Abstracting Rubikrsquos CubeRoice Nelson

ldquoThe art of doing mathematics consists infinding that special case that contains allthe germs of generalityrdquo - David Hilbert

Over the past few decades a growing group ofldquohypercubistsrdquo have been discovering analogues of Ru-biklsquos cube traversing a wide range of mathematicalground Solving puzzles is a core pastime but thisgroup is about much more The explorations havebeen a microcosm of mathematical progress Find-ing and studying natural analogues provides a richway to approach varied topics in mathematics geom-etry (higher-dimensional non-Euclidean projective)group theory combinatorics algorithms topologypolytopes tilings honeycombs and more Eleganceis a core principle in the quest

(a) ldquoMegaminxrdquo uses a do-decahedral shape rather thana cube

(b) The ldquoHelicopter Cuberdquotwists around edges insteadof faces

Figure 1 We begin abstracting Rubikrsquos cube as soon aswe change some property

Those exposed to the twisty puzzle communityknow there are many properties of the classic 3x3x3Rubikrsquos Cube we can change to make new and interest-ing puzzles for example by altering the shape or thetwist centers as in Figure 1 The hypercubing groupbegan by changing a more abstract property namelythe dimension Don Hatch and Melinda Green wrotean exquisite working 4-dimensional 3x3x3x3 (or 34)analogue which they called MagicCube4D Usingdimensional analogy every property of this puzzle isupped a dimension Faces stickers and twisting are3D rather than 2D Using a central 4D rarr 3D pro-jection we see the hyperpuzzle as if you are lookinginto a box with the nearest face hidden Figure 2 putsthe 3D and 4D puzzles side-by-side to emphasize theanalogy

(a) The 33 projected soa 2-dimensional ldquoflatlanderrdquosees 5 of the 6 cube faces

(b) The 34 projected so a3-dimensional being sees 7of the 8 hypercube faces

Figure 2 Dimensional analogy and projection tricks canhelp us understand higher dimensional Rubikrsquos Cubes

The 33 Rubikrsquos Cube has 6 times 32 = 54 stickersthat can live in a mind-boggling 4325times 1019 possi-ble states The hypercubical 34 has 8 times 33 = 216stickers and the number of possible puzzle positionsexplodes to an incomprehensible 1756times 10120 Cal-culating this number is a challenge that will test yourgroup theory mettle

ldquoIn that blessed region of Four Dimen-sions shall we linger on the threshold ofthe Fifth and not enter thereinrdquo - EdwinAbbott Flatland

The group didnrsquot stop there In 2006 a working5-dimensional puzzle materialized with 10 times 34 =810 hypercubical stickers and 7017times 10560 statespushing the boundaries of visualization The pictureon the screen is effectively a shadow of a shadowof a shadow of the 5D object Nonetheless as ofmid 2017 around seventy individuals have solved thispuzzle In June 2010 Andrey Astrelin stunned thegroup by using a creative visual approach to representa 7-dimensional Rubikrsquos Cube Yes it has been solvedCan you calculate the number of stickers on the 37You may also enjoy using dimensional analogy towork out the properties of a 2-dimensional RubikrsquosCube What dimension are the stickers

Figure 3 A shadow of a shadow of a shadow of the 35Stickers are little hypercubes

Of course we can play the same game of chang-ing the shape in higher dimensions to yield a panoplyof additional puzzles There are 5 Platonic solids in

3 dimensions but 6 perfectly regular shapes a dimen-sion up and you can attempt to solve twisty puzzleversions of all of them Figure 4 shows one of themost beautiful in its pristine state

Figure 4 Magic120Cell or the ldquo4D Megaminxrdquo has 120dodecahedral faces It derives from the 120-Cell one of 6Platonic shapes in 4 dimensions

Shapes in arbitrary dimensions are called poly-topes or polychora in 4 dimensions In addition tothe regular polychora there are many uniform poly-chora and quite a few have been turned into twistypuzzles Uniform polychora can break regularity invarious ways They may have multiple kinds of 3Dfaces or the faces may be composed of uniform (akaArchimedean) polyhedra

ldquoFor Godrsquos sake I beseech you give itup Fear it no less than sensual passionsbecause it too may take all your time anddeprive you of your health peace of mindand happiness in liferdquo

No these were not desperate pleas to a hypercu-bist about excessive puzzling adventures Such werethe words of Farkas Bolyai to his son Janos discour-aging him from investigating Euclidrsquos fifth postulateJanos continued nonetheless which led him into thewonderful world of hyperbolic geometry We also didnot heed the advice

