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.