Cubically yours, icosahedron

In the previous article in this blog, Cube on the ball, we saw how a small collection of commercial balls, notably World Cup soccer balls, may help us understand and verify the cube’s subgroup hierarchy. Here we will see how the icosahedron/dodecahedron’s subgroups are related to the cube/octahedron’s subgroups; we will see in particular that, despite the higher order of its symmetry group (120 vs 48), the icosahedron has essentially nothing new to add to the cube in terms of subgroups!

First we need of course to understand the isometries of the icosahedron. We will do that by way of an old fashion soccer ball consisting of 20 white hexagons and 12 black or whatever pentagons, widely available in street kiosks (periptera) and beyond in toy size … and by way of labeling its pentagons exactly as below:


Rather ironically in view of what we saw in World Cup symmetries and Cube on the ball, back in Fall 2001 I had seen it fit to ‘identify’ the soccer ball with the icosahedron — after all, the 2002 ball had not surfaced yet, and I am not old enough to remember the pre-1970 balls ūüôā At any rate, above you see a classification of the icosahedron’s isometries into ten kinds of isometries — exactly as in the case of the cube, but with no ‘grouped group table’ this time — with the new element being fivefold & tenfold isometries that need to be understood before going further. (And a small detail that also needs to be pointed out: D is missing from the diagram above … simply because it is A’s antipodal!)

Following the examples given above, and considering always rotations (and rotoreflections) about the rotation axis A-D¬†(and the ‘equator’ reflection plane perpendicular to it): a short fivefold rotation, like (BGFJK)(CHIEL), is what one might indeed ‘expect’, rotating ten out of twelve pentagons in an ‘upper’ circle (B, G, F, J, K) and in a ‘lower’ circle (C, H, I, E, L), moving each pentagon to the pentagon ‘right next’ to it (by 72 degrees); a long fivefold rotation, like (BFKGJ)(CILHE), ‘jumps’ every other pentagon in both circles, progressing each time by 144 degrees, returning every pentagon to its initial position after two, rather than just one, full circlings of the ball; a short fivefold rotoreflection, like (AD)(BHGIFEJLKC), swaps A & D and ‘rotates’ the other ten pentagons up and down around the ball, moving back and forth from the upper circle to the lower circle, progressing each time by 36 degrees; and a long fivefold rotoreflection, like (AD)(BIJCGEKHFL), swaps A & D and moves every pentagon of the upper circle to a pentagon of the lower circle 108 degrees away, and vice versa, returning every pentagon to its initial position after three circlings of the ball. [Observe that the square of (AD)(BHGIFEJLKC) equals (BGFJK)(CHIEL), whereas the square of (AD)(BIJCGEKHFL) equals the inverse of (BFKGJ)(CILHE), that is (BJGKF)(CEHLI).]


Considering the subgroup diagram above, taken from wikipedia, and comparing it to the analogous wikipedia diagram for the cube, we see that the icosahedron’s ‘large’ subgroups (and their own subgroups) were, in essence, also present in the cube; indeed we see above the teamgeist, [3+, 4], the fevernova, [3, 3]+, the rotation subgroup [5, 3]+, completely analogous to the brazuca‘s [4, 3]+, the 6-T ball, [2+, 6], and a ‘new’ subgroup, [2+, 10], ‘analogous’ to the 6-T ball.

That the teamgeist is a subgroup of the icosahedron becomes clear if we view as its six ‘feet’ the ‘units’ CL, FG, BH, EJ, AK, DI. These ‘units’ give away instantly the three reflections and three twofold rotations, whereas the axes for the threefold rotations are ABG-DEL, AFJ-CHD, BCK-EFI, GHI-JKL; the same axes work for sixfold rotoreflections like ¬†(ALBDGE)(CHIFJK) (axis ABG-DEL). Notice that all the isometries discussed here either preserve or swap/rotate the six ‘units’, the sixfold rotoreflection (ALBDGE)(CHIFJK) for example has the effect AK —>¬†CL —>¬†BH —>¬†DI —> FG —>¬†EJ —>¬†AK.

