Cube on the ball

Expanding on a 4/21/15 workshop with 10th grade students at the Experimental High School of the University of Macedonia (at the invitation of their teacher, Nikos Terpsiadis), I delivered a 5/20/15 lecture at Aristotle University (at the invitation of Mathematics Department chairperson, Nikos Karampetakis) on Cubical Symmetries on World Cup soccer balls; both activities were of course closely related to my World Cup symmetries article (2/14/14) in this blog (and to the introduction of cubical and icosahedral symmetries to MAT 103 following my 2000-2001 sabbatical at SUNY Oswego).

During the workshop students were exposed to the isometries of the cube, and were subsequently asked to determine which isometries were present on the World Cup balls of 2002, 2006, 2010, 2014, whereas during the lecture I showed how answering the hardest of these questions may naturally lead to the concept of group, and how these soccer balls, and other balls as well, correspond to various subgroups of the cube. The goal of this article is to present these activities and ideas in some detail, translating from Greek where necessary and elaborating on parts of both the ppt and the video (part 1 & part 2).


Following a brief overview of World Cup soccer balls and cubical vs icosahedral stichings and designs (slides #4 through #8), the introduction to the isometries of the cube is illustrated by way of a die and permutation notation (slide #9). A new idea is the introduction of fourfold rotoreflection (S4) by way of a two-colored tennis ball (part 1, 21:30-23:20 — downward black ‘tongue’ to upward green ‘tongue’ to downward black ‘tongue’ and so on, four times up-and-down around the tennis ball, and about the axis defined by the middles of the two ‘tongues’) and likewise the introduction of sixfold rotoreflection (S6) by way of a certain 6-T ball (part 1, 26:30-28:15 — downward T to upward T to downward T and so on, six times up-and-down around the 6-T ball): this is done not only because the cube’s magnificent  ‘equator’ (sixfold rotoreflection plane corresponding/perpendicular to diagonal threefold rotation axis)


is so much harder to understand than the corresponding sixfold rotoreflection plane of the 6-T ball, but also because fourfold rotereflection is ‘too easy to see’ on the cube — too much symmetry may at times obscure the symmetries!

[Previously, and much in the same spirit, ‘simple’ balls (with minimal symmetry) were used (part 1, 14:10-16:30) to illustrate the difference between inversion and twofold rotation (two concepts that coincide in dimension two); in particular, the ball with the ‘back-to-back-thinking couple’ has no inversion (P)  … because, although at the antipodes of the couple we get the couple again, the man’s head is antipodal of the woman’s head and vice versa 🙂 (Note that inversion and twofold rotation may easily coexist, in the case of two antipodal isosceles triangles along the ball’s equator, for example — with reflection along the equator being a ‘bonus’, mind you 🙂 ) ]

In their worksheet (slide #10) the students had to provide a “yes” (NAI) or “no” (OXI) answer to seven questions, regarding the existence, in each one of the four World Cup soccer balls, of inversion, reflection, twofold rotation, fourfold rotation, threefold rotation, fourfold rotoreflection, and sixfold rotoreflection; further, and in the form of two ‘footnote’ challenging questions, they had to distinguish between 4-edge reflection (ME) and 4-vertex reflection (MV), and likewise between 2-side twofold rotation (RS) and 2-edge twofold rotation (RE) — if there to begin with, of course.

The answers appear in slides #11 through #14. Looking here at the ‘footnote’ questions, we see that the easier ones to answer are the ones about reflection, thanks to the presence of threefold rotation (R3): since the corresponding axes are contained in reflection planes in the case of the jabulani (2010) but not in the case of the teamgeist (2006), we conclude that the jabulani has 4-vertex reflection (MV) whereas the teamgeist has 4-edge reflection (ME).

At a higher level of difficulty, unless one resorts to counting axes — for the cube has three axes of 2-side twofold rotation (RS) and six axes of 2-edge twofold rotation (RE) — we decide that, whereas the brazuca (2014) has obviously both kinds of twofold rotation, both the teamgeist and the jabulani have only 2-side twofold rotation (RS): this follows from the rather subtle observation (slide #15) that their twofold rotation axes are intersections of the same kind of reflection planes (ME in the case of teamgeist, MV in the case of jabulani); indeed the axes of the cube’s 2-side twofold rotation may be obtained either as intersections of two 4-vertex reflections or as intersections of two 4-edge reflections, whereas the axes of the cube’s 2-edge twofold rotations require one 4-vertex reflection and one 4-edge reflection. These thoughts lead naturally to the concept of composition of perpendicular reflections, either in a geometrical context (slide #16, not easy to extend to compositions of other types of isometries) or in an algebraic context, by way of permutation multiplication (slide #17, easy to extend to other isometries).

