In the previous post we determined *geometrically* all finite three-dimensional rotation groups, that is all finite symmetry groups that contain only three-dimensional rotations and rotoreflections; these are the ‘general’ groups **[n]+**, **[2+, (2n)+]**, **[2, n]+**, and the ‘special’ groups **[3, 3]+**, **[4, 3]+**, **[5, 3]+**. Here we complete the job by allowing reflections in: as previous posts, notably this one, have indicated, our task is to determine how reflection planes can coexist with rotation and rotoreflection axes, and the Conjugacy Principle makes that relatively easy.

Starting with the case of a single n-fold rotation, **[n]+**, we observe that any reflection plane can either contain the rotation axis or be perpendicular to it (otherwise the n-fold rotation would be reflected to another n-fold rotation): in the first case we end up with the group **[n+, 2]** (single reflection perpendicular to the n-fold rotation with implied ‘trivial’ n-fold rotoreflection along the axis of the n-fold rotation), in the second case we end up with the group **[n] **(n reflection planes intersecting along the n-fold rotation axis). (Note here that, as a trivial special case of either group at , we obtain the reflection-only group **[1]**.) So, ‘departing’ from **[n]+** we end up with the groups **[n+, 2]**, **[n]**, and, of course, **[n]+** (with **[1]+** being the identity-only, fully asymmetrical group). [The groups **[n]+** and **[n]** are isomorphic to the only finite two-dimensional symmetry groups, the cyclic group **Cn** and the dihedral group **Dn**, respectively.]

In the case of a single 2n-fold rotoreflection, **[2+, (2n)+]**, with implied n-fold rotation — equivalently, single n-fold rotation with a ‘square root’ — we note that no reflection plane can be perpendicular to the rotoreflection axis, for the product of the 2n-fold rotoreflection with the reflection would yield a 2n-fold rotation. (More precisely, a reflection plane perpendicular to the rotoreflection axis leads to the group **[(2n)+, 2]** obtained above.) Less obviously, no reflection plane could contain the rotoreflection axis: indeed the product of a 2n-fold rotoreflection with a reflection containing it is a twofold rotation perpendicular to the rotoreflection and making an angle of with the reflection plane; this fact has been experimentally verified for in Lemmas 5, 9, 14 here and can be rigorously established by expressing the 2n-fold rotoreflection as product of three reflections — the first of them the said reflection that contains the rotoreflection axis (to be canceled out when we take the product), the second a reflection making an angle of with the first one (so that their product will be the rotational part of the given 2n-fold rotoreflection, that is a rotation), and the third one perpendicular to the first two (and being none other than the reflectional part of the rotoreflection). (More precisely, a reflection plane containing the rotoreflection axis leads to the group **[2+, 2n]** obtained below.) So, in this case we are ‘stuck’ with what we started with, namely the group **[2+, (2n)+]**. (Note that at we obtain the inversion-only group **[2+, 2+]**: no problem, inversion is after all a ‘free’ *twofold rotoreflection*!)

Starting with **[2, n]+**, that is one ‘central’ n-fold rotation with n twofold rotations perpendicular to it, the Conjugacy Principle tells us that any reflection that may wish to join the party must either contain one of the twofold rotations (reflecting it to itself) or lie half way between two twofold rotations (reflecting them to each other); at the same time, as in the previous two cases, any ‘intruding’ reflection must either contain the n-fold rotation axis or be perpendicular to it. But note here that, by Lemma 1 here, a reflection containing the n-fold axis and one twofold axis implies a reflection perpendicular to the n-fold axis containing all twofold axes, and vice versa; in other words, the only way to avoid a reflection perpendicular to the n-fold axis is to have reflection planes containing the n-fold axis and lying half way between adjacent twofold axes. We finally arrive at only two new groups: the *‘half-way’* group **[2+, 2n]** (n reflections containing the n-fold rotation axis, each of them half way between two twofold rotation axes and perpendicular to the plane defined by them) and the *‘right-on’* group **[2, n]** (one reflection perpendicular to the n-fold rotation axis and containing the twofold rotations, plus n reflections containing the n-fold axis and one twofold axis each); **[2+, 2n]** has genuine 2n-fold rotoreflection along the n-fold rotation axis, whereas **[2, n]** has trivial n-fold rotoreflection along the n-fold rotation axis. (Note that is not possible in this case, or rather yields a group already found, **[2+, 2]**, whereas **[2, 1]** = **[2]** and **[2, 1]+** = **[2]+** ‘hold trivially’.)

Putting everything together, we obtain seven general finite symmetry groups: **[n]+**, **[n+, 2]**, **[n]**, **[2+, (2n)+]**, **[2, n]+**, **[2+, 2n]**,** [2, n]**. Marvelously, these seven three-dimensional groups are in one-to-one correspondence with the seven one-dimensional groups known as border patterns, as indicated here: the reader may refer to that post for visualizations.

Speaking of visualizations and border pattern representation of three-dimensional symmetry groups, the following diagram captures our arguments so far:

(Blank = central n-fold rotation [translation], dotted line = central 2n-fold rotoreflection [glide reflection], dots = twofold axes perpendicular to the n-fold rotation [twofold centers], vertical lines = reflections containing the central axis [vertical reflections], horizontal line = reflection perpendicular to the central axis [horizontal reflection].)