It is easy now to see how the fevernova is a subgroup of the teamgeist: the regular tetrahedron with vertices at the ‘centroids’ of the ‘triangles’ ABG, CHD, EFI, JKL has reflections (and fourfold rotoreflections) that do not work for the icosahedron, but its threefold rotations are the teamgeist‘s threefold rotations discussed above, and its twofold rotations, of axes¬†AK-DI, BH-EJ, CL-FG, work for the teamgeist, too.

Due to their fourfold symmetries, there is no room in the icosahedron for either the jabulani or the brazuca, but the latter has a close relative inside the icosahedron, consisting of all the icosahedral rotations; it is not easy to ‘represent’ this subgroup on our labeled soccer ball, but there are, thankfylly, other intriguing examples:


The example on the left is an origami-like creation by Krystyna Burczyk, hosted by George Hart at the Museum of Mathematics. The example on the right is a pendant light by Tom Dixon which I first saw during a late evening walk in a Thessaloniki store last summer. And right in the middle you see a precious ball the internet location of which I unfortunately no longer recall: there is fivefold rotation at the middle of every regular pentagon, threefold rotation wherever three irregular pentagons come together symmetrically, and twofold rotation in the middle of every two-pentagon unit. (There are not that many examples of commercial balls with partial icosahedral, yet not subcubical, symmetry!)

The 6-T ball subgroup, [2+, 6], is easily recreated on our labeled soccer ball either by keeping only the ABG and DEL triangles or by keeping only the remaining triangles, FKH and CIJ: in both cases there are three reflection planes (each of them bisecting precisely two pentagons, antipodal of each other), one threefold rotation axis (defined by the triangles above) that also acts as a sixfold rotoreflection axis (of effect either (ALBDGE) or (CHIFJK)), three twofold rotations (with axes that may be written either as¬†AE-DB, AL-DG, BL-EG¬†or as¬†FJ-CH, JK-HI, FI-CK), and, last but not least, inversion (either in the form (AD)(BE)(GL) or in the form (CF)(IK)(HJ)). [Alternatively, we may also represent [2+, 6] as either¬†AEL-LAB-BLD-DBG-GDE-EGA¬†or FIJ-JFK-KJC-CKH-HCI-IHF, with the six-up-and-down-isosceles-‘triangles’ set remaining invariant under the isometries already discussed, respectively.]

If instead of removing the ‘north pole’ (ABG) and the ‘south pole’ (DEL) corresponding to a threefold axis, as we did in the case of the 6-T ball subgroup above (second possibility), we remove the ‘north pole’ and the ‘south pole’ corresponding to a fivefold axis like A-D …¬†we end up with a subgroup where all threefold rotations and sixfold rotoreflections, as well as all fivefold rotations and rotoreflections, save for those about A-D, are gone; as above, there are five — rather than three — reflection planes (each of them bisecting two pentagons) and five — rather than three — twofold rotations (with axes BH-EJ, HG-JL, GI-LK, IF-KC, FE-CB). ¬†All together — don’t forget inversion and, goes without saying, the identity — we end up with a 20-element subgroup known as [2+, 10]: its similarities to the 12-element subgroup [2+. 6] are clear! (And both groups are of course very similar to the tennis ball ([2+, 4]) — a subgroup of the cube but not of the icosahedron — with the notable difference that the latter has no inversion.) We may also represent [2+, 10] as¬†ACH-DBG-AHI-DGF-AIE-DFJ-AEL-DJK-ALC-DKB (with the ten-up-and-down isosceles-‘triangles’ set remaining invariant under the short and long tenfold rotoreflections (AD)(BHGIFEJLKC) & (AD)(BCKLJEFIGH) and (AD)(BIJCGEKHFL) & (AD)(BLFHKEGCJI) — and the short and long fivefold rotations they ‘create’ — and under the reflections bisecting each isosceles ‘triangle’, as well as under the five twofold rotations listed already).

