Polyhedra Helmer ASLAKSEN Department of Mathematics National University

Polyhedra Helmer ASLAKSEN Department of Mathematics National University of Singapore aslaksen@math. nus. edu. sg www. math. nus. edu. sg/aslaksen/polyhedra/

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

A quick course in Greek 3 4 5 6 7 Tri Tetra Penta Hexa Hepta 8 9 10 12 20 Octa Ennea Deca Dodeca Icosa

More about polygons n The vertex angle in a regular n-gon is 180 (n-2)/n. To see this, divide the polygon into n triangles. n 3: 60 n 4: 90 n 5: 108 n 6: 120

Polyhedra n What is a polyhedron? n Platonic solids n Deltahedra n Archimedean solids n Colouring Platonic solids n Stellation

What is a polyhedron? n Solid or surface? n A surface consisting of polygons.

Polyhedra n Vertices, edges and faces.

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

Properties of Platonic solids Faces Edges Vertices Sides Faces at of face vertex Tet 4 6 4 3 3 Cub 6 12 8 4 3 Oct 8 12 6 3 4 Dod 12 30 20 5 3 Ico 30 12 3 5 20

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

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

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

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

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

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

Why rhombicuboctahedron?

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

Why no snub tetrahedron? n It’s the icosahedron!

The rest of the rest n Prism and antiprism.

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

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

Nested Platonic Solids

How to make models n Paper n Zome n Polydron/Frameworks n Jovo

Web n http: //www. math. nus. edu. sg/aslaksen/