## 2 Symmetries of symmetrical compound systems
As in the last post, we consider a 3 part compound system with 4 edges: $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, $\mathbf{D}$.
The number of total rotations of $\mathbf{D}$ is
$$\rho_I = \frac{\lcm(AC,BC,BD)}{BD}$$
And the total number of rotations of $\mathbf{B}$ and $\mathbf{C}$ is
$$\rho_O = \frac{\lcm(AC,BC,BD)}{CB}$$
First consider a symmetrical system, i.e. where the middle piece is a ring having inner and outer curves parallel. The resulting curve will have rotational symmetry order $\rho_I$ and reflectional symmetry in $\rho_I$ axes in the same way as a simple system.
These systems are useful for introducing patterns which don't lie on whole tooth divisions of the outmost part.
For the remainder of the post we will consider asymmetrical systems where $\mathbf{B}$ and $\mathbf{C}$ are not parallel, i.e. wheels with off-centre cut-outs.
## 3 Rotational Symmetry of compound systems
As $\mathbf{B}$ completes $n$ rotations inside $\mathbf{A}$, $\mathbf{D}$ traverses $n$ full revolutions in $\mathbf{C}$, leaving it offset by $nC \pmod{D}$ teeth. Consider
$$V = \\{nC \mid n \in \mathbb{Z}/{D}\\} = \left\langle \gcd(C,D) \right\rangle $$
$$|V| = \frac{D}{\gcd(C,D)} = r(D,C)$$
Each member of $V$ corresponds to a distinct offset that $\mathbf{D}$ can have when $\mathbf{C}$ has rotated a whole number of times. These offsets will be evenly distributed, thus the number of times each offset will occur in one complete curve is
$$\frac{\rho_O}{|V|} = \frac{\rho_O}{r(D,C)}$$
This gives the order of rotational symmetry of the curve.