Recall at this point (Cube on the ball) that the three subgroups of order 6 of the 6-T ball ([2+, 6]) were the equilateral triangle or triangular pyramid ([3]), the three-‘wings’ ball ([2, 3]+), and the six-up-and-down-slanted-isosceles-‘triangles’ ball ([2+, 6+]). On the icosahedron these groups are represented by ABG (with the three-pentagons set remaining invariant under¬†the threefold rotations about¬†ABG-DEL and under the reflections that bisect the sides of the equilateral ‘triangle’ ABG), CH-JK-IF (with the three-‘wings’ set remaining invariant¬†under the threefold rotations about¬†ABG-DEL and under the twofold rotations about CH-FJ, JK-HI, IF-KC), and AHI-LIF-BFJ-DJK-GKC-ECH¬†(with the six-up-and-down-slanted-isosceles-‘triangles’ set remaing invariant under under the threefold rotations and sixfold rotoreflections about¬†ABG-DEL, as well as under inversion), respectively.

The corresponding subgroups of order 10 of [2+, 10] are, in complete analogy to the subgroups of order 6 of [2+, 6] (or the subgroups of order 4 of the tennis ball for that matter), the regular pentagon or pentagonal pyramid ([5]), the five-‘wings’ ball ([2, 5]+), and the ten-up-and-down-slanted-isosceles-‘triangles’ ball ([2+, 10+]). Replacing the threefold axis ABG-DEL by the fivefold axis A-D we get analogous representation of these ‘new’ groups by BGFJK (with the five-pentagons set remaining invariant under the short and long fivefold rotations about A-D and under the reflections that bisect the sides of the regular ‘pentagon’ BGFJK), BC-GH-FI-JE-KL (with the five-‘wings’ set remaining invariant under the short and long fivefold rotations about A-D and under the twofold rotations about BC-EF, GH-LJ, FI-CK, JE-HB, KL-IG),¬†and BJL-HLK-GKC-ICB-FBH-EHG-JGI-LIF-KFE-CEJ (with the ten-up-and-down-slanted-isosceles-‘triangles’ set remaing invariant under¬†the short and long fivefold rotations and rotoreflections about A-D, as well as under inversion), respectively.

[2, 5]+Messi

[2, 5]+ … in Thessaloniki … and in Lionel Messi’s tender hand

Sitting inside all three subgroups of [2+, 10] discussed above is the group [5]+, that is the rotation subgroup of the regular pentagon, consisting of two short fivefold rotations (inverses of each other), two long fivefold rotations (inverses of each other), and, of course, the identity. If one insists on a representation on our labeled soccer ball, a beautiful way to do that is BCL-GHC-FIH-JEI-KLE: it is easy to check that the only non-trivial isometries leaving this five-slanted-isosceles-‘triangles’ set invariant are (BGFJK)(CHIEL), (BKJFG)(CLEIH), (BFKGJ)(CILHE), (BJGKF)(CEHLI).

An analogous representation for the group [3]+, the rotation subgroup of the equilateral triangle, sitting inside the subgroups [3], [2, 3]+, and [2+, 6+], is¬†AIE-BJL-GCD: the only non-trivial isometries leaving this three-slanted-isosceles-‘triangles’ set invariant are (ABG)(FKH)(CIJ)(DEL) and (AGB)(FHK)(CJI)(DLE).

