This problem builds on work done in finding Semi-Regular
Tessellations and offers a great opportunity to create the
Archimedean solids using students' knowledge of interior
angles of polygons and angles around a point.

The problem has been structured in such a way that a class can
spend some time answering each of the questions in bold, with time
set aside to discuss approaches, thoughts and solutions.

The problem builds up to the search for Archimedean solids
using what students learn about the angle deficit of Platonic
solids. One way of introducing the concept of angle deficit could
be to make a solid out of Polydron and then unfold it into its net
and look at the gap at a vertex.

You might also want to make a link between exterior angles in
two dimensions and angle deficit in three dimensions:

"In two dimensions, the interior
angles of convex polygons are always less than $180^{\circ}$. Can
you explain why?"

"In three dimensions, the sum of
the interior angles at each vertex of convex polyhedra must be less
than $360^{\circ}$. Can you explain why?"

"The exterior angles of polygons
are a measure of how far short the angles are from
$180^{\circ}$."

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

Once students are happy with the concept of angle deficit, you
could organise their results into a table like this:

Vertex Form | Angle Sum | Angle Deficit | Number of Vertices | Total Angle Deficit | |

Cube | $4, 4, 4$ | $270$ | $90$ | $8$ | $720$ |

Tetrahedron | |||||

Octahedron | |||||

Icosahedron | |||||

Dodecahedron |

This leads nicely to the observation that the total angle
deficit is constant.

For teachers with access to Polydron or similar construction
sets, the last part of the problem provides a fruitful opportunity
for much rich mathematical thinking, maybe spanning more than one
lesson!

One way of scaffolding the search for more Archimedean solids
is to list factor pairs of 720, and then challenge students to find
potential angle deficits which are factors of 720. This could be
done in small groups, using Polydron once they've come up with a
possibility to check it can be made.

NB Not all angle deficits
which are factors of 720 can be made into solids - this is the
three dimensional analogy of patterns such as {10, 5, 5} in the problem Semi-Regular Tessellations.

A list of all the Archimedean solids, their vertex forms and
their names can be found in this article.

NB This list would be
infinitely long if we included prisms and antiprisms so this is why
they are excluded from the definition of Archimedean
solids.

Why do the interior angles meeting at a vertex always add up
to less than $360^{\circ}$?

If you had a small number of vertices, would you expect the
angle deficit to be large or small? (Hint: think about how "pointy"
a tetrahedron is compared with a dodecahedron)

Once students have suggested a possible vertex form, challenge
them to explain how they could work out how many of each type
of polygon will be required to make the solid.

Starting with some two-dimensional work on Semi-Regular
Tessellations may help to prepare students for the
three-dimensional thinking required in this task.

Explore the total angle deficit of prisms and antiprisms
before moving onto more complicated solids.