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## 'Which Solids Can We Make?' printed from http://nrich.maths.org/

This problem follows on
from Semi-regular
Tessellations.
Here are pictures of the five

Platonic Solids - solids made
from just one type of polygon, with the same number of polygons
meeting at each vertex.

Can you convince yourself that
there are no more?
The

angle deficit at a
vertex of a polyhedron is a measure of how far short each angle sum
is from $360^{\circ}$.

For example, in a dodecahedron, three pentagons with interior
angles of $108^{\circ}$ meet at each vertex, so the angle sum is
$324^{\circ}$ and the angle deficit is $36^{\circ}$:

Can you work out the angle
deficit at the vertices of the other Platonic solids?

The total angle deficit is the sum of the deficits at each
vertex.

What do you notice about the
total angle deficit for the Platonic solids?

Archimedean Solids have
two properties:

- They are formed by two or more types of regular polygons, each
with the same side length
- Each vertex has the same pattern of polygons around
it.

Here is a picture of an

Archimedean Solid with 24
vertices. Its

vertex form
{3, 3, 3, 3, 4} is defined by the polygons that meet at each
vertex: triangle, triangle, triangle, triangle, square.

Calculate the total angle deficit for this solid. Does it match
your observations about the Platonic solids?

Try to suggest some other vertex
forms which might give rise to Archimedean solids, assuming
all solids share the property you have discovered. If you have
access to construction sets such as Polydron, you could test out
your ideas.

Below are some vertex forms you might like to try: some of them
give rise to solids and some of them don't. Can you decide which
will work before testing them out?