Space and Counterspace II

 The previous post introduced the dual euclidean plane and compared it to the euclidean plane on the basis of a hexagonal grid pattern.

In this post we'll continue with this example, but bring it into motion.

As you know there are two kinds of euclidean motions in EP: translations and rotations. We're going to explore subjecting the pattern to a euclidean translation and see how the corresponding dual translation moves the dual pattern.

A translation is an isometry that fixes an ideal point and moves all other euclidean points in the direction perpendicular to this fixed ideal point. The dual translation will fix the corresponding dual ideal line, perpendicular to the ideal point of the translation. Here's what we see when the translation is in the x-direction:

The hexagonal figure has been translated to the right so that the ideal point of the DEP now lies halfway towards the cyan point on the left boundary. The dual pattern of lines has undergone a dramatic shift and appears very unsymmetric. It is crowded into the left hand side and stretched out towards the right.  How can we understand this?

Remember that the star point (0,0,1) maps under duality to the ideal line [0,0,1]. And that the dual of a point appears on the opposite side of the origin as the point.  Since the hexagon grid is off-center and most of it is now to the right of the origin, most of the dual pattern will be seen on the left side; on the right there is only one vertical line between the cyan line on the right and the ideal line. 

If we continue to move the pattern to the right, eventually the star point will lie outside the grid pattern:

The whole pattern of dots is now to the right of the origin (and star point) so that the whole of the line pattern will lie to the left. 

In fact we won't show it here but the six boundary lines of the dual pattern viewed with euclidean eyes always lie on a conic section; at the above stage that is, not surprisingly, a hyperbola. But perhaps stranger is that viewed with dual euclidean eye, they remain a circle throughout!  Just as the grid of points moves rigidly in the euclidean plane when it's translated, the grid of lines viewed in the dual euclidean plane remain rigidly connected to each other.  The proof is trivial: the calculations involved in confirming this fact for the euclidean pattern are identical to those in the dual euclidean case; the only difference is that the results are interpreted once in EP and the other occasion in DEP.

Space and Counterspace I

The principle of duality in projective geometry creates a partner for any configuration or statement.  It is based on a dictionary of duality that allows any description to be translated into its dual partner.

For 2D geometry, this dictionary starts with the entries:

• point             line
• join               meet
• lies on           passes through
• move along.  rotate around

So for example, "Two points have a unique joining line" is dual to "Two lines have a unique meeting point." (Only true in PG since parallel lines meet in ideal points). The dictionary is symmetric, so that you can look up a word in either column and replace it with its partner.  Words that don't appear in the dictionary are left alone. The above dictionary applies to projective geometry in general. These blog posts [xxx] deal with this basic version of duality.

In this post we want to explore a special extension of this dictionary that allows us to dualize euclidean geometry to produce dual euclidean geometry. Restricted to 2D this leads to the euclidean plane (EP) and the dual euclidean plane (DEP).   We then begin to explore and compare these two partner geometries.

This blog isn't the place to give the details of this construction.  We can give some hints however.  The euclidean plane is derived from the projective plane by identifying a special line called the ideal line (sometimes called the line at infinity by careless speakers).  The points of this line are ideal points and can be identified with directions or vectors, that form the foundation of euclidean measurement.  In DEP, there is correspondingly a single ideal point, along with all the lines passing through it, the ideal lines of the point, which play an analogous role in dual euclidean measurement.

While the ideal line of the euclidean plane appears to be built into the nature of the world, the ideal point of each DEP can be chosen independently.  

We want to compare EP and DEP on the basis of a simple geometric pattern. The following image shows a regular hexagon containing within it a hexagonal pattern of colored dots centered on the origin of the coordinate system.  We choose the ideal line of the DEP to be this origin.  It's marked by a star icon.  What will the dual partner of this pattern look like in DEP?  The simple answer is (ignoring the six colored segments bounding the hexagon for the moment): each colored point in the central grid corresponds to a line of the same color in the outer figure. For example, the red point on the right side of the hexagon corresponds to the vertical red line on the left hand side close to the cyan vertex. Why: the point coordinate (.9, 0, 1) produces the line coordinate [.9, 0, 1] which corresponds to the line x = -1.11... The dual line appears on the opposite side of the origin from the original point, with a slope perpendicular to the line joining the origin to the given point.  The center point (0,0,1) corresponds to the ideal line z = 0. This equation reflects the fact that ideal points have z-coordinate 0. 

One characteristic feature of the line pattern is that there are many points where several different-colored lines meet.  To find the pendant in the original pattern, we have to look for lines upon which several different-colored points lie.  These are easy to find; the hexagon grid is criss-crossed with such lines -- although the lines themselves haven't been drawn.  For example the point on the left where red, yellow, magenta, and 2 olive green lines meet corresponds to the vertical row of dots running from the yellow dot on the upper right of the hexagon boundary to the red dot on the lower right. 

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.