Triangle to or from Square Transformation, Dudeney's Dissection by lgbu

Last crawled date: 3 years, 1 month ago

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
Two models for TPU flexible filaments based on an equilateral triangle of side length 60mm.
A small loose square model (40mm x 40 mm) with a height of 10mm.
A large loose square model (100 mm x 100 mm) with a height of 45mm.
A large loose square model (100 mm x 100 mm) with a height of 90 mm.
Reference
Dudeney, H. (1907). The Canterbury puzzles. Available at https://bestforpuzzles.com/bits/canterbury-puzzles/index.html
Steinhaus, H. (1950/1969). Mathematical snapshots (3rd ed.). New York, NY: Oxford University Press.
https://mathworld.wolfram.com/HaberdashersProblem.html