Activity 7 - Regular Polyhedra

You need to have done Activity 6 before you do this Activity.

Polyhedron is a Greek word meaning "many faces." A regular polyhedron has the following important properties. It:

• is a convex solid (fills space)
• has all faces the same shape and size (one of the regular polygons)
• has all internal angles between adjacent faces the same.

As mentioned in the essay, the Pythagoreans recognized that there are only five regular polyhedra, having the following properties:

• TETRAHEDRON - four equilateral triangular faces, four vertices, and six edges.
• CUBE - six square faces, eight vertices, and twelve edges.
• OCTAHEDRON - eight equilateral triangular faces, six vertices, and twelve edges.
• DODECAHEDRON - twelve pentagonal faces, twenty vertices, and thirty edges.
• ICOSAHEDRON - twenty equilateral triangular faces, twelve vertices, and thirty edges.

Polyhedron Animator Click on a buttons at left to see the solids. Click on the solid to stop the rotation. Click to start again.

It is natural to wonder why there should be exactly five regular polyhedra, and whether there might conceivably be some that simply haven't been discovered yet. However, it is not difficult to show that there must be five, and that there cannot be more than five. It is thought that the Pythagoreans used the following investigations to discover the regular polyhedrons and to demonstrate that there can be only five. Before we start, you should construct and cut out of card, 20 equilateral triangles of the same size, 6 squares of the same size, 12 pentagons of the same size.

Pythagoras' Investigation

The essential observation to make is that a solid can only be made out of regular polygons if the sum of the interior angles of all the regular polygons meeting at a point (vertex of a polyhedron) add to less than 360°. In other words, if we were to lay flat on a table the faces meeting at any vertex of a regular polyhedron, there will always be a gap left. To see this, recall the result from Activity 6 which stated; "When regular polygons fill the plane, the sum of all the interior angles of the polygons meeting at the vertex would add to exactly 360°".

It is thought that Pythagoras started with the the simplest regular polygon (the equilateral triangle) and progressively added more triangles at a point like in Activity 5, to see if a regular polyhedron could be produced. We can start with three faces together at each vertex (point), for if only two were used they would collapse against one another and we would not get a solid.

Here are the possibilities for the equilateral triangles:

We will start the investigation by laying three of the triangles on our table so that each triangle is touching at a common point and so that adjacent sides are also touching (as in Activity 5). Each time we add a triangle, we will see if we can "fill the plane" or "fill the space" (form a regular polyhedron).

• 3 equilateral triangles.

  Obviously these do not "fill the plane". Now take two pieces of adhesive tape and stick neighbouring pairs of the triangles together so that the tape acts as a hinge. Tie a piece of cotton to the common vertex of the three and slowly lift the string pulling the vertex upwards away from the table. The three triangles will close in on each other forming a closed three faced shape with an open triangular base - they form a solid which has as its exposed base another equilateral triangle. Stick another equilateral triangle to the base so that the sides touch. This means that we have a shape with 4 identical faces and identical internal angles. In fact, four is the smallest number of faces a regular polyhedron must have. In summary, three triangles will not "fill the plane" but they will join at a point forming a solid - the first regular polyhedron, the tetrahedron. Click the button "4 faces" on Polyhedron Animator above.

• 4 equilateral triangles

  Question 1:

  Repeat the same process as above but this time use four equilateral triangles. Do four equilateral triangles "fill the plane" or form a regular solid?

  Question 1 - Answer

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• 5 equilateral triangles

  Question 2:

  Now repeat the same process as above for five equilateral triangles. Do five equilateral triangles "fill the plane" or form a regular solid?

  Question 2 - Answer

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• 6 equilateral triangles

  Question 3:

  Now repeat the same process as above for six equilateral triangles. Do six equilateral triangles "fill the plane" or form a regular solid?

  Question 3 - Answer

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Once again we will start with 3 squares and increase the number of squares each time.

• 3 squares

  Question 4:

  Repeat the same process as above but this time use three squares. Do three squares "fill the plane" or form a regular solid?

  Question 4 - Answer

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• 4 squares

  Question 5:

  Repeat the same process as above but this time use four squares. Do four squares "fill the plane" or form a regular solid?

