Transcript The Beauty of Polyhedra - Department of Mathematics, NUS

The Beauty of Polyhedra
Helmer ASLAKSEN
Department of Mathematics
National University of Singapore
[email protected]
www.math.nus.edu.sg/aslaksen/polyhedra/
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
3
4
5
6
7
Tri
Tetra
Penta
Hexa
Hepta
8
9
10
12
20
Octa
Ennea
Deca
Dodeca Icosa
Polyhedra
 Vertices, edges and faces.
Platonic solids
 Euclid: Convex polyhedron with
congruent, regular faces.
Properties of Platonic solids
Faces Edges Vertices Sides Faces at
(F)
(E)
(V)
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
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
Web
 http://www.math.nus.edu.sg/aslaksen/