To verify now that [2, 2] (‘four-feet’ ball, parallelepiped) is indeed a subgroup of [3+, 4] (teamgeist), one simply needs to ‘shrink’ the latter’s representation from¬†CL-FG-BH-EJ-AK-DI¬†to¬†CL-FG-BH-EJ: the threefold rotations (and associated sixfold rotoreflections) no longer work, and we are left with three reflections, three twofold rotations, the inversion and the identity.¬†A further reduction to¬†CL-FG¬†— or merely to C-L or C-G — yields the subgroup [2] (rectangle), whereas C-F yields [2+, 2], a subgroup corresponding to [2+. 6] and [2+, 10] and¬†consisting of one twofold (rather than threefold or fivefold) rotation, one reflection, the inversion, and the identity. (But … which reflection, and which twofold rotation? There are five choices for each, and they must be paired in such a way that the product of each pair is the inversion; in other words, the reflection and the twofold rotation must be perpendicular to each other, like for example (AG)(DL)(HK)(IJ) and (AL)(BE)(CF)(DG)(HI)(JK).)

It should be clear by now that, due to the very rich symmetry of our soccer ball, it gets increasingly difficult to represent on it the smaller subgroups, like [2]+ (one twofold rotation — parallelogram), which may also be viewed as the rotation subgroup of the rectangle ([2]), analogously to [3]+ (rotation group of the fevernova‘s¬†‘unit’, also rotation subgroup of the equilateral triangle, [3]), [4]+ (rotation group of the brazuca‘s cross-like ‘unit’, also rotation subgroup of the square ([4]), a subgroup of the cube but not of the icosahedron), [5]+ (rotation subgroup of the regular pentagon ([5])), and other cyclic groups beyond the cube and the icosahedron. We have already obtained representaions for [3]+ and [5]+, a representation for [2]+ could be a ‘parallelogram’ like ACDG, invariant under only the twofold rotation (AD)(BH)(CG)(EJ)(FL)(IK). Likewise, the ‘isosceles triangle’ ABL represents [1], the smallest dihedral group, remaining invariant only under the reflection (AB)(CJ)(DE)(FH).

How could we represent the group [2+, 2+] (inversion only)? We need two antipodal, totally asymmetrical¬†([1]+) sets on the 12-pentagon ball: one working possibility is ABLE-DEGB. (It seems at first that [2+,2+] may also be represented on a ball by two antipodal slanted isosceles triangles: this is not quite right, as the isosceles triangle’s inner reflection line turns into a reflection — about a ‘new’ equator plane defined by the two antipodal (hence parallel) reflection lines — for the entire pair of antipodal isosceles triangles; we end up, once again, with [2+, 2].)

[At the risk of going slightly off topic, let us mention here that [2+, 4+] — a subgroup of order 4 of the tennis ball and the cube but not of the icosahedron — may be represented on a ball by four up-and-down slanted isosceles triangles, in full analogy to [2+, 6+], [2+, 10+] and even [2+, 2+] right above, but with a curious twist, that is the lack of inversion: it consists of fourfold rotoreflection and twofold rotation about a single axis. (This explains the lack of ‘inclusion segment’ from [2+, 2+] to [2+, 4+] in the cube’s subgroup diagram, but not from [2+, 2+] to either [2+, 6+] or [2+, 10+] in the icosahedron’s subgroup diagram above: the corresponding ‘inclusion segments’ have apparently been omitted by accident (just like the [2]+ to [2+, 6+] ‘connection’, in both¬†subgroup diagrams this time).)]

And — finally — how to represent [2, 2]+, that is that good old brazuca II, geometrically equivalent to a ‘two-wings’ ball, which features only three mutually perpendicular twofold rotations? Just as in the case of [5, 3]+, the icosahedron’s full rotation subgroup, I do not see a solution on the 12-pentagon ball — when you start with so many reflections … it may not be easy to have a rotations-only party!


I would like to dedicate this post to the manager of Jo-Ann Fabric and Craft Stores, Oswego (summer 2001): when she understood that I needed more than the 4 or 5 black-and-white toy soccer balls available at her store … she suggested that she could probably get more balls from other regional Jo-Ann stores she had to visit anyway … and surely she ended up providing me with no fewer than 28 — a perfect number for MAT 103’s last lab!


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