# Swirl product

Swirl product
Symbol

The swirl product is an operation that takes two 3D point groups (more precisely, groups of unit quaternions) to form a 4D point group. Every 4D point group is representable as a swirl product of two 3D point groups, forming a keystone in classifying the 4D point groups. Although the underlying operation has been professionally studied, the term "swirl product" is a coinage of the enthusiast community as part of the study of swirlprisms.

## Preliminaries

4D rotations can be described in terms of quaternions as follows: given a 4D rotation (member of ${\displaystyle {\text{SO}}(4)}$), there is a pair of quaternions ${\displaystyle l}$ and ${\displaystyle r}$, both of unit norm, such that the function ${\displaystyle x\mapsto {\bar {l}}xr}$ induces that rotation on the input quaternion x. (x in this case is identified with ${\displaystyle \mathbb {R} ^{4}}$ in the obvious way.) Furthermore, there is a two-to-one mapping from pairs of unit quaternions ${\displaystyle (l,r)\in S^{3}\times S^{3}}$ to 4D rotations, as the pairs ${\displaystyle (l,r)}$ and ${\displaystyle (-l,-r)}$ denote the same 4D rotation, but other than this equivalence it is an isomorphism. To ensure that we're specifically talking about rotations and not pairs of unit quaternions, we use angle brackets ${\displaystyle [l,r]}$ to denote the 4D rotation corresponding to both these pairs.

For general, possibly orientation-reversing transformations in ${\displaystyle {\text{O}}(4)}$, we also notate quaternion conjugation as ${\displaystyle *:x=(x_{1},x_{2},x_{3},x_{4})\mapsto {\bar {x}}=(x_{1},-x_{2},-x_{3},-x_{4})}$. An improper rotation, or rotation-with-reflection, may be notated as ${\displaystyle *[l,r]:x\mapsto {\bar {l}}{\bar {x}}r}$, which is the result of first applying ${\displaystyle [l,r]}$ and then conjugation. Again, ${\displaystyle *[l,r]=*[-l,-r]}$.

## Chiral point groups

A chiral point group is simply a finite subgroup ${\displaystyle G\leq SO(4)}$. Given the set of unit quaternion pairs

${\displaystyle A=\{(l,r):[l,r]\in G\}}$

whose cardinality is double the order of ${\displaystyle G}$ (the set is more specifically a double cover of ${\displaystyle G}$). The sets of quaternions ${\displaystyle l}$ and ${\displaystyle r}$ form the left and right groups of ${\displaystyle G}$:

• ${\displaystyle L=\{l:(l,r)\in A\}}$
• ${\displaystyle R=\{r:(l,r)\in A\}}$

Together with the operation of quaternion multiplication, these are both finite groups of unit quaternions. If we are given two such groups of unit quaternions, we can reconstruct G, allowing us to notate G as ${\displaystyle G=\pm [L\times R]}$.

## Achiral point groups

An achiral point group is a finite subgroup ${\displaystyle G\leq O(4)}$ that has at least one reflection. A chiral point group ${\displaystyle H=\pm [L\times R]}$ can be converted into an achiral one by adding an orientation-reversing element of the form ${\displaystyle *[a,b]}$. In order for this to work:

• We require that the left and right groups of ${\displaystyle H}$ have to be equal for this to work: ${\displaystyle L=R}$
• The orientation-reversing element can always be chosen to have the form ${\displaystyle *[1,c]}$

The swirl product of two 3D point groups ${\displaystyle G_{1},G_{2}\in O(3)}$ with corresponding quaternion groups ${\displaystyle Q(G_{1}),Q(G_{2})\in \operatorname {Spin} (3)}$, is the group ${\displaystyle \pm [Q(G_{1})\times Q(G_{2})]}$ (extended by quaternionic conjugation ${\displaystyle *}$ if possible) and is denoted by ${\displaystyle G_{1}\bullet G_{2}}$

## Bibliography

• Rastanawi, Laith; Gunter, Rote (11 May 2022), Towards a Geometric Understanding of the 4-Dimensional Point Groups (PDF)