Strange Geometries

Summary

This webpage is about the shapes and surfaces you can make if you put equal numbers of equilateral triangles together at a vertex. 3 triangles gives us a tetrahedron, 4 gives us an octahedron, 5 an icosahedron, and 6 gives us the triangulated plane. And you might think that that was the end of it. But you would be wrong!

If you put 7 (or more) triangles together at a vertex, you get a triangulated hyperbolic surface. It is this that the projects aim to get you studying, together with familiarising you with the properties of the other shapes mentioned.

This material was first used in videoconferences with students from schools in various parts of the UK. Any student or teacher is now welcome to use it. Comments or requests for help should be sent to Jenny Gage (jag55@cam.ac.uk).

Talk

• Strange Geometries. Transcript of Keith Carne's talk.

Project work

• Additional notes to help students, teachers, parents.
• Extension work. Additional geometric problems.
• Project 3. Making other surfaces.
• Project 2. Triangles on the surfaces.
• Project 1: Lines and circles on the surfaces.
• Preliminary project work. Introductory work for all projects.
• Presentation no. 1 by Davison High school. A PowerPoint presentation made by Davison High School students during the second videoconference. 224kb
• Presentation no. 2 by Davison High school. A presentation by students at Davison High School for Girls. 32kb
• Project work by Wednesfield High School, Wolverhampton. 27 kb

Additional material

• References and resources. Books and web links about geometry.