  Question 5 - Answer

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Now let us try the next regular solid after the square - the pentagon:

Once again we will start with 3 pentagons and increase the number of pentagons each time.

• 3 pentagons

  Question 6:

  Repeat the same process as above but this time use three pentagons. Do three pentagons "fill the plane" or form a regular solid?

  Question 6 - Answer

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• 4 pentagons

  Question 7:

  Repeat the same process as above but this time use four pentagons. Do four pentagons "fill the plane" or form a regular solid?

  Question 7 - Answer

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Now let us try the next regular solid after the pentagon - the hexagon:

Once again we will start with 3 hexagons and increase the number of hexagons each time.

• 3 hexagons

  Question 8:

  Repeat the same process as above but this time use three hexagons. Do three hexagons "fill the plane" or form a regular solid?

  Question 8 - Answer

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Question 9:

This accounts for the five known regular solids. But can there be others? Make a conjecture and try to prove it.

Question 9 - Answer

As we already know from Activity 5 the four triangles do not "fill the plane". If we repeat the previous process - tape them, and tie the cotton onto the common vertex and lift the four triangles, they will also form a solid, but this time the exposed base is a square. If we now make another of these using another 4 equilateral triangles similar to the first, the two can be joined at the square base, making an 8 faced regular shape. This solid also has identical internal angles. In summary, four triangles will not "fill the plane" but they will join at a point forming a solid - the third regular polyhedron, the octahedron. Click the button "8 faces" on Polyhedron Animator above.

Back

These do not "fill the plane" but do fit together at a point forming a solid. They form a regular pentagon as their exposed base. If you make four of these, with a bit of adjusting you can make a 20 faced regular shape. This also has identical internal angles. This is the last regular polyhedron - the icosahedron. Click the button "20 faces" on Polyhedron Animator above.

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As we have seen in Activity 5, six equilateral triangles "fill the plane" so they cannot fit together at a point forming a solid. Also, no more equilateral triangles (beyond the 6) are needed since the 6 "fill the plane".

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3 squares don't "fill the plane" but they do fit together at a point. If you put another 3 squares together at a point you can join then to make the hexahedron or cube. Click the button "6 faces" on Polyhedron Animator above.

Back

As we have seen in Activity 5, four squares "fill the plane" so they cannot fit together at a point forming a solid. Also, no more squares (beyond the 4) are needed since the 4 "fill the plane".

Back

3 pentagons don't "fill the plane" but do fit together at a point. If you make four of these, with a bit of adjusting you can make a 12 faced regular shape. This also has identical internal angles. This is the fourth regular polyhedron - the dodecahedron. Click the button "12 faces" on Polyhedron Animator above.

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As we have seen in Activity 5, four pentagons "exceed the plane" so they cannot fit together at a point forming a solid. Also, no more pentagons are needed.

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As we have seen in Activity 5, three hexagons "fill the plane" so they cannot fit together at a point forming a solid. Also, no more hexagons (beyond the 3) are needed since the 3 "fill the plane".

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My conjecture is "There are only five regular polyhedrons possible". My proof is as follows:

Firstly we have demonstrated that a regular polyhedron must have at least 4 faces, and therefore at least 3 faces must meet at a vertex.

Secondly, we have established that the sum of the interior angles of the faces meeting at each vertex must be less than 360°, for otherwise they would "fill or exceed the plane" and not be able to form a solid.

Now, since each interior angle of an equilateral triangle is 60°, we could fit together three, four, or five of them at a vertex, and these correspond to to the tetrahedron, the octahedron, and the icosahedron.

Also each interior angle of a square is 90°, so we can fit only three of them together at each vertex, giving us a cube.

The interior angles of the regular pentagon are 108°, so again we can fit only three together at a vertex, giving us the dodecahedron.

The interior angles of the regular hexagon are 120°, so if we fit three of them together at a vertex the angles sum to precisely 360°, and therefore they "fill the plane".

And finally, no polygon with more than six sides can be used either, because the interior angles just keep getting larger causing less than three faces to meet at a vertex.

Therefore there are only five regular polyhedra possible

Q.E.D.

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