4, 5 q = 3 q = 3 Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Solutions of the inequality Inequality Faces are polygons {p} Exactly q faces meet at each vertex 1 1 1 < + 2 p q Solutions p = 3 p = 4 p = 5 q = 3, 4, 5 q = 3 q = 3 But do the corresponding polyhedrons really exist? (±1, ±1, ±1) Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions {p, q} = {4, 3} Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Cube {p, q} = {4, 3} (±1, ±1, ±1) (±1, 0, 0) (0, ±1, 0) (0, 0, ±1) Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions {p, q} = {3, 4} Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Octahedron {p, q} = {3, 4} (±1, 0, 0) (0, ±1, 0) (0, 0, ±1) Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions {p, q} = {3, 3} Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Tetrahedron {p, q} = {3, 3} Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Tetrahedron {p, q} = {3, 3} (+1, +1, +1) (+1, −1, −1) (−1, +1, −1) (−1, −1, +1) Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions {p, q} = {3, 5} Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Icosahedron {p, q} = {3, 5} Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Icosahedron {p, q} = {3, 5} (0, ±τ, ±1) (±1, 0, ±τ) (±τ, ±1, 0) where √ 1 + 5 τ = 2 Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions {p, q} = {5, 3} Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Dodecahedron {p, q} = {5, 3} Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Dodecahedron {p, q} = {5, 3} (±1, ±1, ±1) 1 (0, ±τ, ± τ ) 1 (± τ , 0, ±τ) 1 (±τ, ± τ , 0) where √ 1 + 5 τ = 2 Introduction Two dimensions Three dimensions Schl¨afli symbol Four dimensions Five and more dimensions Five Platonic solids Cube Tetrahedron Icosahedron 4, 3 3, 3 3, 5 8 < 8 < 8 < Octahedron Dodecahedron 3, 4 5, 3 8 < 8 < Desired properties of a Schl¨afli symbol of a regular polytope Πn 1 Schl¨afli symbol is an ordered set of n − 1 natural numbers 2 If Πn has Schl¨afli symbol {k1, k2 . , kn−1}, then its Facets have Schl¨afli symbol {k1, k2 .