The last workshop question concerns the nature of the twofold rotation in the fevernova (2002): this is again a 2-side twofold rotation (RS), either because there are three twofold rotation axes in the fevernova or because the fevernova‘s twofold rotations are, unlike 2-edge twofold rotations (RE), closed under multiplication/composition (slide #18, right below); indeed always RS * RS = RS (or perhaps RS * RS = I, where I stands for the identity tansformation), whereas RE * RE= RS is possible, for example (15)(26)(34) (12)(34)(56) = (16)(25). (A third way of answering this question would be an ‘external’ one, requiring the observation that the product of two threefold cubical rotations (R3) may be a 2-side twofold rotation (RS), but not a 2-edge twofold rotation (RE).)


The above notion of closedness under isometry multiplication leads naturally to the concepts of (sub)group and group table. The closedness of 2-side twofold rotations is in particular celebrated in the brazuca II ball (slide #19, complete with full group table, right below), which has no other isometries (except of course for the identity (I)), and whose subgroup-of-cube status is illustrated in slide #20, by way of grouped group table (employed for the same purpose for the four balls in slides #21 through #24, as well as in World Cup symmetries — where, of course, all answers to the workshop questions may also be found 🙂 ). [Note here that the brazuca II (part 2, 3:32-5:20) may be viewed as a tennis ball with asymmetrical ‘tongues’ — much in the same way the tennis ball ‘tongues’ may be viewed as less symmetrical versions of the die’s 3-1-4 and 2-6-5 ‘sections’.]


Despite this affinity, the brazuca II appears at first not to be a subgroup of the tennis ball, as the latter contains only two 2-side twofold rotations (slide #25): this is not correct, and the subgroup status of the brazuca II within the tennis ball becomes clear if one observes that the latter contains three mutually perpendicular twofold rotations that are closed under isometry composition. (The distinction between 2-edge twofold rotations and 2-side twofold rotations — and likewise between 4-edge reflections and 4-vertex reflections — becomes … counterproductive in the absence of threefold rotation!)

The tennis ball’s full 8 x 8 group table is exhibited in slide #26, whereas the 16 x 16 group table of a ‘four-ovals’ ball (part 2, 13:15-14:02, slides #28 and #29), geometrically/symmetrically equivalent to a square prism, that does contain the tennis ball as a subgroup (slide #30) is shown in full in slide #27; and two more subgroups of order 8, the ‘four-feet’ ball (part 2, 14:50-16:02), equivalent to the rectangular parallelepiped, and the ‘four-wings’ ball (part 2, 16:05-16:56), each of them containing the brazuca II as a subgroup, are shown in slides #31 through #35. (An interesting observation here is that the brazuca II has the same isometries as a ‘two-wings’ ball, where the two ‘wings’ (rectangles) are antipodal of each other.)


Slide #36 (right above), taken from wikipedia and employing the Coxeter notation, shows the cube’s full subgroup structure (which we are merely reproducing here ‘experimentally’). We see that the soccer balls of the last three World Cups correspond, remarkably, to the three subgroups of the cube of order 24 (teamgeist = [3+, 4], jabulani = [3, 3], brazuca = [4, 3]+), each of them containing the fevernova = [3, 3]+ as a subgroup of order 12 (slide #37).

Slides #38 through #43 are devoted to the 6-T ball ([2+, 6], the only other subgroup of the cube of order 12) and its three subgroups of order 6: these are the only subgroups of the cube of order 6 and are ‘orphan’ … in the sense that I have not (yet) found commercial balls corresponding to them, so they are provided only by way of group tables — but check toward the end of this post, too! The 6-T ball is not a subgroup of any World Cup soccer ball: it is not a subgroup of either the jabulani or the brazuca because it has, as we have already seen, sixfold rotoreflection (S6); and it is not a subgroup of the teamgeist because it has four-vertex reflection (MV) — or, in broader terms, its reflection planes contain a threefold rotation axis 🙂 [A non-convex solid isomorhic to the 6-T ball is obtained by raising two regular tetrahedra off opposite sides of a Star of David, each one having one of the equilateral triangles as its base.]

Slide #44 — shown below with a couple of corrections involving the tennis ball — conludes the show with a partial cube diagram (subgroup relations), showing all the balls discussed here … in pictures rather than scientific notation. Two subgroups of order 8 and all subgroups of order lower than 6 but one (brazuca II = [2, 2]+) have not been included, but they are easily found by way of group tables etc The ‘four-ovals’ ball = [2, 4] is the only subgroup of order 16 and contains all subgroups of order 8, including the tennis ball = [2+, 4] (also a subgroup of the jabulani), the ‘four-feet’ ball = [2, 2] (also a subgroup of the teamgeist), and the ‘four-wings’ ball = [2, 4]+ (also a subgroup of the brazuca). [The die at the top may of course be replaced by any ball with full cubical symmetry, like this one for example.]


