Square-Octagon Dissection, Hinged and Flexible Models: Functional and Playful
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####Square-Octagon Dissection, Hinged and Flexible Models: Functional and Playful. The square can be dissected into five pieces, including four congruent pentagons and a small square, and then re-rearranged into an octagon: It is the very interesting math problem. The key is to figure out the side length of the center square: * \text{InnerSquareDim}= \frac{1}{2}\sqrt{2 \sqrt{2} - 2} \cdot \text{OuterSquareDim}. * A while ago, I designed a simple model (https://www.thingiverse.com/thing:2946095 ). Using hinges, the present design has taken the problem into another level yielding functional caddies and playful flexible models, great for *teachers*. The center square is not really necessary but is included for those who prefer to have it in the model. As shown in the figures, the structure is a chain of congruent quadrilateral, which can be easily manipulated between a square (prism) or an octagon (prism). The large one is 120mm ×120mm × 80mm, fully functional for a pencil organizer (for teachers). All models can be printed at a resolution of 0.2mm or higher (<0.2mm). The big one does take some time to print due to its size (about 13 hours on a home printer). ####Among the Files 1. A flexible model to be printed using **flexible filament**. 2. Hinged models of three sizes and two styles. The latch version has a hole (diameter 4mm) in the last hinge so the box can be somehow latched with a pin, a small screw, or a piece of wire. **After printing, please use a small flat screwdriver to loosen the hinge.** 3. Three optional center squares (prisms). Note that the dimension references in the filenames refer to the dimension of the corresponding outer square (prism). ####References 1. Bu, Lingguo. https://www.thingiverse.com/thing:2946095 2. Cundy, H. M., & Rollett, A. P. (1961). Mathematical models (2nd ed.). London, UK: Oxford University Press.
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