Facing Up to Football 1
Modern school Mathematics doesn’t deal much with solid shapes beyond a cube or a square-based pyramid, but you’ll find it’s easy to work out how many pieces of leather you need to make a traditional football (or ‘truncated Icosahedron’, as I’m sure your Mum calls it…).
So here goes!
How Solids Work – an example:
A cube is made from squares put together, three at a time around each vertex.
The fact that 3 x 900 makes 2700, rather than 3600, means that when you stick the edges
of the three squares together, they won’t lie flat, but instead begin to fold around to make a solid, rather than a
How quickly the faces of the cube ‘curl’ around to join back up again depends on the ‘angle deficit’ (or
‘gap’!) at each vertex: instead of a nice flat 3600, we have only 2700, so there is an
‘angle deficit’ of 900 at each vertex of a cube.
A cube has eight vertices, making a Total ‘angle deficit’ of 8 x 900 = 7200
**** This Total a.d. of 720 turns out to be important…****
Let’s look at a slightly less familiar solid – the Octahedron:
It’s one of the simplest solids – made from (usually equilateral) triangles put together, four at a time
around each vertex. Each angle of the faces is 600.
The fact that 4 x 600 makes 2400, rather than 3600, means that when you stick
the edges of the four triangles together, they won’t lie flat, but instead start to fold around…
There is an ‘angle deficit’ of 3600 – 2400 = 1200 at each vertex.
To make the required total ‘angle deficit’ of 7200 requires 7200
1200 = 6 vertices.
To find how many triangular faces you need altogether, you can add up the 6 lots of 4 corners-at-each-vertex = 24
corners, remembering that each triangle provides 3 corners…
So we need 24
3 = 8 triangles to make an Octahedron…Surprise!
The largest of the perfectly symmetrical (Platonic) solids is the Icosahedron, made by putting 5 equilateral triangles
together at each vertex.
Follow the train of argument above to find how many triangles are needed to make the Icosahedron.
A Cuboctahedron is made by cutting all the corners off a cube, by joining the mid-points of the original edges
that meet at each corner of the cube, forming triangles.
Sketch it and find how many squares and how many equilateral triangles the Cuboctahedron is formed from. You
should also identify the number of vertices.
Each vertex has 2 squares and 2 triangles meeting. Calculate the ‘angle deficit’ at each vertex, and hence
check what the Total ‘angle deficit’ comes to for the Cuboctahedron.
Imagine, or sketch, a Pentagonal prism. The angles of each regular Pentagon are 108
How many vertices are there altogether?
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