In a cube, three orthogonal sides can connect four corners of one face such that all sides and corners lie in one plane, or they can connect four corners of two faces sharing two corners such that they do not.
In the former conformation the mirror image is the same (symmetric), but in the latter conformation it is not (antisymmetric). A mirror swaps front and back, so the side that projects away in the cube projects towards in the mirror.
The symmetric conformation can be superimposed on its mirror image, but the antisymmetric conformation cannot – it has two mutually exclusive (chiral) conformations.
The two different conformations emerge in the mind of an observer through perception, which breaks symmetry, dividing out a cube, faces, sides and corners, and the symmetric and antisymmetric conformations that the sides and faces can adopt.
The mirror images of chiral conformations are in general referred to as enantiomers. However, in the absence of an alternative term to identify this specific enantiomer pair consisting of three orthogonal connecting four corners, the term chiralkine is being used.
When symmetry is restored by joining up the two end corners of pair of enantiomers in the antisymmetric conformation, a skew hexagon is produced.
A skew hexagon has six sides and six corners, so restoring symmetry eliminates a pair of corners. A skew hexagon has bilateral symmetry like the human body.
When a cube is observed along an axis connecting two opposed corners, what is seen is one central corner surrounded by six corners arranged in a hexagon that appears to be a planar symmetric hexagon, but is in fact a skew hexagon.
The brain can perceive the hexagon as being in a cube (Necker cube effect).
Two skew hexagons can be superimposed six different ways. One can always be rotated in space relative to the other so that the two fit together. As a whole a skew hexagon is symmetric.
In order to better visualise how conformations rotate in 3D, it is helpful to use symbols to represent whether a side is projecting to or from the observer. From some viewpoints all three sides and four corners corners can be seen. In this case black and white arrows can be used to indicate projection towards or away from the observer.
From other viewpoints, only some of the sides and corners are observed.
If two corners overlap, three sides and three corners will be perceived to be in a triangle. This can be depicted using three arrows, because all of the sides are now projecting either towards or away from the observer.
If two corners overlap because a terminal side is projecting towards or away from the observer such that two sides and three corners are perceived, + and – signs can be used towards and away respectively. If the central side is projecting towards and way from the viewer, the side above can be indicated with a solid line and that below with a dotted line. In the drawings below, the same enantiomer is being shown from the different viewpoints. The mirror opposite enantiomer has the opposite conformation.
Chiralkines cannot be oriented such that less than two sides and three corners are perceived. This contrasts with the case where three sides and four corners lie in a plane. In that case it is possible to orient the conformation such that only one side and two corners can be seen, by looking within the plane.
A skew hexagon has three planes of symmetry, so can be divided three ways into a pair of chiralkine enantiomers. Each enantiomer can be superimposed three different ways on its own kind in the skew hexagon, but not in any of the three ways in which its opposite enantiomer can be superimposed. Each chiralkine and its oriented surface on the skew hexagon interact like lock and key or ligand and receptor.
One chiralkine superimposed on a skew hexagon so that its ends overlie opposed corners in the skew hexagon cannot be pivoted around one end to superimpose the other side, and it cannot be superimposed on the same side if it is moved one step from the pivot in one direction around the ring. However it can be superimposed if it is moved one step in the opposite direction. Altogether it can be superimposed if it is moved in one direction (say clockwise) 2, 4 or 6 steps around the ring, but not 1, 3 or 5 and if it is moved in the opposite direction (say anticlockwise) 1, 3 or 5 steps, but not 2, 4 or 6 steps. The act of perception, in breaking the bilateral symmetry of the skew hexagon, draws a distinction between clockwise and anticlockwise. The binding and antibinding sites for chiralkine enantiomers on a skew hexagon oscillate.
Breaking the symmetry of a (bilaterally symmetric) skew hexagon does not simply generate two static chiral conformations, but also a kinetic oscillation – a pulse.