Symmetries of Columns

We define a column by a real-valued function f on the cylinder ${\cal C}={\bf S}^1\times{\bf R}$. Let $(\varphi,z)\in{\cal C}$. The function $f(\varphi,z)$ measures the height of the column in the direction normal to the cylinder at the point $(\varphi,z)$.

The group of symmetries of the cylinder is

$$\Gamma={\bf D}_2(\tau,\kappa)\dot+({\bf SO(2)}\oplus{\bf R})$$

where $\Gamma$ acts on $(\varphi,z)\in{\cal C}$ by

$$\begin{eqnarray*}(\theta,t) (\varphi,z) & = & (\varphi+\theta, z+t) \qquad (\the... ...,z) & = & (-\varphi,z) \\ \kappa(\varphi,z) & = & (\varphi,-z). \end{eqnarray*}$$

Multiplication in $\Gamma$ follows from the definition of the action. Suppose that $(A_j,(\theta_j,t_j))$ is in $\Gamma$ for j=1,2, where $A_j\in{\bf D}_2$, $\theta_j\in{\bf SO(2)}$ and $t_j\in{\bf R}$. Then multiplication is given by

$$(A_2,(\theta_2,t_2))\cdot(A_1,(\theta_1,t_1))=(A_2A_1\,,\, A_2(\theta_1,t_1)+(\theta_2,t_2)).$$ (1)

We wish to classify columns by their symmetries. A symmetry of the column $f:{\cal C}\to{\bf R}$ is $\gamma\in\Gamma$ such that

$$f(\gamma(\varphi,z)) = f(\varphi,z) \quad \forall (\varphi,z)\in{\cal C}.$$

The symmetry group $\Sigma_f\subset\Gamma$ is the collection of all symmetries of f. We classify all subgroups $\Sigma$ which are symmetry subgroups for some column f.

Our classification proceeds as follows. To each subgroup $\Sigma \subset \Gamma$, we can associate the normal subgroup

$$\Sigma_0=\Sigma\cap({\bf SO(2)}\oplus{\bf R}).$$ (2)

(So $\Sigma _0$ consists of the pure `translations' in $\Sigma$.) Thus it suffices to

(i)

classify the closed subgroups $\Sigma _0$ of ${\bf SO(2)}\oplus{\bf R}$,

(ii)

for each subgroup $\Sigma _0$ in (i), compute the subgroups $\Sigma \subset \Gamma$ that satisfy (2.2).

The calculation in (ii) is simplified by observing that $\Sigma$ is contained in the normalizer of $\Sigma _0$.

As usual, we identify conjugate subgroups of $\Gamma$. In addition, we identify subgroups that are related by axial scalings. More precisely, we define the scaling transformation $s_\alpha:\Gamma\to\Gamma$ by

$$s_\alpha(A,\theta,t)=(A,\theta,\alpha t), \qquad (A,\theta,t)\in\Gamma.$$

Provided $\alpha\neq0$, this is an isomorphism. We say that two subgroups $\Sigma$, $\Sigma'$ are related by a scaling if $s_\alpha\Sigma=\Sigma'$for some nonzero $\alpha$.