The three ~ signs above would push the lecture into a next level and concept, that of group isomorpism: the two ‘orphan’ groups of order 6 on the right (slides #42 and #43) are isomorphic by way of mapping every 4-vertex reflection of #42 to the unique 2-edge twofold rotation perpendicular to it, for example (13)(46) to (14)(25)(36), and every threefold rotation to itself; the tennis ball (slide #29) is isomorphic to the ‘four-wing’ ball (slide #34) by multiplying every reflection and rotoreflection by (16) and leaving everything else invariant; and the jabulani is isomorphic to the brazuca by mapping 4-vertex reflections to 2-edge twofold rotations as above (extending the isomorphism between the two subgroups of order 6) and every fourfold rotoreflection to the inverse of the fourfold rotation naturally associated with it, like (16)(2453) to (2354), and leaving everything else invariant.

An important lesson here is that two subgroups may be geometrically distinct yet algebraically identical. It follows in particular, by way of the ‘four-wings’ ball and the brazuca, that the tennis ball is an algebraic subgroup of the jabulani:


This isomorphism, mapping reflections to twofold rotations and vice versa, may create the impression that, just as the brazuca is isomorphic to the jabulani algebraically but not geometrically, the tennis ball is an algebraic but not geometrical subgroup of the jabulani: this is quite easily seen not to be correct, especially in case one views the jabulani as a symmetrical version of the fevernova as seen further above, that is consisting of four symmetrically placed curvy equilateral triangles A, B, C, D, and subsequently views the A-B, C-D pairs as equivalent to the ‘tongues’ of the tennis ball; the tennis ball subgroup consists then of those jabulani isometries that happen to either swap or preserve the two pairs (OR the 3-1-4 and 2-6-5 ‘sections’ previously mentioned 🙂 ):

(AB) = (24)(35), (CD) = (23)(45), (AB)(CD) = (25)(34), (AC)(BD) = (16)(34), (AD)(BC) = (16)(25), (ADBC) = (16)(2354), (ACBD) = (16)(2453)

It is clear here that, consistently with the previously stated principle against distinguishing — in the absence of threefold rotation — between 2-edge and 2-side twofold rotations, and likewise between 4-edge and 4-vertex reflections, we have replaced the tennis ball’s twofold rotations (16)(23)(45), (16)(24)(35) by the twofold rotations (16)(25), (16)(34), and likewise we have replaced the tennis ball’s reflections (25), (34) by the reflections (24)(35), (23)(45):


It follows from the observations above and the well known full isomorphism between the jabulani and the regular tetrahedron (just join the centroids of the jabulani‘s four curvy equilateral triangles) that the tennis ball is a subgroup of the tetrahedron (a well known fact mentioned for example here or here in passing). (That the tennis ball is a subgroup of the jabulani also becomes clear if one removes either two antipodal ‘units’ or the other four ‘units’ from the fully isomorphic to the jabulani six-‘units’ ball exhibited at the end of the lecture (part 2, 33:40-37:15), receiving a ball fully isomorphic to the tennis ball.)

It should be easy now to verify that one of the two subroups of order 8 that has not been discussed so far, namely the square pyramid (or merely … the square), {I, (25), (34), (23)(45), (24)(35), (2453), (2354), (25)(34)} = [4], is algebraically isomorphic to the tennis ball (therefore to the ‘four-wings’ ball as well); it can be represented on a ball by four equally spaced congruent isosceles triangles, all pointing toward the north pole … whereas inverting every other triangle leads, once again, to the tennis ball — but geometrically, too, this time — and renders the jabulani a special case of it (with isosceles triangles turning equilateral)!  (This last idea, applied to six isosceles triangles, leads once again to the 6-T ball, [2+, 6], whereas … slanting the six up-and-down isosceles triangles — in such a way that reflections and twofold rotations are gone but inversion, threefold rotation and sixfold rotoreflection survive — leads to the ‘orphan’ subgroup of slide #41, [2+, 6+]; as for slides #42 and #43, those correspond to the three-straight-isosceles-triangles ball, that is the triangular pyramid (or merely the equilateral triangle), [3], and to a ‘three-wings’  ball, [2, 3]+, analogous to the ‘four-wings’ ball, respectively.)

[As for the last subgroup of order 8, {I, (16)(25)(34), (16), (2453), (2354), (16)(2453), (16)(2354), (25)(34)} = [4+, 2], that one may be represented on a ball by four equally spaced congruent isosceles triangles along the equator, all pointing in the same direction. (Note here that if “four” is replaced by “three” … we simply depart from the cube, ending up with a ball, which — just like a ball already existing out there as a basketball ball, basically a ball with the equator and three meridians, an ‘aligned’ version of the 6-T ball also known as triangular bipyramid 🙂  — has reflection and threefold rotation perpendicular to each other!)]

… Only a few days before their fourth birthday, I would like to dedicate this post to my partner’s twin nieces, Sophia and Sandy: their two-colored tennis ball toy, complete with ‘magnetic’ racket, inspired me to delve into its symmetries — and realize that a rotoreflection is what ‘mirrors’ one half of the tennis ball to the other half — in July 2013!


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