Experiment objective

Make rhombus tiles and use them to make various designs.

Read more background about this activity in Donald (H. S. M.) Coxeter's bio...

You need

  • Cardboard or construction paper.
  • A straight edge or ruler.
  • A protractor to measure angles.
  • Scissors or a utility knife.

A rhombus is a squashed square. Try making some rhombus tiles. Make two diamond-shaped rhombuses, each having four sides of equal length. One rhombus has two angles of 36 degrees and two of 144 degrees. The other one has two angles of 72 degrees and two angles of 108 degrees. It’s very important that you use the protractor to measure these exact angles when you draw your rhombus shapes or else they won’t fit together properly to make patterns. You can use this pattern below as a guide. Just draw it onto some coloured paper.

Measure, draw several dozen of these rhombus shapes. Or click on the pattern below and print it onto coloured paper. Cut out the shapes. Then move them around on a table to make patterns and shapes or glue them in various designs to another sheet of paper.

A rhombus is a two-dimensional or 2D shape. You can make shapes that look like three-dimensional (3D) cubes projected onto the 2D surface of the paper. Try to make 3D cube projections from these 2D rhomboids. How many different ones can you make? What other shapes can you make?

Roger Penrose, the brilliant English mathematician and physicist (b. 1931 and currently professor of Mathematics at Oxford University in England), has shown how these shapes can be used to make an aperiodic tiling of the plane. That means you can tile a flat area with these shapes in a pattern that never repeats. Here’s what one looks like. Try it.

If you love this and it gets you thinking and dreaming about shapes and numbers, consider this: multiples of the same 36 degree angle make up all the angles in the two shapes! That is 36 x 2 = 72; 36 x 3 = 108; 36 x 4 = 144; all the angles we started out with! If you feel this is cool, you may have a future in math and geometry.