Transcript Is IT it? - National University of Singapore

Tilings and Polyhedra
Helmer ASLAKSEN
Department of Mathematics
National University of Singapore
[email protected]
www.math.nus.edu.sg/aslaksen/polyhedra/
Why are we interested in this?
 They look nice!
 They teach us mathematics.
 Mathematics is the abstract study of
patterns.
 Be conscious of shapes, structure and
symmetry around you!
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 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
 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/