David Hilbert wrote, The art of doing

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1 Fe 1. The u i u, warped! Abtracting the Rubik Cube Roice Nelon David Hilbert wrote, The art of doing mathematic conit in finding that pecial cae that contain all the germ of generality. Over the pat few decade, a growing group of puzzle enthuiat known a hypercubit have generalized the Rubik Cube in way that travere a wide expane of mathematical ground. The exploration have been a microcom of mathematical progre. Finding and tudying thee puzzle provide a rich way to approach varied topic in mathematic: geometry (higher dimenional, non-euclidean, projective), group theory, combinatoric, algorithm, topology, polytope, tiling, honeycomb, and more. For thi group of people, twity puzzle are more than jut a caual patime. Elegance i a core principle in their quet. group began by channg a more abtract property, namely, the dimenion. Don Hatch and Melinda Green created an exquiite working four-dimenional (or 3 4 ) analogue, which they called MacCube4D. Every property of thi puzzle i upped a dimenion: Face, ticker, and twit are three-dimenional rather than twodimenional. Fe 3 how the ordinary Rubik cube and the hyperpuzzle uing a central projection that reduce the dimenion by one; it i a if we are looking into a box, with the nearet face hidden. Hypercube We can change many propertie of the claic Rubik Cube, uch a it hape or twit center, to make new and intereting puzzle (ee fe 2). But the hypercubing 18 April 2018 : : Math Horizon : : g in u o r l r t r t n u li o t r u t it roun g int o

2 ro tion tri n l u iu li i nion l tl n r fi o t i u no t ig t r u ig r i The 33 Rubik Cube ha ticker that can live in a mind-boggling poible 4 tate. The hypercubical 3 ha ticker and the number of poible puzzle poition explode to an incomprehenible Calculating thi number i a challenge that will tet your group theory mettle! But a Edwin Abbott wrote in Flatland, In that bleed reon of Four Dimenion, hall we linger on the threhold of the Fifth, and not enter therein? Fe 4. A o o o o 5 o o t ti r r littl r u nion l u i u 4, ro t o t r 3 i, ro t o t o nion l ing The group didn t top at four dimenion. In 2006, a working five-dimenional puzzle materialized with hypercubical ticker and tate, puhing the boundarie of viualization. Fe 4 how a hadow of a hadow of a hadow of the five-dimenional object. Nonethele, a of mid2017, around 70 people have olved thi puzzle. In June 2010, Andrey Atrelin tunned the group by uing a creative viual approach to repreent a even-dimenional Rubik Cube. Ye, it ha been o on o i r l l toni ll, or t g in, t ri ro t ll, in our i nion : : Math Horizon : : April

3 u i u ro t r i ll onto r i l t o i nion l tiling o t r t i t n t r ogr i ll ro t onto t l n olved. Can you calculate the number of ticker on the 3 7? You may alo enjoy trying to work out the propertie of a two-dimenional Rubik Cube. What dimenion are the ticker? Of coure, we can play the hape-channg game in higher dimenion too, yielding a panoply of additional puzzle. There are five Platonic olid in three dimenion, but ix perfectly regular hape a dimenion up, and you can attempt to olve twity puzzle verion of all of them! Fe 5 how one of the mot beautiful in it pritine tate. Shape in arbitrary dimenion are called polytope, or polychora in four dimenion. In addition to the regular polychora, there are many uniform polychora, and quite a few have been turned into twity puzzle. Uniform polychora can break regularity in variou way. They may have multiple kind of three-dimenional face, or the face may be compoed of uniform (that i, Archimedean) polyhedra. Curved Twity Puzzle For God ake, I beeech you, ve it up. Fear it no le than enual paion becaue it too may take all your time and deprive you of your health, peace of mind and happine in life. No, thee were not deperate plea to a hypercubit about exceive puzzling adventure. Such were the word of Farka Bolyai to hi on Jáno, dicourang him from invetigating Euclid fifth potulate. Jáno continued nonethele, which led him into the wonderful world of hyperbolic geometry. We will alo not heed the elder Bolyai advice. Let ue topology to abtract away a different property of Rubik Cube it cubene. To do o, project the cube radially outward onto a phere (ee fe 6a). Notice that all the important combinatorial propertie remain. Furthermore, what were planar lice of the Rubik 20 April 2018 : : Math Horizon : : Cube are now circle on the phere urface. A twit imply rotate the portion of the urface inide one of thee twiting circle. In hort, we are viewing the Rubik Cube a a tiling of the phere by quare, liced up by circle on the urface. Inpired by thi example, we can conider other colored regular tiling, and a huge number of new twity puzzle become poible, ome living in the world of hyperbolic geometry! For two-dimenional urface, there are three geometrie with contant curvature: pherical, Euclidean, and hyperbolic. Thee geometrie correpond to whether the interior angle of a triangle um to greater than, equal to, or le than 180 degree, repectively. Intuitively, we can think of the urface of a phere, a flat plane, and a Pringle potato chip a repreentative urface for thee geometrie. Each urface of contant curvature can be tiled with regular polygon. The Schläfli ymbol encode regular tiling with jut two number, {p,q}. Thi denote a tiling by p-gon in which q uch polygon meet at each vertex. The value determine the geometry: Euclidean when equal to 4, pherical when le, and hyperbolic when greater. For example, {4,3} denote a tiling by quare with three arranged around each vertex, that i, the cube. A we aw in fe 6a, thi ve a tiling of the phere, and indeed, Euclidean geometry i the only one of the three geometrie that can live on the plane without any ditortion. A lovely way to repreent the other geometrie on the plane i via conformal, or angle preerving, map. The tereographic projection i a conformal map for pherical geometry. Fe 1 and 6b how the tereographic projection of the pherical Rubik Cube onto the plane. For hyperbolic geometry we ue the Poincaré dik, which quahe the infinite expane of the hyperbolic plane into a unit dik (ee fe 10). One challenge of turning Euclidean and hyperbolic tiling into twity puzzle i that unlike pherical tiling, which are finite, tiling of thee two geometrie go on forever. To overcome thi hurdle, we ben with a tiled urface, called the univeral cover; chooe a certain ubet of tile, called the fundamental domain; and identify it edge to form a quotient urface. Intuitively, we glue the edge of thi reon together to turn the infinite tiling into finite puzzle. Fe 7, 8, and 9 how a few example. One of the crown jewel of thi abtraction i the Klein quartic Rubik Cube, compoed of 24 hepata-

