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
‘plane’ shape.
·
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 Problems:
1
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.
2a
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.
b
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.
c
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.
3
Imagine, or sketch, a Pentagonal prism. The angles of each regular Pentagon are 108
0
.
How many vertices are there altogether?
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