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

Updated versions of these notes are at https://www.foldworks.net/mathsconf33

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)

Pythagorean tiling

• The Platonic Solids
• Reverse engineering
• Cube
• Skeletal Octahedron
• 1:sqrt(2) paper (ISO A standard paper)
• Golden rectangle by folding
• Exact https://www.geogebra.org/material/show/id/bc9rmbqe
• 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)