Print-It-Yourself Puzzles

Currently Playing: Robin's Caltrop - Can all caltrops be canonicalised? (Tough question for mathematicians)

Print it yourself at https://oskarvandeventer.nl/Print-It-Yourself/. Read more at https://twistypuzzles.com/forum/viewtopic.php?t=40787 Robin's Caltrop is a design (invention? discovery?) by Robin Houston. It is his answer to my question "what is the lowest number of spikes on a caltrop that has no intrinsic symmetry". His answer is 10 spikes. The object has the property that for each of the ten ways that the caltrop can stand, a spike stands out vertically, and each of the ten spikes will stay up vertically. And it has no intrinsic symmetry. To find this design, Robin made a systematic search of convex polyhedra with an equal number of faces and vertices. However, many of those have intrinsic symmetry. Robin found that one could design a whole family of asymmetric four-spike caltrops with vertices A, B, C and D. They are caltrops, i.e. each spike orthogonal to opposite face, when AB^2+CD^2=AC^2+BD^2=AD^2+BC^2 (https://en.wikipedia.org/wiki/Orthocentric_tetrahedron, https://encyclopediaofmath.org/wiki/Tetrahedron,_elementary_geometry_of_the). However, those are not intrinsically asymmetric, as one can take A=B=C=D and get the classic caltrop. Robin proved that no 5-spike caltrop exists, symmetric or not. He continued searching for 6, 7, 8 and 9, without finding a solution. For 10 spikes, he found two graphs/topologies that satisfied the requirement. He turned these into "canonicalised" convex decahedra (https://en.wikipedia.org/wiki/Midsphere, https://mathworld.wolfram.com/CanonicalPolyhedron.html, https://www.georgehart.com/virtual-polyhedra/canonical.html), which have their edges on a unit sphere. Diogo Sousa developed a program to "jiggle" the positions of the vertices in order to converge the design to a state where all vertices are orthogonal to their opposing faces. Surprisingly, the program was not needed at the canonicalised decahedra already. Of the two spike solutions, we choose the one that looked most elegant, without any spikes too close together. Copyright (c) 2025, M. Oskar van Deventer. Frequently Asked Question: http://oskarvandeventer.nl/FAQ.html Buy Oskar puzzles at https://www.puzzlemaster.ca/browse/inventors/oskar/ (USA, CA) and puzzleguy.store/collections/oskar-van-deventer (EU).