The Beauty of Polyhedra Helmer ASLAKSEN Department of Mathematics National University of Singapore

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Polyhedra Helmer ASLAKSEN Department of Mathematics

Presentation transcript:

The Beauty of Polyhedra Helmer ASLAKSEN Department of Mathematics National University of Singapore

What is a polyhedron? A surface consisting of polygons.

What is a polygon? Sides and corners. Regular polygon: Equal sides and equal angles. For n greater than 3, we need both.

How many sides? Where in Singapore is this? How many aisles?

A quick course in Greek TriTetraPentaHexaHepta OctaEnneaDecaDodecaIcosa

Polyhedra Vertices, edges and faces.

Platonic solids Euclid: Convex polyhedron with congruent, regular faces.

Properties of Platonic solids Faces (F) Edges (E) Vertices (V) Sides of face Faces at vertex Tet46433 Cub Oct Dod Ico Notice that V – E + F = 2 (Euler’s formula)

Duality Tetrahedron is self-dual Cube and octahedron Dodecahedron and icosahedron

Colouring the Platonic solids Octahedron: 2 colours Cube and icosahedron: 3 Tetrahedron and dodecahedron: 4

Euclid was wrong! Platonic solids: Convex polyhedra with congruent, regular faces and the same number of faces at each vertex. Freudenthal and Van der Waerden, 1947.

Deltahedra Polyhedra with congruent, regular, triangular faces. Cube and dodecahedron only with squares and regular pentagons.

Archimedean solids Regular faces of more than one type and congruent vertices.

Truncation Cuboctahedron and icosidodecahedron. A football is a truncated icosahedron!

The rest Rhombicuboctahedron and great rhombicuboctahedron Rhombicosidodecahedron and great rhombicosidodecahedron Snub cube and snub dodecahedron

Why rhombicuboctahedron? It can be inscribed in a cube, an octahedron and a rhombic dodecahedron (dual of the cuboctahedron)

Why snub? Left snub cube equals right snub octahedron. Left snub dodecahedron equals right snub icosahedron.

Why no snub tetrahedron? It’s the icosahedron!

The rest of the rest Prism and antiprism.

Are there any more? Miller’s solid or Sommerville’s solid. The vertices are congruent, but not equivalent!

Stellations of the dodecahedron The edge stellation of the icosahedron is a face stellation of the dodecahedron!

How to make models Paper Zome Polydron/Frameworks Jovo

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