The observation presented below is definitely not new, as the ubiquitous Marjorie Senechal came up with it 40 years ago in Point Groups and Color Symmetry; I know this thanks to Alain Bossavit’s Point Groups (1995). Still, I think it is worth discussing it here, in some detail and color, due to its great didactic potential.

Unaware of Senechal’s 1975 paper, and having not noticed it on first reading of Bossavit’s 1995 paper, I rediscovered the connection between ‘monoaxial’ cubical/icosahedral subgroups and border patterns through my observation of various commercial balls; indeed most ideas presented here are also present in Cube on the ball. Further, this border pattern connection extends to crystallographic point groups (briefly discussed below) and beyond, that is to all finite subgroups of the sphere (not discussed below); note here that point groups are finite spherical subgroups where only rotations of orders two, three, four, six and rotoreflections of orders two, four, six are allowed. (This extension may of course remind my readers of Cubically yours, icosahedron 🙂 )

Our presentation is going to be succint and visual: for each type of border pattern, *finite *versions (bands) are shown, and a brief explanation is made on how the two-dimensional isometries of the border pattern turn into three-dimensional isometries of some cubical/icosahedral group or point group … simply by *wrapping *the band! (Note here that every cubical subgroup happens to be a point group, but not vice versa, whereas some icosahedral subgroups (specifically, those consisting of five or ten ‘units’) are not point groups: details will be provided along as needed.) Please excuse various drawing imprecisions!

**p111**: **[2]+**, **[3]+**, **[4]+**, **[5]+**, **[6]+**

The band’s finite/circular n-unit translation turns into n-fold rotation about the axis of the cylinder created by wrapping the band (i.e., by joining its right and left bands). Note that **[6]+** is a point group but not a subgroup of either the cube or the icosahedron (which possess no sixfold rotation); conversely, **[5]+** is an icosahedral subgroup, but not a point group.

**p1a1**: **[2+, 2+]**, **[2+, 4+]**, **[2+, 6+],** **[2+, 10+]**

The band’s glide reflection (along the band’s middle line, featuring 2n units) turns into 2n-fold rotoreflection (with implied n-fold rotation). Of special is interest is the first case above (**[2+, 2+]**), where the two units (one upward, one downward) yield twofold rotoreflection, that is inversion, without rotation. Inversion is absent from the next case, **[2+, 4+]**, which is now equipped with twofold rotation, but it reappears in the last two cases. Since eightfold and twelvefold rotoreflection are not allowed in either cubical/icosahedral subgroups or point groups, the spherical subgroups **[2+, 8+]** and **[2+, 12+]** are not listed.

**p1m1**: **[2+, 2]**, **[3+, 2]**, **[4+, 2]**, **[6+, 2]**

The band’s horizontal reflection turns now into reflection perpendicular to the n-fold rotation. (Observe here that, in the same way horizontal reflection (together with translation) yields ‘trivial’ glide reflection in a border pattern, reflection and n-fold rotation perpendicular to each other combine into ‘trivial’ rotoreflection.) Notice that **[3+, 2]** is a point group but not a cubical/icosahedral subgroup: as we have seen in previous posts, no threefold rotation can be perpendicular to a reflection in either the cube or the icosahedron! (For a very similar reason the spherical subgroup **[5+, 2]** is not a cubical/icosahedral subgroup, and, since it is not a point group, either, it is not listed.)

**pm11**: **[2]**, **[3]**, **[4]**, **[5]**, **[6]**

The band’s vertical reflection turns into reflection containing the axis of the n-fold rotation. There are two kinds of vertical reflection in the band, one running through units and one running half way between units, which leads to two kinds of reflection planes when n is even and to reflection planes of non-equivalend ‘ends’ when n is odd; either way, we obtain n reflection planes intersecting each other along the n-fold rotation axis.

**p112**: **[2, 2]+**, **[2, 3]+**, **[2, 4]+**, **[2, 5]+**, **[2, 6]+**

The band’s half turn centers turn into endpoints of axes of twofold rotation, all n of them perpendicular to the axis of the n-fold rotation. As in the previous case, the two kinds of half turn centers lead to two kinds of twofold axes when n is even and to twofold axes of non-equivalent ‘ends’ when n is odd.

**pma2**: **[2+, 2]**, **[2+, 4]**, **[2+, 6]**, **[2+, 10]**

The band’s glide reflection, n vertical reflections (of one kind only), and n half turns (of one kind only) yield 2n-fold rotoreflection (and implied n-fold rotation along the same axis), n reflections (of one kind only) containing the rotoreflection axis, and n twofold rotations (of one kind only) perpendicular to the rotoreflection axis, respectively. Note that **[2+, 2]** has been obtained already out of **p1m1**! (Again, there is inversion at 2n = 6 and 2n = 10, but not at 2n = 4 (tennis ball).)

**pmm2**: **[2, 2]**, **[2, 3]**, **[2, 4]**, **[2, 6]**

The band’s horizontal reflection, vertical reflection (of two kinds), and half turn (of two kinds) yield reflection perpendicular to the n-fold rotation (and ‘trivial’ rotoreflection along the same axis), n reflections (of two kinds when n is even) containing the n-fold rotation axis, and n twofold rotations (of two kinds when n is even) perpendicular to the n-fold rotation axis, respectively. Again, **[2, 3]**, the only group above without inversion, is a point group, but not a cubical/icosahedral subgroup; and **[2, 5]**, being neither of the two, is not listed.

… There exist 32 crystallographic point groups, how many have we listed above? Subtracting 1 for the one and only repetition (**[2+, 2]**), and not including the five icosahedral, non-point-groups listed (**[5]+**, **[2+, 10+],** **[5]**, **[2, 5]+**, **[2+, 10]**), we count (4 + 3 + 4 + 4 + 4 + 3 + 4) – 1 = 25 groups. To these we must obviously add **[1]+** (the trivial identity-only group that needs no representation) and **[1]** (the reflection-only group represented by the symmetry group of the isosceles triangle — especially when on a wrapped band or sphere rather than on a flat surface!), so we move from 25 to 25 + 2 = 27 groups. The remaining five groups are the so-called *cubical point groups*, that is the ‘polyaxial’ symmetry groups of the cube and … the 2002-2014 World Cup soccer balls!

Tags: Alain Bossavit, border patterns, cubical subgroups, finite spherical subgroups, icosahedral subgroups, Marjorie Senechal, point crystallographic groups

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