MathsConf33: Origami and Mathematics with Sue Pope and Tung Ken Lam

Updated versions of these notes are at

Session description: Folding paper is an accessible entry to many aspects of mathematics. In this practical workshop, we’ll explore fractions, angles, symmetry and proof.

Here are some of the ideas that we might explore:

  • Proof of the angle sum of a triangle by folding
    • and the area of a triangle

      Angle sum of a triangle by folding. Pinch the midpoints of two edges and join with a fold. Then fold the other two corners of the triangle to meet.
  • Folding an angle of 60 degrees
  • Fractions 4. Dividing lengths into equal parts

    Dividing a square into thirds: Here are eight methods for dividing a square into thirds. How can you prove each method works? What are the advantages and disadvantages of each method? Which methods can be generalised to oblongs and other divisions e.g. fifths and sevenths?
  • Proofs of Pythagoras Theorem (including fractions, perpendicular bisector and rotational symmetry)
    • Tilted square
      Pythagorean Theorem: Proof Using a Tilted Square

      Pythagorean tiling

      Pythagorean Tiling and Proof of Pythagoras Theorem
  • The Platonic Solids
    • Reverse engineering
      • Cube
      • Skeletal Octahedron
  • 1:sqrt(2) paper (ISO A standard paper)
  • Golden rectangle by folding
    • Exact
    • Excellent approximation using A4 paper: trim a long strip that is 1/8 of the short edge. 7(√2)/8 is √5 – 1 with 0.1/% error.
    • Rarely used in origami, but two models using golden rectangles are
      • Skeletal Icosahedron (Kasahara and others)

        Skeletal Icosahedron made from 30 golden rectangles
      • Great Dodecahedron (Shapcott and others)

        Great Dodecahedron made from 30 golden rectangles
  • Lesson plans, presentations and origami instructions for KS2 and KS3 at
  • Other ideas are in the ATM publication Learning Mathematics with Origami. Some sample materials is in Mathematics Teaching 254.