The 21st century does not seem to be too kind towards pentagons and pentagonal symmetries, at least when it comes to World Cup soccer balls: indeed the traditional (1970-1998) icosahedral ‘surface design’ (and stiching) consisting of 12 pentagons and 20 hexagons is giving way to other designs which we will briefly discuss here; more to the point, fourfold symmetries remininscent of the cube rather than the icosahedron have surfaced and deserve a closer look.

(I) fevernova (Korea/Japan 2002)

The pentagons and hexagons are still there, obscured however by four partly asymmetrical triangles dominating the ball’s surface and drastically reducing its symmetry:

This ball has the same symmetry as the famous Escher fish sphere: threefold rotations (with axes joining the meeting point of three tails and the meeting point of three fins) and twofold rotations (with axes ending at meeting chins at both ends); there is no antipodal symmetry, that is there is no *inversion*.

The * fevernova*‘s isometries are all present in the standard soccer ball’s icosahedral symmetry group (as presented for example here). The

*‘s symmetry group is, however, a*

**fevernova***subgroup*not only of the standard soccer ball symmetry group but also a subgroup of the cube’s symmetry group (as presented for example here). The following ‘grouped group table’ illustrates this fact, showing in green the cubical isometries that are also present in the

*:*

**fevernova**[In addition to threefold rotation (along a main diagonal) and fourfold rotation, the cube has two kinds of twofold rotation: the first kind, the 2-side twofold rotation (**RS**), also present in the * fevernova* (and the Escher fish sphere), has axes connecting the centers of two opposite sides (working for fourfold rotation as well); the second kind, the 2-edge twofold rotation (

**RE**), not present in the

*, has axes connecting the midpoints of two antipodal edges.*

**fevernova**Note here that, as the cube’s group table indicates, the presence of threefold rotations (**R3**) in the **fevernova***implies* the existence of 2-side twofold rotations (**RS**): indeed the product of two cubical threefold rotations is, depending on axes and angle senses, either the identity isometry **I** or a 2-side twofold rotation **RS** or a threefold rotation **R3**.

(The reader should probably think at this point about the cube formed by meeting points of tails and meeting points of fins in the Escher fish sphere!)]

(II) teamgeist (Germany, 2006)

The pentagons and the hexagons are gone, but there are now six ‘feet’ on the ball’s surface:

The * teamgeist*, a distant echo of the 1934-1966 balls, is in fact symmetrically equivalent to the pyritohedron and also to the notched-prisms toy (which I first encountered in a Montreal store in late November 2007):

As a subgroup of the cube, the * teamgeist* has inherited, in addition to threefold rotations and 2-side twofold rotations, inversion

**P**(antipodal symmetry), 4-edge reflections

**ME**(with reflection planes cutting four edges of the cube perpendicularly), and sixfold rotoreflections

**S6**(combinations of a 60-degree rotation about a threefold axis and an ‘equator’ reflection perpendicular to that axis):

(III) jabulani (South Africa, 2010)

This ball is … just the * fevernova* with some reflectional symmetry thrown in for more (or perhaps less) fun, that is with the partly asymmetrical triangles turned into fully symmetrical ones. The

*‘s 4-edge reflections*

**teamgeist****ME**are now replaced by 4-vertex reflections

**MV**(with reflection planes defined by two antipodal edges). Less obviously, the

*‘s sixfold rotoreflections*

**teamgeist****S6**have been replaced by fourfold rotorerflections

**S4**(along the axes of the 2-side twofold rotations

**RS**). The resulting symmetry group is that of the good old regular tetrahedron:

That the symmetry group of the regular tetrahedron is a subgroup of the symmetry group of the cube is not at all surprising if we recall how the former is placed inside the latter:

(IV) brazuca (Brazil 2014)

This dazzling ball is an all-rotations and no-(roto)reflections slick design: YES, the elusive 2-edge twofold rotation **RE** has finally arrived (with axes connecting antipodal ‘kissing tongues’), along with the more evident (and thoroughly cubical) fourfold rotation **R4** (with axes connecting antipodal ‘cross’ centers); NO, there is no full antipodal symmetry, for, as the group table below shows, the existence of inversion (**P**) would imply, in composition with the rotations, the existence of reflection:

This post is dedicated to my friend of 4+ decades Agis Zamanis, a mathematician, active goalkeeper and late-start marathon runner among other things: it was our trip to a sports store three days ago, with fast-pace-walking socks in mind, that made me realize that the soccer ball I had seen somewhere online less than a month ago — and saved because of its sublime symmetries — was so special, namely the 2014 World Cup soccer ball!

Tags: cube, group table, icosahedron, notched-prisms, pyritohedron, reflection, rotation, rotoreflection, soccer ball, symmetry group, tetrahedron, World Cup

June 15, 2014 at 6:42 pm |

This is exactly the information I was looking for about this beautiful ball (Brazuca)–without having one in my hands I couldn’t quite visualise the symmetry. I can see the black “triangles” where three patches come together have 3-fold rotational symmetry, and the centres of the white areas have 4-fold rotational symmetry. There are 6 white patches, 2 each of blue, orange and green boundaries arranged like opposite faces of a cube–there must 8 black triangles, at the 8 cubic vertices? Thanks google, and thanks George.

June 16, 2014 at 6:09 pm |

You are welcome, Lloyd! Yes, you do have the right ‘vision’ of the ball.

[Let me add, by the way, that colors (blue, green, orange) are not taken into account — or are all considered black, if you prefer — because that would further reduce the brazuca’s symmetry. (One might protest that threefold rotation should (?) take blue to blue rather than orange, etc)]