By Frank A. Farris

Mathematical art is everywhere these days. If you’ve been wondering how to get involved, I have an idea for you, a way to start making your own art today. You can involve your students too, especially if your course has anything to do with geometry.

The heavy lifting is done by a web-based app written by Swiss mathematician and mathematical artist Peter Stampfli. You choose any photograph you may have handy, manipulate the controls until you find something you find beautiful, and then save your artwork. You might enjoy trying now, before you read what I have to say about turning it into homework for students in geometry classes. Just click on http://geometricolor.ch/images/geometricolor/sphericalKaleidoscopeApp.html.

Your choices determine how a portion of the photo—roughly a triangle—are reflected in a wonderful mathematical kaleidoscope. Figure 1 shows what you might see when you try it out. The default settings give you a triangle with angles π/5, π/3, and π/2, as indicated in the numbers 5, 3, and 2 in the dialog boxes. Wait a minute! Those angles add up to more than π! This triangle belongs to the world of spherical geometry because the surface of a sphere is the natural home for such a triangle.

If you change the 5 to a 6 in the dialog box, the pattern changes dramatically. We now see a wallpaper pattern in the plane, like the one on the left in Figure 2. Increasing that 6 to a 7 takes us into a new realm, the hyperbolic plane. Whether you have taught a course in “the three geometries,” or are new to this whole circle of ideas, you can see how reflecting across the sides of a triangle with greater than, equal to, or less than 180◦ leads to remarkably different visual appearances.

You may wish to return to the app and try some more experiments to see what you can find. When you do, notice the drop-down box labeled View. If you have chosen a spherical pattern, try the Stereographic with Equator option to see how the pattern looks under stereographic projection, with a convenient equator inserted to help you orient yourself. If your pattern has hyperbolic geometry, you can see the pattern in the Poincaré Upper Halfplane or a model popularized by Vladimir Bulatov, which Stampfli calls the Bulatov strip.

The artistic dimension

A little experimentation shows that the pattern depends heavily on which photograph you chose to open and where the triangle is positioned on the photo. This is where artistic choice comes in. The controls allow you to move the triangle around the photo, searching for a pattern that you find beautiful enough to want to share with others. In addition to translating the triangle, you can dilate and rotate it. The positions of the corners of the triangle on the photo seem critical: The viewer’s eye will be drawn to the points, which turn out to be centers of rotational symmetry in the pattern.

When I created the image on the right in Figure 1, I put the centers of 3-fold rotational symmetry—the corners where the triangle has an angle of π/3—on the dark placemat in order to create a negative space to try to balance the image. I also enjoyed the small blue hearts created as the triangle passes across my mother’s blue vase.

For the purpose of illustration, I did no such manipulation to seek better versions of the images in Figure 2. I simply changed the smallest angle to π/6 and then π/7. Without human intervention, the resulting patterns are not as nice. Especially in the hyperbolic pattern, the range of colors from the picture is not used to great advantage. Before I ask my students to create their own patterns, I show them a demonstration like this and ask them to find their own answers to the question of what makes one pattern more beautiful than another. In my book [1], I use a completely different approach to making patterns: Fourier series. When it comes to deciding which image among myriad possibilities to offer as an artistic creation, the issues are the same, so I refer you there to read my further thoughts about that rich issue.

Geometry assignments

To couple the assignment to find beautiful patterns with something more substantial mathematically, I enlist another technological tool, Geogebra. This free product makes it easy to draw compass-and-straightedge constructions on top of images produced by Stampfli’s kaleidoscope. At an elementary level, students could draw lines to confirm that it really is a 30-60-90 triangle in the Euclidean pattern. For my classes, Survey of Geometry and Differential Geometry, I used more advanced geometric concepts.

For the survey course, we read Michael Hitchman’s intriguing text, Geometry with an Introduction to Cosmic Topology [2]. Students learn how to construct the great circle joining two points in the stereographic model of the sphere.(It’s not hard to construct the antipodal point of a point z.) This means that they can draw great circles—the analog of lines on the sphere—and draw the angles in the triangle that produced the pattern.

For a richer exercise, students can figure out how to create a tool to draw circles in spherical geometry and so test the regularity of the pattern. For instance, Figure 3 was created from a triangle with settings 2-3-4 and a photograph featuring red cherries and green chard leaves. Geogebra allows us to draw circles to show that four of the red centers of 3-fold rotational symmetry are indeed all the same distance from the green center of 4-fold symmetry. The cream-colored equator passes through four 4-centers, which are vertices of a regular octahedron blown out onto the surface of the sphere and then moved to the plane through stereographic projection.

Similar exercises are possible in the Upper Halfplane, where the mathematical construction of a hyperbolic compass tool is rather more difficult. For the course in differential geometry, students can use the local Gauss Bonnet Theorem to compute the areas of these hyperbolic triangles. My point in this short blog post is that exercises that range from rather easy to quite difficult can use Stampfli’s kaleidoscope as an artistic starting point.

Frank A. Farris is Professor of Mathematics and Computer Science at Santa Clara University, where he has taught since 1984. His book, Creating Symmetry, The Artful Mathematics of Wallpaper Patterns (Princeton, 2015), grew from his service as Benedict Distinguished Visiting Professor at Carleton College in 2011.

References

[1] Farris, F. 2015. Creating Symmetry: The Artful Mathematics of Wallpaper Patterns. Princeton, NJ : Princeton University Press.

[2] Hitchman, M. 2009. Geometry with an Introduction to Cosmic Topology. Open source. https://mphitchman.com/