## A hyperbolic kaleidoscope

In “creating symmetry” Frank Farris shows a wallpaper for hyperbolic space. He uses the Poincaré plane to project the hyperbolic space to our Euclidean drawing surface. The wallpaper then results from mirror symmetries at vertical lines at x=0 and x=0.5 together with inversion at the unit circle. The mirror symmetries generate a periodic drawing in x-direction with periodic length 1. The inversion at the circle makes an increasingly complex design going from above to the x-axis. This is an example I created:

Farris uses algebra and group theory to discuss the structure of these images. To me this is fascinating and challenging. To see if I really got it I tried to explain these images using geometry.

First, the most important properties of hyperbolic space: The Poincaré plane occupies the upper half plane. Equal distances in the hyperbolic space seem to shrink going to the x-axis because of the projection. A particle with constant speed in the hyperbolic plane would never reach our x-axis. Thus the x-axis is infinity far away in the hyperbolic plane. Straight lines of the hyperbolic space are projected as circles in the Poincaré plane. Their centers lie on the x-axis. If the radius of the circle goes to infinity we get a straight vertical line. A mirror images at a straight line in hyperbolic space becomes an inversion at the corresponding circle in the Poincaré plane. Note that the inversion at a circle maps circles into circles. It has to be like that, because straight lines in the hyperbolic plane have straight lines as mirror images. Similarly, inversion at a circle preserves the angle between intersecting circles.

I have drawn the two vertical lines and the unit circle in another image of the same symmetries:

Note how the vertical lines generate the mirror symmetry and periodicity in x-direction. At the point (0,1) the circle intersects the vertical line at a right angle. Thus we get a two-fold rotational symmetry with mirror symmetries at this point. Going away from this point distortions set in. Repeated copies lie at the line y=1, which is not a straight line in hyperbolic space. Inversion at the unit circle maps this line into a circle of radius 0.5 with its center at the point (0,0.5). Thus images of this center of two-fold rotational symmetry appear on this circle. The first one is at (0.5,0.5). At (0.5,0.866) there is an intersection angle of 60 degrees between the vertical line and the unit circle. This makes a center of three-fold rotational symmetry. Again, copies appear on a horizontal line and on circles.

Let us compare this with kaleidoscopes in Euclidean space, as discussed in “Geometry of kaleidoscopes with periodic images“. In general they have three straight lines that serve as mirror axis and form a triangle. Reflections of its inside cover the plane. Thus the kaleidoscopic image arises. Here we have something similar. The two straight lines and the circle make a triangle in the hyperbolic space if you accept that the parallel lines meet at infinity with a vanishing intersection angle. Again, we can use mirror symmetry and inversion to cover the plane.

To show the structure more clearly I filled the basic triangle region with a dark brown color. Each reflection or inversion switches the color from dark brown to light blue or inversely. In the resulting image we can easily see the different images of the basic region:

Look out for centers of two-fold and three-fold rotational symmetries. How are they mapped by the inversion at the unit circles?

I have created my images using iterated reflections of points and not with symmetric bundles of wave function as Farris. Thus I get discontinuities in the first derivative of the images and my images are not so smooth at the unit circle. But I can generate images much faster and easily. Thus I can rapidly change the elements of symmetry.

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