Turing's Symmetries

Question

In Alan Turing's classic paper on the Chemical Basis of Morphogenesis, the following bilateral and left-right symmetries are defined as follows:

Might be an misunderstanding of English from my side, but I can't really visualize this non-parallel upright strokes and what makes it different from the bilateral symmetry. This is what I'm imagining

Is that what he means? He goes further to define P- and F-symmetries, which is also a bit confusing with the 'coal scuttle' example:

Any suggestions on where I could learn more about this? Are there more mathematically "better" ways of defining such types of symmetries?

Answer 1

Imagine looking through the $P$ that's closer to you (colored in green below) and right at the $P$ that's on the far side (in red). Namely, you line of sight roughly aligns with the straight line formed by connecting the two $P$s.

Apologies for the choice of colors. I tried to make the green $P$ thicker to represent that it is closer to your eyes.

The red $P$ is an image of the green (thicker) $P$ via moving along the surface of cylinder (equivalent to rotating in space).

The upright stroke (the "vertical" bar when P is written this way) of one is not parallel to the other in the 3-dim Euclidean space.

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^{I don't know how to produce using Mathjax the diagram with two $P$s. The all-powerful Detexify returns nothing.}