Tilings and Polyhedra

Helmer ASLAKSEN

Department of Mathematics

National University of Singapore

aslaksen@math.nus.edu.sg

www.math.nus.edu.sg/aslaksen/polyhedra/

They look nice! They teach us mathematics. Mathematics is the abstract study of

patterns. Be conscious of shapes, structure and

symmetry around you!

Why are we interested in this?

What is a polygon?

Sides and corners. Regular polygon: Equal sides and equal

angles. 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

The vertex angle in a regular n-gon is 180 (n-2)/n. To see this, divide the polygon into n triangles.

3: 60 4: 90 5: 108 6: 120

What is a tiling?

Tilings or tessellations are coverings of the plane with tiles.

Assumptions about tilings 1

The tiles are regular polygons. The tiling is edge-to-edge. This

means that two tiles intersect along a common edge, only at a common vertex or not at all.

Assumptions about tilings 2

All the vertices are of the same type. This means that the same types of polygons meet in the same order (ignoring orientation) at each vertex.

Regular or Platonic tilings

A tiling is called Platonic if it uses only one type of polygons.

Only three types of Platonic tilings. There must be at least three

polygons at each vertex. There cannot be more than six. There cannot be five.

Archimedean or semiregular tilings There are eight tilings that use

more than one type of tiles. They are called Archimedean or semiregular tilings.

Picture of tilings

More pictures 1

More pictures 2

More pictures 3

A trick picture

Polyhedra

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

What is a polyhedron?

Solid or surface? A surface consisting of polygons.

Polyhedra

Vertices, edges and faces.

Platonic solids

Euclid: Convex polyhedron with congruent, regular faces.

Properties of Platonic solids

Faces Edges Vertices Sides

of face

Faces at

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 20 30 12 3 5

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?

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!

Nested Platonic Solids

How to make models

Paper Zome Polydron/Frameworks Jovo

Web

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