Letrsquos use topology to abstract away a differentproperty of Rubikrsquos Cube - its cubeness To do this

we project the cube faces radially outward onto asphere Mathematicians label the sphere S2 becausethey consider it a 2-dimensional surface rather thana 3-dimensional object Notice in Figure 5a that al-though the familiar cubeness is gone all of the impor-tant combinatorial properties remain Furthermorewhat were 2-dimensional planar slices of the Rubikrsquoscube are now 1-dimensional circles on the sphericalsurface A twist simply rotates the portion of thesurface inside one of these ldquotwistingrdquo circles

In short we are considering the Rubikrsquos cube asa 2-dimensional tiling of the sphere by squares slicedup by circles on the surface Why Because we canthen consider other colored regular tilings and a hugenumber of new twisty puzzles become possible someliving in the world of hyperbolic geometry

(a) Radially projected to thesphere S2

(b) Stereographically pro-jected from the sphere to thecomplex plane

Figure 5 The Rubikrsquos cube viewed as a 2-dimensionaltiling on a surface

In 2 dimensions there are three geometries withconstant curvature spherical Euclidean and hyper-bolic and each can be tiled with regular polygonsThese geometries correspond to whether the interiorangles of a triangle sum to greater than equal to orless than 180 degrees respectively The Schlafli sym-bol efficiently encodes regular 2-dimensional tilings inall of these geometries with just two numbers p qThis denotes a tiling of p-gons in which q such poly-gons meet at each vertex For example 4 3 denotesthe tiling of squares with three arranged around eachvertex ie the cube The value of (p minus 2)(q minus 2)determines the geometry Euclidean when equal to 4spherical when less and hyperbolic when greater

Euclidean geometry is the only one of the threethat can live on the plane without any distortion Alovely way to represent the others on the plane is via

conformal or angle preserving maps Stereographicprojection is a conformal map for spherical geometryIts analogue for hyperbolic geometry is the Poincaredisk which squashes the infinite expanse of the hyper-bolic plane into a unit disk These models have manybeautiful properties and the isometries (transforma-tions which preserve length) of all 3 models can bedescribed via a simple mathematical expression that

(a) Torus Rubikrsquos cube on theEuclidean universal cover

(b) Torus Rubikrsquos cubemapped to the Clifford torus

(c) Klein bottle Rubikrsquos cubeon the Euclidean universalcover

(d) Klein bottle Rubikrsquos cubemapped to a Lawson Klein bot-tle

(e) Hemi-icosahedron (or realprojective plane) Rubikrsquos cubeon the spherical universalcover

(f) Hemi-icosahedron Rubikrsquoscube mapped to the BryantndashKusner parametrization ofBoyrsquos surface

Figure 6 Example tiling analogues Note that there areother tilings that can also map to the surfaces on the right

acts on the complex plane the Mobius transforma-tions

f(z) =az + b

cz + d

You may have noticed that we have with anotherproblem to make puzzle analogues workable for Eu-clidean and hyperbolic tilings Spherical tilings arefinite but tilings of the other two geometries go onforever To overcome this final hurdle we take a fun-damental set of tiles and identify edges to be gluedup into a quotient surface This serves to make theinfinite tilings into finite puzzles Figure 6 show but afew examples We can even glue up a subset of tileson the sphere as in Figure 6e

One of the crown jewels of this abstraction isthe Klein Quartic Rubikrsquos cube composed of 24 hep-atagons three meeting at each vertex It has ldquocenterrdquoldquoedgerdquo and ldquocornerrdquo pieces just like Rubikrsquos cubeThe universal cover is the 7 3 hyperbolic tilingand the quotient surface it is living on turns out to be a3 holed torus This results in some solution surprisesif you solve layer-by-layer as is common on the Ru-bikrsquos cube yoursquoll find yourself left with two unsolvedfaces at the end instead of one

Figure 7 Klein Quartic Rubikrsquos cube on the hyperbolicuniversal cover The quotient surface is a 3 holed torus

All of these puzzles and more are implemented inprogram called MagicTile The puzzle count recentlyexceeded a thousand with literally an infinite numberof possibilities remaining

ldquoWe live on an island surrounded by asea of ignorance As our island of knowl-edge grows so does the shore of our ig-norancerdquo - John Archibald Wheeler

