Square-Octagon Dissection, Hinged and Flexible Models: Functional and Playful

####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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