Variations on the hyperbolic kaleidoscope

In the last post I have presented a hyperbolic kaleidoscope with two- and three-fold rotational symmetries. Could we have other rotational symmetries?

Yes, we simply move the vertical lines! To get an n-fold rotational symmetry the circle has to intersect a straight line at an angle of π/n. The distance between the line and the center of the unit circle is then cos(π/n). We can freely choose the rotational symmetry at the left and at the right of the unit circle as long as the sum of the angles of the kaleidoscopic triangle is less than π.

Here is an example with 2- and 4-fold rotational symmetry:

And here comes a symmetric kaleidoscope with two 3-fold rotational symmetries:

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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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Five fold rotational symmetry: Tuning the harmonics

In “better images from higher harmonics ?” I have replaced the basic sine and cosine functions by Fourier series approaching a symmetric triangular wave. This gave images with more details and somewhat smaller bulls-eyes. Here I want to show similar results for an image with 5-fold rotational symmetry using a different input image.

With only the basic sine and cosine waves I get:


The center of perfect 5-fold rotational symmetry lies slightly at the right of the center of the image. Note the concentric shapes inside a pentagon and the large orange bulls-eyes.

Adding the third harmonic I get more details:


At the center we now have star like shapes instead of the pentagon and the circle.

Using the basic wave together with the third and the fifth harmonics gives smaller changes:


Finally, the triangle wave makes:


Now, straight line segments appear. The image has a more crystalline appearance. The rather large orange blobs remain, but they have now a more interesting polygon shape.

I like that I can tune the images changing the wave function, but I am disappointed that the large blobs do not disappear.


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Color symmetry using the length scale of the inflated lattice

I have shown some images with 2-color symmetry upon rotation shown in “images of 10-fold rotational …“. But the fast color changes they hacked them into small pieces. We can get better images if we use a color changing function with slower oscillating waves. A good choice results from the self-similarity discussed in “self-similarity and color modification“. Thus I used sin(x/1.618) as basic wave to build a color changing function for quasi-periodic design with 10-fold rotational symmetry.

This is a basic result that shows mainly the color-changing structure:

And here is a more subtle image that shows increasingly complex stars of 5-fold symmetry

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images with 5-fold symmetry and color change indicating self-similarity

And now for more images with 5-fold rotational symmetry and color change derived from self-similarity as discussed in “self-similarity and color modification“.

Zoom in to see the molten watch faces in this image:

Here I used the portrait of a damselfly, but you wont recognize much:


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images of 8-fold rotational symmetry and color changing mirror symmetry

Here I am showing some quasi-periodic designs of eight-fold rotational symmetry. They have a color change upon mirroring at the x-axis and 7 other mirror axis generated by the rotational symmetry. Note that these images have a rather large scale and have a grey appearance because the color change generates complimentary colors. To see more you should use the zoom function of your browser.

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images of 10-fold rotational symmetry and 2-color symmetry upon rotation

Here are some images of 10-fold rotational symmetry and 2-color symmetry upon rotation. They have an additional mirror symmetry. Thus you can discover local mirror symmetries with and without color change. Again, they are of large size and you can zoom in to see details:


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