Chiral symmetries in E[SUP]^n[/SUP]

[Mark Harder]

As a biochemist, I deal with chirality of molecules all the time. If you have a tetrahedral molecule, for example a carbon atom, and all 4 vertices are labeled differently, as in different atoms on each one, then that molecule has a mirror-symmetric one that cannot be superimposed on the original. This makes a big difference when it comes to binding of small chiral molecules to big ones, esp. enzymes. What about 2-D geometry? Are there figures, triangles for example, that are mirror-asymetric? The answer is yes, provided they are confined to the plane they are in. Flipping over is not an allowed move, since the triangles are confined to E^{^2}.
Try it. Draw a triangle with each side of a different length. Draw a straight line parallel to one of the edges (not necessary, but it makes visualizing easier) and outside the triangle. Now 'reflect' the triangle about this 1-D 'mirror', i.e. draw a triangle with vertices at the same distances from the mirror as the original. Now try to mentally rigidly rotate and translate the triangles so that they superimpose. You can't do it (without breaking the law and rotating in the 3rd dimension.) This is a 2D analog of 3D chiral figures and 2D mirrors. Not that the angles (disregarding their directions) and included sides are the same, satisfying Euclid's definition of congruency.
Now try the 1D case. Draw a directed line segment on the number line. Locate a point (a 0-D figure) on the line outside the directed segment and 'reflect' the directed segment about the point. The result will be a directed line segment pointed in the opposite direction from the original - a 1-D, 0-D analog of higher dimensions. Obviously, there is no example for 0-D figures.
So, my question is, can these examples be extended to higher dimensions? To n-D Euclidean spaces in general? What about other geometries?

 

[pasmith]

The mappings from $\mathbb{R}^n$ to itself which preserve the euclidean inner product are $x \mapsto Ax + b$ where $A^{-1} = A^T$, so $|\det A| = \pm 1$. Those with $\det A = -1$ are reflections, which require "flipping" in an additional dimension.

In more general geometries, you can have distance- and angle-preserving mappings, which will either preserve orientation or reverse it.