There are quite a few intriguing analogues thatI have not been able to describe here Let me justmention two of my favorite abstractions shown inFigure 8 The first is another astonishing set of puz-zles by Andrey are based on the 6 3 3 honeycombin 3-dimensional hyperbolic space H3 The facesare hexagonal 6 3 tilings with 3 faces meeting ateach edge Gluing via identifications serve to makethe underlying honeycomb finite in two senses thenumber of faces and the number of facets per face Ifwe take a step back and consider where we startedthis puzzle has altered the dimension the geometryand the shape compared to the original Rubikrsquos cube

The second is a puzzle written by Nan Ma basedon the 11-cell an abstract regular polytope composedof eleven hemi-icosahedral cells You might considerthis a higher dimensional cousin of the Boyrsquos surfacepuzzle we met earlier The 11-cell can only live geo-metrically unwarped in ten dimensions but Nan wasable to preserve the combinatorics in his depiction

With so many puzzles having been uncoveredone could be forgiven for suspecting there is not muchmore to do On the contrary there are arguably moreavenues to approach new puzzles now than ten yearsago For example there are no working puzzles inH3 composed of finite polyhedra There are not yetpuzzles for uniform tilings of euclidean or hyperbolicgeometry in 2 or 3 dimensions Uniform tilings arenot even completely classified so further mathemat-ics is required before some puzzles can be realizedMelinda has been developing a physical puzzle thatis combinatorially equivalent to the 24 The idea offractal puzzles have come up but no one has yet beenable to find a good analogue

In addition to the search for puzzles countlessmathematical questions have been asked or are ripefor investigation How many permutations do thevarious puzzles have What checkerboard patternsare possible Which nd puzzles have the same numberof stickers as pieces How many ways can you colorthe faces of the 120-Cell puzzle What is the nature ofGodrsquos number for higher dimensional Rubikrsquos cubesThe avenues seem limited only by our curiosity

(a) Magic Hyperbolic Tile 6 3 3 This is an in-space viewof the puzzle in 3-dimensional hyperbolic space

(b) Magic 11-Cell Here we see the puzzle scrambled

Figure 8 Two extremely exotic Rubikrsquos cube abstractions

Furthur Exploration

MagicCube4D websiteContains links to all the puzzles in this article and thehypercubing mailing list

Burkard Polster (Mathologer) produced wonderfulintroductory videos to MagicCube4D and MagicTileCracking the 4D Rubikrsquos Cube with simple 3D tricksCan you solve THE Klein Bottle Rubikrsquos cube

The following papers are freely available online

Kamack H J and T R Keane rdquoThe RubikTesseractrdquo (1982)

Stillwell John rdquoThe Story of the 120-cellrdquo No-tices of the AMS 481 (2001)

Sequin Carlo H Jaron Lanier and UC CET rdquoHy-perseeing the regular Hendecachoronrdquo Proc ISAMA(2007) 159-166

Roice is a software developer with a passion for ex-ploring mathematics through visualization He enjoysspending time with his soul-mate Sarah and their threecats and prefers traveling on two or fewer wheels

This is a preprint of an article published by Taylor ampFrancis Group in Math Horizons on March 7 2018available online Abstracting the Rubikrsquos Cube

The 33 Rubikrsquos Cube has 6 times 32 = 54 stickersthat can live in a mind-boggling 4325times 1019 possi-ble states The hypercubical 34 has 8 times 33 = 216stickers and the number of possible puzzle positionsexplodes to an incomprehensible 1756times 10120 Cal-culating this number is a challenge that will test yourgroup theory mettle

ldquoIn that blessed region of Four Dimen-sions shall we linger on the threshold ofthe Fifth and not enter thereinrdquo - EdwinAbbott Flatland

The group didnrsquot stop there In 2006 a working5-dimensional puzzle materialized with 10 times 34 =810 hypercubical stickers and 7017times 10560 statespushing the boundaries of visualization The pictureon the screen is effectively a shadow of a shadowof a shadow of the 5D object Nonetheless as ofmid 2017 around seventy individuals have solved thispuzzle In June 2010 Andrey Astrelin stunned thegroup by using a creative visual approach to representa 7-dimensional Rubikrsquos Cube Yes it has been solvedCan you calculate the number of stickers on the 37You may also enjoy using dimensional analogy towork out the properties of a 2-dimensional RubikrsquosCube What dimension are the stickers

Figure 3 A shadow of a shadow of a shadow of the 35Stickers are little hypercubes

