**We recommend that you use "Introduction: Use, Safety and the Rhombus" first, with each new group of participants.**

- Introduction: Use, Safety and the Rhombus Short activity to introduce the triangles to a new group of learners, show how to connect them together, review basic geometric vocabulary and possibly explore properties of the rhombus. Finally there is an option to examine two dimensional symmetry, reflection and rotation, for the rhombus and the regular
*n*-gons. - Strips and Tunnels Triangles are assembled into strips. Two strips are attached to make a double width strip. This is then folded to make a tunnel roof. Investigate how the tunnel can turn a corner.
- Pyramids Build pyramids with regular bases (triangle, square, pentagon) and compare properties.
- Regular Polyhedra Extend pyramids with regular bases to form the regular Platonic polyhedra with triangular faces. Then find the other regular polyhedra and demonstrate there are only 5. Count faces, edges and vertices and discover the Euler formula. Note: Nets for polyhedra are intentionally
**NOT**used in this activity. - Symmetry Review 2 dimensional symmetry. Identify and mark with tape all reflection planes on a tetrahedron. Find rotational symmetry axes using the turn and stop game. Investigate compositions of reflections and rotations. Investigate other polyhedra.
- Colour Patterns Investigate which polyhedra can be made with a fixed set of colours of face and different rules for how the colours can coincide at edges or vertices. Symmetries of the octahedron are related to colour-swapping transformations.
- Space Fillers Construct parallelepipeds from rhombi. See how they stack. Construct a parallelepiped from an octahedron and two tetrahedra. Use this to see how tetrahedra and octahedra together can fill space.
- Double Edge Length Tetrahedron Construct a net for a double edge length tetrahedron, noting that the sum of the first n odd numbers is n squared. Fill the double edge length tetrahedron as far possible with single edge length tetrahedra and add a ‘mystery shape’, the octahedron. Using scaling and volume subtraction deduce the 4 to 1 ratio of volumes of a regular octahedron to a regular tetrahedron of equal edge length.
- Stella Octangula Either stellate an octahedron or the middle triangles of the faces of the double edge length tetrahedron. Observe the fact that it is the union of two double edge length tetrahedra. Use tape to reveal the cube dual of the octahedron.
- Stellated Polyhedra and Duality Make stellated versions of polyhedra and connect coplanar vertices with tape to show the dual polyhedron. Compare symmetries of a polyhedron and its dual.
- Faces and Edges Count the numbers of faces and edges on a range of polyhedra. Tabulate and graph results, look for patterns in the table and graph and express as an equation, determining the ratio of faces to edges, and reasons for the ratio. Examine how varying definitions of face can give varying results. For advanced students, discuss different types of proof.
- Angle Deficit Define and record angle deficit at a vertex. Investigate the sum of angle deficits over all vertices in a polyhedron. Discover its constant value for a range of polyhedra.
- Torus Build and explore a torus, looking at symmetry, the inside and outside, and going round the inside tunnel. Compute total angle deficit and F-E+V, and find they are zero for a torus. Identify symmetries of the constructed torus.

The proof of the Angle Deficit Theorem can be expressed in two accessible ways.