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.*

last modified june 2009