Of course we can play the same game of chang-ing the shape in higher dimensions to yield a panoplyof additional puzzles There are 5 Platonic solids in

3 dimensions but 6 perfectly regular shapes a dimen-sion up and you can attempt to solve twisty puzzleversions of all of them Figure 4 shows one of themost beautiful in its pristine state

Figure 4 Magic120Cell or the ldquo4D Megaminxrdquo has 120dodecahedral faces It derives from the 120-Cell one of 6Platonic shapes in 4 dimensions

Shapes in arbitrary dimensions are called poly-topes or polychora in 4 dimensions In addition tothe regular polychora there are many uniform poly-chora and quite a few have been turned into twistypuzzles Uniform polychora can break regularity invarious ways They may have multiple kinds of 3Dfaces or the faces may be composed of uniform (akaArchimedean) polyhedra

ldquoFor Godrsquos sake I beseech you give itup Fear it no less than sensual passionsbecause it too may take all your time anddeprive you of your health peace of mindand happiness in liferdquo

No these were not desperate pleas to a hypercu-bist about excessive puzzling adventures Such werethe words of Farkas Bolyai to his son Janos discour-aging him from investigating Euclidrsquos fifth postulateJanos continued nonetheless which led him into thewonderful world of hyperbolic geometry We also didnot heed the advice

Letrsquos use topology to abstract away a differentproperty of Rubikrsquos Cube - its cubeness To do this

we project the cube faces radially outward onto asphere Mathematicians label the sphere S2 becausethey consider it a 2-dimensional surface rather thana 3-dimensional object Notice in Figure 5a that al-though the familiar cubeness is gone all of the impor-tant combinatorial properties remain Furthermorewhat were 2-dimensional planar slices of the Rubikrsquoscube are now 1-dimensional circles on the sphericalsurface A twist simply rotates the portion of thesurface inside one of these ldquotwistingrdquo circles

In short we are considering the Rubikrsquos cube asa 2-dimensional tiling of the sphere by squares slicedup by circles on the surface Why Because we canthen consider other colored regular tilings and a hugenumber of new twisty puzzles become possible someliving in the world of hyperbolic geometry

(a) Radially projected to thesphere S2

(b) Stereographically pro-jected from the sphere to thecomplex plane

Figure 5 The Rubikrsquos cube viewed as a 2-dimensionaltiling on a surface

In 2 dimensions there are three geometries withconstant curvature spherical Euclidean and hyper-bolic and each can be tiled with regular polygonsThese geometries correspond to whether the interiorangles of a triangle sum to greater than equal to orless than 180 degrees respectively The Schlafli sym-bol efficiently encodes regular 2-dimensional tilings inall of these geometries with just two numbers p qThis denotes a tiling of p-gons in which q such poly-gons meet at each vertex For example 4 3 denotesthe tiling of squares with three arranged around eachvertex ie the cube The value of (p minus 2)(q minus 2)determines the geometry Euclidean when equal to 4spherical when less and hyperbolic when greater

Euclidean geometry is the only one of the threethat can live on the plane without any distortion Alovely way to represent the others on the plane is via

conformal or angle preserving maps Stereographicprojection is a conformal map for spherical geometryIts analogue for hyperbolic geometry is the Poincaredisk which squashes the infinite expanse of the hyper-bolic plane into a unit disk These models have manybeautiful properties and the isometries (transforma-tions which preserve length) of all 3 models can bedescribed via a simple mathematical expression that

(a) Torus Rubikrsquos cube on theEuclidean universal cover

(b) Torus Rubikrsquos cubemapped to the Clifford torus

(c) Klein bottle Rubikrsquos cubeon the Euclidean universalcover

(d) Klein bottle Rubikrsquos cubemapped to a Lawson Klein bot-tle

(e) Hemi-icosahedron (or realprojective plane) Rubikrsquos cubeon the spherical universalcover

(f) Hemi-icosahedron Rubikrsquoscube mapped to the BryantndashKusner parametrization ofBoyrsquos surface

Figure 6 Example tiling analogues Note that there areother tilings that can also map to the surfaces on the right

acts on the complex plane the Mobius transforma-tions

f(z) =az + b

cz + d

You may have noticed that we have with anotherproblem to make puzzle analogues workable for Eu-clidean and hyperbolic tilings Spherical tilings arefinite but tilings of the other two geometries go onforever To overcome this final hurdle we take a fun-damental set of tiles and identify edges to be gluedup into a quotient surface This serves to make theinfinite tilings into finite puzzles Figure 6 show but afew examples We can even glue up a subset of tileson the sphere as in Figure 6e

