## SHAPES - PLATONIC AND ARCHIMEDEAN SOLIDS

PLATONIC SOLIDS
The 'Platonic solids' are three dimensional shapes formed by putting together identical regular faces. Every corner and every edge is identical to every other corner and edge. There are five such 'solids': the tetrahedron, cube, octahedron, dodecahedron and icosahedron.

 SHAPE FACES CORNERS EDGES tetrahedron 4 triangles 4 6 cube 6 squares 8 12 octahedron 8 triangles 6 12 dodecahedron 12 pentagons 20 30 icosahedron 20 triangles 12 30 Some Platonic and Archimedean solids. Do you know which is which ?

To make your own 'Platonic solids' print out the following nets, cut them out and use the tabs to glue them together, click below:

ARCHIMEDEAN SOLIDS
These are three dimensional shapes that are formed when more than one type of repeating face is used. The corners are still identical with each other. Shapes include the famous truncated icosahedron (better known as the football) composed of 12 pentagons and 20 hexagons.

EULERS LAW
Eular discovered a simple mathamatical law that relates the number of Faces, Edges and Corners in many different shapes including the Platonic and also the Archimedian solids:

in words:

number of Faces + number of Corners - number of Edges = 2

F + C - E = 2

Can you work out the answers to the "?'s" in the table below:

 SHAPE num. of F's num. of C's num. of E's F+C-E=? cube 6 8 12 ? Egyption pyramid 5 5 8 ? icosahedron 20 12 ? 2 Truncated icosahedron (football) 12+20=32 ? 90 2

RE-EXPRESSING EULERS LAW:
We can re-write the simple formula shown above in terms of just the different types and numbers of faces:

in words:

3 x number of trangles + 2 x squares + 1 x pentagons - 1 x heptagons ...etc. = 12

3.T + 2.S + 1.P - 1.Hep. ... = 12

NOTE: This formula only works when each edge has only two corners and where each of the corners has exactly three edges radiating from them. Many things found in nature follow this rule including soap bubbles and animal skeletons to name a few.

For shapes like the football (the truncated icosahedron) for example, which only contains pentagons and hexagons we find that the number of triangles, squares and heptagons is zero and so the formula simplies to:

P = 12

Which actually means that to make a ball shape (or cage) you can have as many hexagons as you like but only if there are exactly 12 pentagons!

QUESTION
If the cage like skeleton of a sea creature has 60 hexagons and 18 pentagons how many heptagons (7 sided polygons) does it need to form a closed cage?

THE CREATIVE SCIENCE CENTRE

Dr Jonathan Hare, The University of Sussex
Brighton, East Sussex. BN1 9QJ.