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Currently Playing: 76 Spike Caltrop - What other asymmetric caltrops exist mathematically?

Print it yourself at https://oskarvandeventer.nl/Print-It-Yourself/. Read at https://twistypuzzles.com/forum/viewtopic.php?t=40768 76-Spike Caltrop is caltrop with 76 spikes. It was suggested by Ben Streeter after seeing my Asymmetric Caltrop, which has 31 spikes. Regular caltrops have four spikes in a tetrahedral configuration. This object always falls such that one spike is straight up, ready to puncture tires of feet. 76-Spike Caltrop also has this same property. There will always be a spike straight up. Conversely, one can point each of the spikes straight up, and the object can rest on a flat surface. One can easily design caltrops that have some symmetry. The challenge was to have one that has no overall symmetry. The geometry of the object is based on an isohedral variation of the pentagonal hexecontahedron https://en.wikipedia.org/wiki/Pentagonal_hexecontahedron ). That object has 60 faces and 92 vertices, where each faces has an exactly-opposing vertex, but not vice versa. By adding 16 faces and removing 16 vertices, the resulting object has 76 faces and 76 vertices, each face exactly opposing a vertex, AND vice versa. The object has several local 3-fold and 5-fold rotation symmetries, but no over-all symmetry. This particular arrangement was found by Ben Streeter through a semi-automated search. The goal was to purge 16 of the 92 spikes, such that the resulting caltrop was as balanced as possible. An open question is whether there exist asymmetric caltrops, other than the 31- and 76-spike ones found so far? Copyright (c) 2025, M. Oskar van Deventer. Frequently Asked Question: http://oskarvandeventer.nl/FAQ.html Buy mass-produced Oskar puzzles at https://www.puzzlemaster.ca/browse/inventors/oskar/ (USA, CA) and https://www.sloyd.fi/brain-teasers/inventors-designers/oskar-van-deventer (EU) Buy exclusive 3D-printed Oskar puzzles at https://i.materialise.com/shop/designer/oskarpuzzle, https://www.chewiescustompuzzles.com/puzzleshop/oskars-puzzles