One of the crown jewels of this abstraction isthe Klein Quartic Rubikrsquos cube composed of 24 hep-atagons three meeting at each vertex It has ldquocenterrdquoldquoedgerdquo and ldquocornerrdquo pieces just like Rubikrsquos cubeThe universal cover is the 7 3 hyperbolic tilingand the quotient surface it is living on turns out to be a3 holed torus This results in some solution surprisesif you solve layer-by-layer as is common on the Ru-bikrsquos cube yoursquoll find yourself left with two unsolvedfaces at the end instead of one

Figure 7 Klein Quartic Rubikrsquos cube on the hyperbolicuniversal cover The quotient surface is a 3 holed torus

All of these puzzles and more are implemented inprogram called MagicTile The puzzle count recentlyexceeded a thousand with literally an infinite numberof possibilities remaining

ldquoWe live on an island surrounded by asea of ignorance As our island of knowl-edge grows so does the shore of our ig-norancerdquo - John Archibald Wheeler

There are quite a few intriguing analogues thatI have not been able to describe here Let me justmention two of my favorite abstractions shown inFigure 8 The first is another astonishing set of puz-zles by Andrey are based on the 6 3 3 honeycombin 3-dimensional hyperbolic space H3 The facesare hexagonal 6 3 tilings with 3 faces meeting ateach edge Gluing via identifications serve to makethe underlying honeycomb finite in two senses thenumber of faces and the number of facets per face Ifwe take a step back and consider where we startedthis puzzle has altered the dimension the geometryand the shape compared to the original Rubikrsquos cube

The second is a puzzle written by Nan Ma basedon the 11-cell an abstract regular polytope composedof eleven hemi-icosahedral cells You might considerthis a higher dimensional cousin of the Boyrsquos surfacepuzzle we met earlier The 11-cell can only live geo-metrically unwarped in ten dimensions but Nan wasable to preserve the combinatorics in his depiction

With so many puzzles having been uncoveredone could be forgiven for suspecting there is not muchmore to do On the contrary there are arguably moreavenues to approach new puzzles now than ten yearsago For example there are no working puzzles inH3 composed of finite polyhedra There are not yetpuzzles for uniform tilings of euclidean or hyperbolicgeometry in 2 or 3 dimensions Uniform tilings arenot even completely classified so further mathemat-ics is required before some puzzles can be realizedMelinda has been developing a physical puzzle thatis combinatorially equivalent to the 24 The idea offractal puzzles have come up but no one has yet beenable to find a good analogue

In addition to the search for puzzles countlessmathematical questions have been asked or are ripefor investigation How many permutations do thevarious puzzles have What checkerboard patternsare possible Which nd puzzles have the same numberof stickers as pieces How many ways can you colorthe faces of the 120-Cell puzzle What is the nature ofGodrsquos number for higher dimensional Rubikrsquos cubesThe avenues seem limited only by our curiosity

(a) Magic Hyperbolic Tile 6 3 3 This is an in-space viewof the puzzle in 3-dimensional hyperbolic space

(b) Magic 11-Cell Here we see the puzzle scrambled

Figure 8 Two extremely exotic Rubikrsquos cube abstractions

Furthur Exploration

MagicCube4D websiteContains links to all the puzzles in this article and thehypercubing mailing list

Burkard Polster (Mathologer) produced wonderfulintroductory videos to MagicCube4D and MagicTileCracking the 4D Rubikrsquos Cube with simple 3D tricksCan you solve THE Klein Bottle Rubikrsquos cube

The following papers are freely available online

Kamack H J and T R Keane rdquoThe RubikTesseractrdquo (1982)

Stillwell John rdquoThe Story of the 120-cellrdquo No-tices of the AMS 481 (2001)

Sequin Carlo H Jaron Lanier and UC CET rdquoHy-perseeing the regular Hendecachoronrdquo Proc ISAMA(2007) 159-166

Roice is a software developer with a passion for ex-ploring mathematics through visualization He enjoysspending time with his soul-mate Sarah and their threecats and prefers traveling on two or fewer wheels

