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
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
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,. . . ”
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 A polytope in R n is a finite, convex region enclosed by a finite number of hyperplanes. We denote it by Π n .
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 A polytope in R n is a finite, convex region enclosed by a finite number of hyperplanes. We denote it by Π n . Examples n = 0 , 1 , 2 , 3 , 4 .
Introduction Two dimensions Three dimensions Schl¨ afli symbol Four dimensions Five and more dimensions Definition Regular polytope is a polytope Π n ( n ≥ 3 ) with 1 regular facets 2 regular vertex figures
Introduction Two dimensions Three dimensions Schl¨ afli symbol Four dimensions Five and more dimensions Definition Regular polytope is a polytope Π n ( n ≥ 3 ) with 1 regular facets 2 regular vertex figures We define all Π 0 and Π 1 to be regular. The regularity of Π 2 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 Π n ( n ≥ 3 ) with 1 regular facets 2 regular vertex figures We define all Π 0 and Π 1 to be regular. The regularity of Π 2 is understood in the usual sense. Vertex figure at vertex v is a Π n − 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 Π n ( n ≥ 3 ) with 1 regular facets 2 regular vertex figures We define all Π 0 and Π 1 to be regular. The regularity of Π 2 is understood in the usual sense. Vertex figure at vertex v is a Π n − 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 � 8 3 � 2 � 4 �
Introduction Two dimensions Three dimensions Schl¨ afli symbol Four dimensions Five and more dimensions Kepler-Poinsot solids � 5, 5 � 3, 5 2 � 2 � � 5 � 5 2 , 5 � 2 , 3 �
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 � � 5 � � 4 � � 6 � � 7 � � 8 � � 9 � � 10 �
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.
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. � π − 2 π � q < 2 π p
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. � π − 2 π � q < 2 π p 1 − 2 p < 2 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. � π − 2 π � q < 2 π p 1 − 2 p < 2 q 1 2 < 1 p + 1 q
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 2 < 1 p + 1 q
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 2 < 1 p + 1 q Solutions p = 3 p = 4 p = 5
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 2 < 1 p + 1 q Solutions p = 3 p = 4 p = 5 q = 3 , 4 , 5
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 2 < 1 p + 1 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 2 < 1 p + 1 q Solutions p = 3 p = 4 p = 5 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 { 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)
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 √ τ = 1 + 5 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) (0 , ± τ, ± 1 τ ) ( ± 1 τ , 0 , ± τ ) ( ± τ, ± 1 τ , 0) where √ τ = 1 + 5 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 � Dodecahedron Octahedron � 5, 3 � � 3, 4 �
Introduction Two dimensions Three dimensions Schl¨ afli symbol Four dimensions Five and more dimensions Schl¨ afli symbol � 6 � � 3, 4 �
Introduction Two dimensions Three dimensions Schl¨ afli symbol Four dimensions Five and more dimensions Schl¨ afli symbol � 6 � � 3, 4 � Desired properties of a Schl¨ afli symbol of a regular polytope Π n 1 Schl¨ afli symbol is an ordered set of n − 1 natural numbers
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