To complete the job, we need to see how reflection might be injected into the special rotation groups **[3, 3]+**, **[4, 3]+**, **[5, 3]+**. This is an easy task: **[4, 3]+** and **[5, 3]+** allow only the possibilities **[4, 3]** (cube/octahedron) and **[5, 3]** (dodecahedron/icosahedron), whereas **[3, 3]+** allows the possibilities **[3, 3]** (regular tetrahedron, * jabulani* ball) and

**[3+, 4]**(pyritohedron,

*ball). These limitations are implied by the Conjugacy Principle and the “either half-way or right-on” principle for reflections together with the placement of rotation axes in the rotation groups established in the previous post:*

**teamgeist**— in **[5, 3]+** starting with two fivefold axes like AD and BE we get the four remaining axes: AD and BE rotate each other to GL and KI forming two *equilateral rotation triples* (as shown in the previous post), whereas the product of the fivefold rotations corresponding to AD and BE yields, under the right combination of order and orientation, the two fivefold axes corresponding to FC and JH (by way of (BKJFG)(CLEIH) * (AKCHG)(DIFJL) = (AJEIG)(BKLDH) and (BGFJK)(CHIEL) * (AGHCK)(DLJFI) = (AFELK)(BGIDC), for example);

this placement of fivefold axes makes it clear that every half-way reflection must be right-on (containing two fivefold axes), and from this observation it follows that there exists only one reflection group containing the rotation group **[5, 3]+**, and that is the full symetry group of the icosahedron/dodecahedron, **[5, 3]**.

— in **[4, 3]+** the three orthogonal fourfold axes make it clear that every reflection must be right-on, either containing one fourfold axis and reflecting the other two fourfold axes to each other or containing two fourfold axes being perpendicular to the third one (this is a special case of right-on reflection, with the other two to-be-reflected-to-each-other axes collapsing into one); but it is easy to see, multiplying them by fourfold rotations, that these *4-vertex* and *4-edge* reflections imply each other, therefore the only reflection group containing the rotation group **[4, 3]+** is the full symmetry group of the cube/octahedron, **[4, 3]**.

— in **[3, 3]+** there exist four threefold axes, the four diagonals of a cube basically, forming two X’s perpendicular to each other;

more precisely, let the first X consist of AA’, BB’ and the second X consist of CC’, DD’, with AA’, BB’ rotating each other to CC’, DD’ (and vice versa). It is clear that a half-way reflection with respect to the one X will also be half-way with respect to the other X, whereas any right-on reflection (like one containing AA’) must also contain another axis (like BB’) and reflect the other two axes (CC’, DD’) to each other: we are left with two possibilities, the half-way reflections (containing no threefold axis) of the pyritohedron (**[3+, 4]**) and the right-on reflections (containing two threefold axes each) of the regular tetrahedron (**[3, 3]**).

[We point out here — making yet another analogy with infinite two-dimensional symmetry groups (wallpaper patterns) in the spirit of the previous post and Chapter 8 of *Isometrica* — that **p6** and **p3** behave like **[4, 3]+** and **[5, 3]+**, allowing only the right-on possibility for reflection (**p6m** and **p3m1**, **p31m**, respectively), whereas **p4** behaves like **[3, 3]+**, allowing both the half-way (**p4g**) and right-on (**p4m**) possibilities for reflection. (One may point out here that the case of **p31m** is more complicated, as some threefold centers lie on no reflection axis … and it is for similar reasons that we leave **p2** out of this somewhat eccentric parenthetical discussion altogether!)]

We conclude that there exist fourteen types of finite three-dimensional symmetry groups, the seven general groups **[n]+**, **[n+, 2]**, **[n]**, **[2+, (2n)+]**, **[2, n]+**, **[2+, 2n]**,** [2, n] **and the seven special groups **[3, 3]+**,** [3, 3]**,** [3+, 4]**, **[4, 3]+**, **[4, 3]**, **[5, 3]+**, **[5, 3]**. (For a more traditional (*algebraic*) derivation of this classic Klein-Weyl result and further discussion the reader is referred to Marjorie Senechal’s 1990 paper Finding the Finite Groups of Symmetries of the Sphere.)

This post is dedicated to the gradually disappearing kiosks of Greek streets known as *periptera (wings-all-around,* a word known from* peripteros naos = column-surrounded temple *that plausibly entered Greek through certain *flames of love* in* Song of Songs 8.6 *(*περίπτερα αὐτῆς περίπτερα πυρός, φλόγες αὐτῆς*)* *and the *Septuagint* rendering of* *Hebrew* רִשְׁפֵּי = spark)*: initially conceived as phone-booth-sized roofy stores to be owned by amputated soldiers, they grew in size in recent decades … to the point of selling various symmetry-minded balls that partially inspired this series of posts … and made my Thessaloniki strolls — *talk the walk* — more interesting 🙂

http://greece.greekreporter.com/2015/10/14/the-uniquely-greek-kiosks-soon-to-become-history/

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