This is a preprint of an article published by Taylor ampFrancis Group in Math Horizons on March 7 2018available online Abstracting the Rubikrsquos Cube

we project the cube faces radially outward onto asphere Mathematicians label the sphere S2 becausethey consider it a 2-dimensional surface rather thana 3-dimensional object Notice in Figure 5a that al-though the familiar cubeness is gone all of the impor-tant combinatorial properties remain Furthermorewhat were 2-dimensional planar slices of the Rubikrsquoscube are now 1-dimensional circles on the sphericalsurface A twist simply rotates the portion of thesurface inside one of these ldquotwistingrdquo circles

In short we are considering the Rubikrsquos cube asa 2-dimensional tiling of the sphere by squares slicedup by circles on the surface Why Because we canthen consider other colored regular tilings and a hugenumber of new twisty puzzles become possible someliving in the world of hyperbolic geometry

(a) Radially projected to thesphere S2

(b) Stereographically pro-jected from the sphere to thecomplex plane

Figure 5 The Rubikrsquos cube viewed as a 2-dimensionaltiling on a surface

In 2 dimensions there are three geometries withconstant curvature spherical Euclidean and hyper-bolic and each can be tiled with regular polygonsThese geometries correspond to whether the interiorangles of a triangle sum to greater than equal to orless than 180 degrees respectively The Schlafli sym-bol efficiently encodes regular 2-dimensional tilings inall of these geometries with just two numbers p qThis denotes a tiling of p-gons in which q such poly-gons meet at each vertex For example 4 3 denotesthe tiling of squares with three arranged around eachvertex ie the cube The value of (p minus 2)(q minus 2)determines the geometry Euclidean when equal to 4spherical when less and hyperbolic when greater

Euclidean geometry is the only one of the threethat can live on the plane without any distortion Alovely way to represent the others on the plane is via

conformal or angle preserving maps Stereographicprojection is a conformal map for spherical geometryIts analogue for hyperbolic geometry is the Poincaredisk which squashes the infinite expanse of the hyper-bolic plane into a unit disk These models have manybeautiful properties and the isometries (transforma-tions which preserve length) of all 3 models can bedescribed via a simple mathematical expression that

(a) Torus Rubikrsquos cube on theEuclidean universal cover

(b) Torus Rubikrsquos cubemapped to the Clifford torus

(c) Klein bottle Rubikrsquos cubeon the Euclidean universalcover

(d) Klein bottle Rubikrsquos cubemapped to a Lawson Klein bot-tle

(e) Hemi-icosahedron (or realprojective plane) Rubikrsquos cubeon the spherical universalcover

(f) Hemi-icosahedron Rubikrsquoscube mapped to the BryantndashKusner parametrization ofBoyrsquos surface

Figure 6 Example tiling analogues Note that there areother tilings that can also map to the surfaces on the right

acts on the complex plane the Mobius transforma-tions

f(z) =az + b

cz + d

You may have noticed that we have with anotherproblem to make puzzle analogues workable for Eu-clidean and hyperbolic tilings Spherical tilings arefinite but tilings of the other two geometries go onforever To overcome this final hurdle we take a fun-damental set of tiles and identify edges to be gluedup into a quotient surface This serves to make theinfinite tilings into finite puzzles Figure 6 show but afew examples We can even glue up a subset of tileson the sphere as in Figure 6e

One of the crown jewels of this abstraction isthe Klein Quartic Rubikrsquos cube composed of 24 hep-atagons three meeting at each vertex It has ldquocenterrdquoldquoedgerdquo and ldquocornerrdquo pieces just like Rubikrsquos cubeThe universal cover is the 7 3 hyperbolic tilingand the quotient surface it is living on turns out to be a3 holed torus This results in some solution surprisesif you solve layer-by-layer as is common on the Ru-bikrsquos cube yoursquoll find yourself left with two unsolvedfaces at the end instead of one

Figure 7 Klein Quartic Rubikrsquos cube on the hyperbolicuniversal cover The quotient surface is a 3 holed torus

All of these puzzles and more are implemented inprogram called MagicTile The puzzle count recentlyexceeded a thousand with literally an infinite numberof possibilities remaining

ldquoWe live on an island surrounded by asea of ignorance As our island of knowl-edge grows so does the shore of our ig-norancerdquo - John Archibald Wheeler

There are quite a few intriguing analogues thatI have not been able to describe here Let me justmention two of my favorite abstractions shown inFigure 8 The first is another astonishing set of puz-zles by Andrey are based on the 6 3 3 honeycombin 3-dimensional hyperbolic space H3 The facesare hexagonal 6 3 tilings with 3 faces meeting ateach edge Gluing via identifications serve to makethe underlying honeycomb finite in two senses thenumber of faces and the number of facets per face Ifwe take a step back and consider where we startedthis puzzle has altered the dimension the geometryand the shape compared to the original Rubikrsquos cube

