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?

            What shapes meet at each vertex? What is the Total ‘angle deficit’ for this Prism?

Open the File as a Word Document

 

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Last modified: June 18, 2007