Space and Counter-space III

In a previous post, I introduced the euclidean plane (EP) and its dual partner, the dual euclidean plane (DEP).  I used the example of a hexagonal pattern of dots in EP that gives rise to a pattern of lines in DEP.  In a further post I showed how applying a translation to the pattern in EP leads to a "dual" translation in DEP that looks unfamiliar to our euclidean eyes.

In this post I want to continue with this line of thought, but replace the hexagonal pattern with another one.  Instead of having the euclidean pattern within and the dual pattern on the outside, the pattern consists of a regular 2D "crystal" of rhombi that extends out from the origin in all directions.  The following figure is scaled to show this pattern; the dual pattern is too small to see in this image.

What does the corresponding dual pattern look like?  Notice that a crystallographic pattern is characterized by a finite number of families of parallel lines. That is, all the lines in the pattern has a certain direction that it has in common with many other lines of the pattern. In this case, there are three different directions, or ideal points, that characterize the pattern of lines. The dual of this is: all the points of the dual pattern lies on one of three ideal lines.  Furthermore,  the spacing between neighboring lines in the same family is (in this case) always the same size; one moves from one line to the next by taking identical euclidean steps. 

So in the dual pattern will consist of a pattern of points, arranged on a set of three ideal lines (lines passing through the star pattern), such that you take equal-sized dual euclidean steps when moving from one point to the next.  The above image shows the dual pattern zoomed up; hopefully it's large enough to see that the points are in fact arranged on 3 lines through the origin. But what about the claim they are separated by equal steps?  By the fact that the colors get darker as they approach the origin, it's clear that the points closest to the ideal point correspond to the lines farthest away from the origin: which makes sense, since in both cases they are "approaching" the ideal. But what about the step size?

To understand that, consider the following line equations: x == i for integer i.  These describe a set of parallel vertical lines with a spacing of 1 unit. The line coordinates are [1,0,-i].  These become the point coordinates (1,0,-i) in the dual plane. To render them, we first note  that for i=0, this is the ideal point in the x-direction. Otherwise, we can dehomogenize them to obtain the equivalent points (-1/i, 0, 1).  These represent the sequence of points with x-coordinates (1,1/2,1/3, ...1/n,...) etc.  These points are equally spaced in the dual euclidean way of measuring! They approach but never reach the ideal point (0,0,1).

I close with a question. The following image shows an x-ray diffraction pattern for common table salt NaCl.  The latter lies on a cubic lattice in 3D.  The x-ray image has similarities to the image we encountered above of the counter-space partner of a regular 2D crystal in EP. Is this more than a coincidence? 

George Adams considered this connection in a short essay "The reciprocal lattice and the X-ray analysis of crystals" in the collection "Universal Forces in Mechanics" (Volume 35 of "Mathematische-Astronomische Blätter", Dornach, 1996, p. 213-224).  In the end he concluded that the X-ray crystallography produces another lattice in ordinary space, not in counter-space.  That should be easy to check, since X-ray crystallography implements a Fourier transform on the original crystal lattice.