The second is a puzzle written by Nan Ma basedon the 11-cell an abstract regular polytope composedof eleven hemi-icosahedral cells You might considerthis a higher dimensional cousin of the Boyrsquos surfacepuzzle we met earlier The 11-cell can only live geo-metrically unwarped in ten dimensions but Nan wasable to preserve the combinatorics in his depiction

With so many puzzles having been uncoveredone could be forgiven for suspecting there is not muchmore to do On the contrary there are arguably moreavenues to approach new puzzles now than ten yearsago For example there are no working puzzles inH3 composed of finite polyhedra There are not yetpuzzles for uniform tilings of euclidean or hyperbolicgeometry in 2 or 3 dimensions Uniform tilings arenot even completely classified so further mathemat-ics is required before some puzzles can be realizedMelinda has been developing a physical puzzle thatis combinatorially equivalent to the 24 The idea offractal puzzles have come up but no one has yet beenable to find a good analogue

In addition to the search for puzzles countlessmathematical questions have been asked or are ripefor investigation How many permutations do thevarious puzzles have What checkerboard patternsare possible Which nd puzzles have the same numberof stickers as pieces How many ways can you colorthe faces of the 120-Cell puzzle What is the nature ofGodrsquos number for higher dimensional Rubikrsquos cubesThe avenues seem limited only by our curiosity

(a) Magic Hyperbolic Tile 6 3 3 This is an in-space viewof the puzzle in 3-dimensional hyperbolic space

(b) Magic 11-Cell Here we see the puzzle scrambled

Figure 8 Two extremely exotic Rubikrsquos cube abstractions

Furthur Exploration

MagicCube4D websiteContains links to all the puzzles in this article and thehypercubing mailing list

Burkard Polster (Mathologer) produced wonderfulintroductory videos to MagicCube4D and MagicTileCracking the 4D Rubikrsquos Cube with simple 3D tricksCan you solve THE Klein Bottle Rubikrsquos cube

The following papers are freely available online

Kamack H J and T R Keane rdquoThe RubikTesseractrdquo (1982)

Stillwell John rdquoThe Story of the 120-cellrdquo No-tices of the AMS 481 (2001)

Sequin Carlo H Jaron Lanier and UC CET rdquoHy-perseeing the regular Hendecachoronrdquo Proc ISAMA(2007) 159-166

Roice is a software developer with a passion for ex-ploring mathematics through visualization He enjoysspending time with his soul-mate Sarah and their threecats and prefers traveling on two or fewer wheels

This is a preprint of an article published by Taylor ampFrancis Group in Math Horizons on March 7 2018available online Abstracting the Rubikrsquos Cube

acts on the complex plane the Mobius transforma-tions

f(z) =az + b

cz + d

You may have noticed that we have with anotherproblem to make puzzle analogues workable for Eu-clidean and hyperbolic tilings Spherical tilings arefinite but tilings of the other two geometries go onforever To overcome this final hurdle we take a fun-damental set of tiles and identify edges to be gluedup into a quotient surface This serves to make theinfinite tilings into finite puzzles Figure 6 show but afew examples We can even glue up a subset of tileson the sphere as in Figure 6e

One of the crown jewels of this abstraction isthe Klein Quartic Rubikrsquos cube composed of 24 hep-atagons three meeting at each vertex It has ldquocenterrdquoldquoedgerdquo and ldquocornerrdquo pieces just like Rubikrsquos cubeThe universal cover is the 7 3 hyperbolic tilingand the quotient surface it is living on turns out to be a3 holed torus This results in some solution surprisesif you solve layer-by-layer as is common on the Ru-bikrsquos cube yoursquoll find yourself left with two unsolvedfaces at the end instead of one

Figure 7 Klein Quartic Rubikrsquos cube on the hyperbolicuniversal cover The quotient surface is a 3 holed torus

All of these puzzles and more are implemented inprogram called MagicTile The puzzle count recentlyexceeded a thousand with literally an infinite numberof possibilities remaining

ldquoWe live on an island surrounded by asea of ignorance As our island of knowl-edge grows so does the shore of our ig-norancerdquo - John Archibald Wheeler

