Visit to the
Simon Langton Grammar School
and the
Langton Institute for Young Mathematicians

June 2009

The Platonic Solids

The Platonic solids are the basic building blocks of all polyhedral geometry. Download nets (one copy for each set of polyhedra) for making all of the Platonic solids.

One very famous result concerning convex polyhedra in general is Euler's formula. Here is a worksheet which will help you discover this important result.

Time required: at least 90 minutes to make all the Platonic solids (less to make fewer polyhedra), plus time to investigate Euler's formula.

The Sculpture of George Hart

The Dragonflies sculpture by George Hart is based on the rhombic dodecahedron. Click here to see George's website about this sculpture. You can also download a net for the rhombic dodecahedron.

And exactly how is Dragonflies related to the rhombic dodecahedron? First, we need to learn something about the stellations of the rhombic dodecahedron. Vladimir Bulatov has written an applet to help us understand this relationship.

Click here for a stellation diagram you can print out to design your own sculpture.

You can also trying building two different types of paper models of the sculpture: Model 1 (requires 6 copies for one model) or Model 2 (also 6 copies for one model).

Note: if many models are being built, print a few extra pages in case of mistakes.

Time required: the rhombic dodecahedron will take one student about 30-40 minutes. Model 1 is extremely difficult. Model 2 will take two students about 60 minutes to complete.

Building Slide-Togethers

Slide-togethers are also featured on George Hart's web site: check them out! They make interesting models, and require no glue to assemble.

A very nice model is one based on the rhombic triacontahedron. You will need thirty squares (5 copies) to build it. Be careful when cutting the slits in the squares: continue about 1 mm further to ensure they will fit together well.

An especially nice version can be made with five colours (1 page needed in each of 5 different colours); the rhombic triacontahedron may be inscribed in a cube in five different ways. You may want to build your own rhombic triacontahedron and plan your colour scheme before building.

Two other models are based on the icosahedron: one with triangles (4 copies needed if all one colour; 1 each of 5 different colours for the 5-colour model) and one with hexagons (10 pages needed if all one colour; 2 each of 5 different colours for the 5-colour model). You may make a nice five-colour model such that no two adjacent faces are the same colour; create your colour scheme on an icosahedron first. Again, cut about 1 mm further along the slits for ease in putting the models together.

There are many more models available on George's web site. For a real challenge, try the one with twelve pentagons. Finishing this model is difficult, but rewarding.

Time required: two students can complete a model using 30 squares in 60-75 minutes; the others take somewhat less time.

Trihexaflexagons and Square Flexagons

Flexagons are arguably the most interesting polygons you can make. Click here or here to see videos of flexing hexagons. Here are some instructions for making a trihexaflexagon. And you can find nets for the triangles here (one copy can make 3 flexagons).

Flexagons can also be made from square nets (large) or square nets (small) (one copy for each flexagon). Download a set of instructions for making seven different square flexagons. Note: to make all seven flexagons, one large net and six small nets will be needed.

Many, many more interesting flexagons may be found here, or in Les Pook's excellent book Flexagons Inside Out, ISBN 0-521-52574-8.

Time required: one student can make a flexagon of either type in 10-20 minutes, depending on the complexity of the model. Note: do not print on cardstock, or the flexagons will be too stiff. Use ordinary copy paper.

© 2004-12 vincent j matsko vmatsko(at)imsa(dot)edu
illinois mathematics and science academy

last modified june 2009