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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/
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