SCMP Summer 2009

Wheel of Theodorus Art Project

Theodorus was a Greek mathematician of the fifth century BC, a Pythagorean, a member of a group of devoted followers of Pythagoras, and one of Plato’s teachers who is remembered by mathematicians for his contribution to the development of irrational numbers.

Materials Needed:

  • 12x18 white drawing or construction paper
  • 3x5 index cards
  • Rulers

Assignment:

Create a Wheel of Theodorus neatly and in color. Mark the unit measure of all of your triangle sides. Decorate your wheel in a way that demonstrates this spiral in the real world.

Instructions:

  1. Using a template for a particular unit length and a right angle, create an isosceles right triangle.
  2. Using your template again, add another unit length and right angle to the hypotenuse of your original right triangle.
  3. Repeat the process again, making a right triangle out of a unit length from your template and the previous hypotenuse.
  4. Keep adding another unit length to the previous hypotenuse at the right angles to build new right triangles. All the right triangles should share a central point.
  5. When you get to the stage where your right triangles will overlap previous right triangles, draw your hypotenuse toward the center of the spiral but do not mark over the previous drawings.
  6. Remember to label your figure with all of the dimensions of your successive right triangles. Each leg and hypotenuse should be labeled. If a hypotenuse has a length that can be simplified to a rational number, demonstrate that you recognize this equivalence. (For example, since show this on your project.)
  7. Include a title for your picture.
  8. On the front of your picture, include your signature and the date.
  9. Make sure your project is neat. Use color, unless you mean to emphasize contrast by using black and white.
  10. Attach a lined sheet of paper to your art containing your calculations to find lengths of segments (using the Pythagorean Theorem) for your first 8 triangles.
  11. Answer the following questions on the sheet of paper which shows your calculations:
  12. What patterns do you see in the hypotenuse lengths?
  13. The Wheel of Theodorus can go on forever. List the side lengths (legs and hypotenuse) of the next 6 triangles (without using the Pythagorean Theorem).
  14. Suppose you continue adding triangles to the wheel. Which triangle will have a hypotenuse length of 5 units? 10 units? 15 units? Explain how you determined your answers.

Adapted from Lewis, Leslie, “Irrational Numbers Can ‘In-Spiral’ You.” MTMS, Reston, VA: NCTM, April 2007.