Can a single piece of paper fold into two different shapes?
In short: Imagine a complex geometric net that, when folded, could become either a cube or a tetramonohedron. This isn't just a theoretical puzzle; it is a frontier of computational geometry where mathematicians use integer linear programming to solve the mysteries of how surfaces can occupy space.
In the realm of geometric folding algorithms, researchers are exploring 'common unfoldings'—the ability for one flat shape to fold into multiple distinct polyhedra. One prominent instance features a fractal-style expansion thought to collapse into a cube as well as a tetramonohedron (a tetrahedron possessing uniform sides). While this specific instance has been published at EuroCG, it remains a conjecture because finding an infinite sequence of connected polygons that converge perfectly into such a fractal set is still a significant mathematical challenge. Researchers have also investigated common unfoldings for multiple boxes simultaneously. A specific problem regarding three boxes was recently solved in two separate papers.
The complexity of these shapes is staggering. One specific iteration procedure produces a shape with approximately 10 to the 180 edges. While this number is astronomical, researchers have found it to be within 3 times 10 to the minus 1796 of a regular structure, providing strong evidence that the folding logic holds. These problems are often solved using integer linear programming, where researchers define which 'pixels' in a representation are occupied to determine if a surface can physically fit into a desired volume. Computational methods were used to find a configuration where two boxes meet at an unusual angle. This discovery involved calculating a root 10 value based on the sum of squares of roots 3 and 1.
Beyond pure theory, these principles inform kinetic sculpture and complex machines. MIT's coursework on geometric folding explores how these mathematical rules manifest in the physical world, citing works like Theo Jansen’s Strandbeests and Arthur Ganson’s sculptures. By mastering the constraints of linkages, origami, and polyhedra, designers can bridge the gap between abstract computational geometry and tangible, moving art. The curriculum covers various components including linkages and polyhedra. It also highlights how these concepts relate to machines and specific museum works.
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Sources
- Summary by Intellect · source: Class 19: Refolding & Kinetic Sculpture
- Also: Class 19: Refolding & Kinetic Sculpture | Geometric Folding Algorithms: Linkages, Origami, Polyhedra | Electrical Engineering and Computer Science | MIT OpenCourseWare
- Also: Class 19: Refolding & Kinetic Sculpture | MIT Learn
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