There are quite a few intriguing analogues thatI have not been able to describe here Let me justmention two of my favorite abstractions shown inFigure 8 The first is another astonishing set of puz-zles by Andrey are based on the 6 3 3 honeycombin 3-dimensional hyperbolic space H3 The facesare hexagonal 6 3 tilings with 3 faces meeting ateach edge Gluing via identifications serve to makethe underlying honeycomb finite in two senses thenumber of faces and the number of facets per face Ifwe take a step back and consider where we startedthis puzzle has altered the dimension the geometryand the shape compared to the original Rubikrsquos cube

The second is a puzzle written by Nan Ma basedon the 11-cell an abstract regular polytope composedof eleven hemi-icosahedral cells You might considerthis a higher dimensional cousin of the Boyrsquos surfacepuzzle we met earlier The 11-cell can only live geo-metrically unwarped in ten dimensions but Nan wasable to preserve the combinatorics in his depiction

With so many puzzles having been uncoveredone could be forgiven for suspecting there is not muchmore to do On the contrary there are arguably moreavenues to approach new puzzles now than ten yearsago For example there are no working puzzles inH3 composed of finite polyhedra There are not yetpuzzles for uniform tilings of euclidean or hyperbolicgeometry in 2 or 3 dimensions Uniform tilings arenot even completely classified so further mathemat-ics is required before some puzzles can be realizedMelinda has been developing a physical puzzle thatis combinatorially equivalent to the 24 The idea offractal puzzles have come up but no one has yet beenable to find a good analogue

In addition to the search for puzzles countlessmathematical questions have been asked or are ripefor investigation How many permutations do thevarious puzzles have What checkerboard patternsare possible Which nd puzzles have the same numberof stickers as pieces How many ways can you colorthe faces of the 120-Cell puzzle What is the nature ofGodrsquos number for higher dimensional Rubikrsquos cubesThe avenues seem limited only by our curiosity

(a) Magic Hyperbolic Tile 6 3 3 This is an in-space viewof the puzzle in 3-dimensional hyperbolic space

(b) Magic 11-Cell Here we see the puzzle scrambled

Figure 8 Two extremely exotic Rubikrsquos cube abstractions

Furthur Exploration

MagicCube4D websiteContains links to all the puzzles in this article and thehypercubing mailing list

Burkard Polster (Mathologer) produced wonderfulintroductory videos to MagicCube4D and MagicTileCracking the 4D Rubikrsquos Cube with simple 3D tricksCan you solve THE Klein Bottle Rubikrsquos cube

The following papers are freely available online

Kamack H J and T R Keane rdquoThe RubikTesseractrdquo (1982)

Stillwell John rdquoThe Story of the 120-cellrdquo No-tices of the AMS 481 (2001)

Sequin Carlo H Jaron Lanier and UC CET rdquoHy-perseeing the regular Hendecachoronrdquo Proc ISAMA(2007) 159-166

Roice is a software developer with a passion for ex-ploring mathematics through visualization He enjoysspending time with his soul-mate Sarah and their threecats and prefers traveling on two or fewer wheels

This is a preprint of an article published by Taylor ampFrancis Group in Math Horizons on March 7 2018available online Abstracting the Rubikrsquos Cube

(a) Magic Hyperbolic Tile 6 3 3 This is an in-space viewof the puzzle in 3-dimensional hyperbolic space

(b) Magic 11-Cell Here we see the puzzle scrambled

Figure 8 Two extremely exotic Rubikrsquos cube abstractions

Furthur Exploration

MagicCube4D websiteContains links to all the puzzles in this article and thehypercubing mailing list

Burkard Polster (Mathologer) produced wonderfulintroductory videos to MagicCube4D and MagicTileCracking the 4D Rubikrsquos Cube with simple 3D tricksCan you solve THE Klein Bottle Rubikrsquos cube

The following papers are freely available online

Kamack H J and T R Keane rdquoThe RubikTesseractrdquo (1982)

Stillwell John rdquoThe Story of the 120-cellrdquo No-tices of the AMS 481 (2001)

Sequin Carlo H Jaron Lanier and UC CET rdquoHy-perseeing the regular Hendecachoronrdquo Proc ISAMA(2007) 159-166

Roice is a software developer with a passion for ex-ploring mathematics through visualization He enjoysspending time with his soul-mate Sarah and their threecats and prefers traveling on two or fewer wheels

This is a preprint of an article published by Taylor ampFrancis Group in Math Horizons on March 7 2018available online Abstracting the Rubikrsquos Cube