I'm reading Alan Turing's paper titled The Chemical Basis of Morphogenesis and there is a section in it with mathematical definitions that mystify me. I'm guessing that Turing tried to keep mathematical jargon out of the definitions to perhaps increase the accessibility of the paper, but unfortunately it isn't aiding my understanding.
(I'm not even sure if this is the right platform for my questions; if it isn't then I'd appreciate someone pointing me to the right one.)
Turing first defines bilateral symmetry, which is fine.
An organism is said to have "bilateral symmetry" if it is identical with its own reflection in some plane. This plane of course always has to pass through some part of the organism, in particular through its centre of gravity.
Then Turing goes on to define something called 'left-right symmetry' which is claimed to be a more general notion of bilateral symmetry:
An organism has left-right symmetry if its description in any right-handed set of rectangular Cartesian co-ordinates is identical with its description in some set of left-handed axes. An example of a body with left-right symmetry, but not bilateral symmetry, is a cylinder with the letter P printed on one end, and with the mirror image of a P on the other end, but with the two upright strokes of the two letters not parallel. The distinction may possibly be without a difference so far as the biological world is concerned, but mathematically it should not be ignored.
Then Turing defines P-symmetry and F-symmetry, things which I am unable to understand.
An entity may be described as "P-symmetrical" if its description in terms of one set of right-handed axes is identical with its description in terms of any other of right-handed axes with the same origin. Thus, for instance, the totality of positions that a corkscrew would take up when rotated in all possible ways about the origin has P-symmetry.
The entity will be said to be "F-symmetrical" when changes from right-handed axes to left-handed may also be made. This would apply if the corkscrew were replaced by a bilaterally symmetrical object such as a coal scuttle, or a left-right symmetrical object.
Could someone please provide mathematically precise definitions for these types of asymmetries? From this, could someone also please explain to me why a corkscrew isn't bilaterally symmetric while a coal scuttle is?
Left-right symmetry makes me think of chirality. In fact, that's what Wikipedia redirects to. But I don't understand the example, either.
For the last quote, here are my interpretations of the definitions. Right-handed and left-handed axes are two oriented bases of $\Bbb R^3$ sharing one axis (let's call it the $z$-axis) and exchanging the other two, i.e. $(x,y,z)$ is the right-handed axis, and $(y,x,z)$ is the left-handed one.
A P-symmetrical entity seems to be an entity that is stable under rotations around the $z$ axis. The example is a bit trivial, since Turing is saying that the set of rotations around $z$ is stable by rotations around $z$. Maybe he meant that any rotation of a corkscrew is equal to the corkscrew up to translation along the $z$ axis?
An F-symmetrical entity would be an entity that is stable under rotations around $z$ as well as reflections whose reflection plane contains $z$. A coal scuttle is stable by one reflection, but I don't think it's stable by rotation... at least, according to the images I found of coal scuttles.