4 gon, three meeting at each vertex. It ha center, edge, and corner piece jut like the Rubik Cube. The univeral cover i the {7,3} hyperbolic tiling, and the quotient urface i a three-holed toru. Thi puzzle contain ome urprie; if you olve layer by layer, a i common on the Rubik Cube, you ll be left with two unolved face at the end intead of one. All thee puzzle and more are implemented in a program called MacTile. The puzzle count recently exceeded a thouand, with an infinite number of poibilitie remaining. t it toru n it uni un nt l o in red. u l on t r l o r in i outlin More Puzzle There are even more intriguing analogue that we have not yet een. Let me mention two of my favorite. The firt i another atonihing et of puzzle by Andrey Atrelin baed on the {6,3,3} honeycomb in t it u l on t l in three-dimenional hyperbolic pace, ottl n it uni r l o r (ee fe 11). The face are un nt l o in i outlin in r hexagonal {6,3} tiling, with three face meeting at each edge. Gluing via identification erve to make the underlying honeycomb finite in two ene: the number of face and the number of facet per face. If we take a tep back and conider where we tarted, thi puzzle ha altered the dimenion, the geometry, and the hape compared to the orinal Rubik Cube! t it u l on o The econd i a puzzle created ur t r l ro ti l n by Nan Ma baed on the 11-cell, n it uni r l o r an abtract regular polytope comun nt l o in i outlin poed of 11 hemi-icoahedral cell in red. (ee fe 11). Thi i a higheravenue to approach new puzzle now than 10 year dimenional couin of the Boy urface puzzle in ago. For example, there are no working puzzle in fe 9. The 11-cell can only live geometrically compoed of finite polyhedra. There are not yet unwarped in 10 dimenion, but Nan wa able to prepuzzle for uniform tiling of Euclidean or hyperbolic erve the combinatoric in hi depiction. geometry, in two or three dimenion. Uniform tiling With o many puzzle having been uncovered, one are not even completely claified, o further mathecould be forven for upecting there i not much matic i required before ome puzzle can be realized. more to do. On the contrary, there are arguably more : : Math Horizon : : April

5 l in u rti u i u uni r l o r uoti nt ur toru on t i t r ol Melinda Green ha been developing a phyical puzzle that i combinatorially equivalent to the 24. The idea of fractal puzzle ha come up, but no one ha yet been able to find a good analogue. In addition to the earch for puzzle, countle mathematical quetion have been aked or are ripe for invetigation. How many permutation do the variou puzzle have? What checkerboard pattern are poible? Which nd puzzle have the ame number of ticker a piece? How many way can you color the face of the 120-cell puzzle? What i God number for thee higher dimenional Rubik Cube; that i, what i the minimum number of move in which the puzzle can be olved, regardle of tart- r l ll ing poition? The avenue are limited only by our curioity. A John Archibald Wheeler wrote, We live on an iland urrounded by a ea of ignorance. A our iland of knowledge grow, o doe the hore of our ignorance. n Further Reading The MacCube4D webite (uperliminal.com/cube/ cube.htm) contain link to all the puzzle in thi article and to the hypercubing mailing lit. Burkard Polter (Mathologer) produced wonderful introductory video to MacCube4D and MacTile Cracking the 4D Rubik Cube with imple 3D trick (youtu.be/yhph1369owc) and Can you olve THE Klein Bottle Rubik Cube? (youtu.be/dvznh7-nlo) The following paper are freely available online: H. J. Kamack and T. R. Keane, The Rubik Teeract, (1982) John Stillwell, The Story of the 120-cell, Notice of the AMS 48, no. 1 (2001): Carlo H. Séquin, Jaron Lanier, and UC CET, Hypereeing the Regular Hendecachoron, Proc. ISAMA (2007): Roice Nelon i a oftware developer with a paion for exploring mathematic through viualization. He enjoy pending time with hi oul mate Sarah and their three cat, and prefer traveling on two or fewer wheel. il,, n in i u l in t r o t i nion l 22 April 2018 : : Math Horizon : : /