Circle Limit III

The angles of a triangle must add up to 180°. This is a simple fact that you were probably taught fairly early in your math career. It’s been known for millennia and is pretty simple to prove: for a right triangle, assume we have two parallel lines, one line perpendicular to them, and a fourth line between one of the intersections and an arbitrary non-intersection point on the opposite line as shown below.
triangle-1
This makes a triangle with one right angle, C, and two acute angles, A and B. We also need to consider angle D, the complementary angle to A. We know that A+D has to be 90° since they sum together to make a right angle, so the measure of angle D must be 90° – A. Since D and B are alternate interior angles with respect to the parallel lines and the red transverse line (remember all those awful congruence theorems you learned in your high school geometry class?) they have to be congruent angles. This means that the measure of angle B has to be 90°-A as well. So if we sum up the angles inside the triangle, A + B + C = A + 90° – A + 90° = 180° + A – A = 180°. The proofs for acute and obtuse triangles are similar, but a bit more complicated so we won’t go through them. The point is, we proved it! Triangles have to have 180°, right? Wrong.
The proof we used—and indeed all proofs that triangles must have 180° inside them—relies in some way on an infamous postulate used by Euclid around 300 BCE that says (more or less) that given a line and a point not on that line there is exactly one line through the point that does not intersect the original line. This postulate, though reasonable sounding, foiled mathematicians for thousands of years. Despite attempt after attempt to prove this postulate, no one was ever able to succeed. In fact, it was eventually proven that there IS no proof of this persnickety postulate. The angry mathematicians, having been foiled by this simple-yet-unprovable statement, began to consider what would happen if, indeed, it were not true. What would happen if, for example, there were an infinite number of lines through the point that didn’t intersect the original line? This line of questioning led to the discovery of hyperbolic geometry: a world where there are infinitely many parallels to a line through a given point off the line.
One of the many interesting aspects of hyperbolic geometry is that triangles don’t have to have 180°—In fact, they must have less than 180° (otherwise they could be a triangle in spherical or euclidean geometry). These triangles can still tessellate a plane though! In one particular representation of hyperbolic space, called a Poincaré disk, this tessellation would look like the image below.
The Poincaré disk is a way to show the hyperbolic plane on a circle. The idea is that straight lines are represented as curves from one side of the circle to another with the intention of preserving angles without necessarily preserving lengths. These curves must be circles that intersect the boundary of, or must be diameters of, the disk. The result is that each triangle in the picture above is the same size! From the large-looking central triangles to the itsy bitsy ones on the edge, each triangle would have exactly the same area in a hyperbolic space.
M.C. Escher was a Dutch artist whose graphics are widely known for their otherworldly bizarre mathematics. Stairs that led up to themselves and water that flowed in a ring are just two examples of his pieces, enacted with an almost formulaic mathematical exactness. He is well known in scientific communities for the diagramesque works of art.
You may be asking what this little Dutch artist has to do with our discussion of “curved” triangles. Well, Escher had become somewhat famous for using tessellations in his work. Creating shapes, especially in the shape of animals, which would tessellate all the way across the pieces, forming a lattice of cells that had only to be filled with a clever image. In the early 1950s, he became curious about finding different ways to “draw” infinity on a page. A letter from a friend came to him with some of these Poincaré tilings in the hyperbolic plane and became enamored with them. The images in the letter were a type of tiling denoted by {p,q} that was a tiling of p-gons with q of them meeting at each vertex. These images of hyperbolic tilings inspired Escher to create his Circle Limit series in 1959 and 1960. Circle Limit III was inspired in particular by the {8,3} tiling—4 octagons meeting at every vertex, and is a beautiful reimagining of the tiling with fish in place of the triangles.

Circle Limit III by M.C. Escher. His other work, including the other Circle Limits, can be found at http://www.mcescher.com.

caption

Circle Limit III with the {8,3} tiling overlaid on it. Image by Doug Dunham.

Escher’s works seem to represent the very nature of the hyperbolic plane that we have talked about. After all, in a world where there are an infinite number of parallel lines, why couldn’t I draw infinite fishes on a page?
Anyone wanting to know more can Google hyperbolic geometry, parallel postulate, M.C. Escher, or triangle group.
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