Triangle to or from Square Transformation, Dudeney's Dissection
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####Equilateral Triangle to/from Square, Dudeney's Dissection ##### The large models can be used as a pencil holder or organizer. Square or triangle? You choose what you like. It is fun to go back and forth-- very satisfying like a puzzle. An equilateral triangle can be beautifully dissected into four pieces and rearranged into a square (Dudeney, 1902/1907; Steinhaus, 1950/1969), where there is a minor mathematical error with no physical implications. More interestingly, the pieces can be taped together and looped around between a triangle and a square. Mathematically, if the side length of the equilateral triangle is *s*, then the side length of the corresponding square is * 3^(1/4) s/2 *. In the design process, we use this fact, two midpoints, and two perpendicular lines (see Figure for Design). To make it playful, I tried two designs: (1) Connected dissections for TPU flexible filaments, which demonstrate the idea well but are not perfect due to the twists and tension at the connections. One can start from a triangle or square; both are included. (2) Loose dissections which can be taped or used separate pieces. ####Among the Files 1. Two models for TPU flexible filaments based on an equilateral triangle of side length 60mm. 2. A small loose square model (40mm x 40 mm) with a height of 10mm. 3. A large loose square model (100 mm x 100 mm) with a height of 45mm. 4. A large loose square model (100 mm x 100 mm) with a height of 90 mm. ####Reference 1. Dudeney, H. (1907). The Canterbury puzzles. Available at https://bestforpuzzles.com/bits/canterbury-puzzles/index.html 2. Steinhaus, H. (1950/1969). Mathematical snapshots (3rd ed.). New York, NY: Oxford University Press. 3. https://mathworld.wolfram.com/HaberdashersProblem.html
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