Polygons & Polyhedra
polygons  polyhedra
 polyhedral dice  plaited dice
Polygons
A polygon is a multisided shape. Simple examples are
triangles and rectangles.
A regular polygon is a multisided shape
where all the sides are the same length. A simple example would be an
equilateral triangle or a square. We can also describe a polygon as
convex if all the internal angles are less than 180°.
Polyhedra
A polyhedron is a multifaced object. Simple examples
are boxes and prisms (triangular or rectangular).
A regular
polyhedron is one where all the faces are the same regular
polygon. There are just five convex regular polyhedra, known as
the Platonic
solids, after Greek philosopher Plato, although they were known
of well before his time.
Here are the five Platonic solids.
Swapping the faces and vertices (corners) of a solid makes its dual. In other words, putting a vertex of the new solid in the middle of each face of the original solid. 

The cube and the octahedron are the dual of each other (see the pic), and the dodecahedron and the icosahedron are the dual of each other. The tetrahedron is selfdual – if you join the points at the centre of each face you get another tetrahedron. 

A soccer ball is a truncated (chopped off) icosahedron, which basically means the points have been cut off an icosahedron leaving 20 hexagons and 12 pentagons.
As well as graphite and diamond, carbon can also form a C_{60} molecule – buckminsterfullerene, also known as buckyballs. It's a truncated icosahedron and is naturally occuring in soot. It's the largest particle which has been shown to have displayed waveparticle duality. 

Polyhedral Dice
Since
the faces of each Platonic solid have exactly the same area, we can
use the Platonic solids as dice. The Platonic solids give us the
standard six faced die, as well as dice with four, eight, twelve and twenty
faces.
We can also use nonPlatonic solids as dice. A ten sided polyhedra (which has kiteshaped sides)
is great for a die with ten faces. All the faces are the same shape and
have the same area, but the faces are not regular polygons because
their sides are not all the same length. In the picture at the right,
there are two 10sided dice, one with 010 and the other with 0090.
24 sided dice can be made with triangular faces (they look like a small pyramid has been placed on each face of a cube) or from an icositetrahedron which has kite shaped faces.
A die with 30 diamond shaped faces is formed if each of the 30 edges of either an icosahedron (20 faces) or dodecahedron (12 faces) is turned into a flat face. 
Plaited Dice
Making a die out of paper is easy enough, but it normally involves putting glue on little paper tabs, making a mess, and making a cube that looks a mess – in that order.
Plaiting a paper die, on the other hand, involves no glue, no mess, and a very neat die at the end of it.
Download the PDF (13 KB) – complete with instructions. 
