Snell and Escher

A few weeks ago, Grant Sanderson posted a video on the brachistochrone, with guest Steven Strogatz.

The video explains Johann Bernoulli’s solution to the problem of finding the brachistochrone, which is a clever application of Snell’s Law. I immediately wondered if a similar application could be used to explain the behavior of geodesics in the hyperbolic plane, which it turns out is true. I’m not the first to think of this, but it doesn’t seem to be well-known, so that’s what I’ll try to explain in this post. This may become my standard way of introducing hyperbolic geometry in informal settings, i.e., when formulas aren’t needed. (As an example of another exposition that describes hyperbolic geodesics this way, see the lecture notes for this geometry course.)

Snell’s Law, as represented in the above diagram (image source), applies to light traveling from one medium to another, where the interface between the two is horizontal. If light travels at speed \(v_1\) in the first medium and \(v_2\) in the second medium, and its trajectory meets the interface at an angle of \(\theta_1\) and leaves at an angle of \(\theta_2\) (both angles measured with respect to the vertical), then \[ \frac{\sin\theta_1}{v_1} = \frac{\sin\theta_2}{v_2}. \] This is the case of two distinct media. Snell’s Law has a continuous version (derived from the discrete one by a limiting process, as suggested in the video). Suppose light is traveling through a medium with the property that the speed of light at each point depends on the vertical position of the point. That is, the speed of light in this medium at a point \((x,y)\) is a function \(v(y)\), which may vary continuously. At each point of a trajectory of light in this medium, let \(\theta\) be the angle formed by the direction of the trajectory (i.e., the tangent line) and the vertical. Then the quantity \[ \frac{\sin\theta}{v(y)} \] is constant along the trajectory.

So suppose we are looking at a medium that covers the half-plane \(y > 0\), in which light travels at a speed proportional to the distance from the \(x\)-axis: \(v(y) = cy\). (The constant \(c\) may be thought of as the usual speed of light in a vacuum, so that along the line \(y = 1\) light moves at the speed we expect. As we shall see, this is analogous to the fact that distances along the line \(y = 1\) in the hyperbolic metric match Euclidean distances. Of course, it also means that light moves faster than \(c\) above this line, which is physically impossible, but we’re doing a thought experiment, so we’ll allow it.) If we imagine someone living inside this medium trying to look at an object, what direction should they face?

From our outside perspective, it seems that the observer should look “directly at” the object, in a straight (Euclidean) line. However, in this medium light does not travel along Euclidean line segments, but instead along curved arcs, as illustrated below.

Click on the graph to go to an interactive version.
It’s not too surprising that light follows a path something like this if it’s trying to minimize the time it takes to travel from the object to the observer: the light travels faster at higher vertical positions, so it’s worth going up at least slightly to take advantage of this property, and it’s also worth descending somewhat sharply so as to spend as little time as possible in the lower, slower regions.

What may come as a surprise is that the path of least time is precisely a circular arc. With Snell’s Law, however, this fact can be derived quickly. We have that \(v(y) = cy\), and so along a light trajectory \[ \frac{\sin\theta}{cy} = \text{constant}. \] Multiplying both sides by \(c\), we find that \(\frac{\sin\theta}{y}\) is also a constant. If this constant is zero, then \(\theta = 0\) constantly, so the path is a vertical segment. Otherwise, call this constant \(\frac{1}{R}\). Then \(y = R \sin\theta\). Now set \(x = a + R \cos \theta\). The curve \[ (x,y) = (a + R \cos\theta, R \sin\theta) \] parametrizes a circle centered at \((a,0)\) by the angle between the \(x\)-axis and the diameter. It remains to see that this angle \(\theta\) is the same as the angle between the vertical direction and the tangent line at the corresponding point of the circle. This equality can be shown in any number of ways from the diagram below.

Click on the graph to go to an interactive version.
This is not to say that this parametrization describes the speed at which light moves along the path. As previously observed, light slows as it approaches the horizontal boundary, that is, the \(x\)-axis.

But perhaps we’ve been prejudiced in assuming our perspective is the right one. We’ve been looking with our Euclidean vision and supposing light moves at different speeds depending on where it is in this half-plane. Thus it seems to us that light covers Euclidean distances more quickly the further it gets from the \(x\)-axis. But relativity teaches us that distance isn’t absolute: instead, the speed of light is what’s absolute. So perhaps we could gain greater insight by measuring the distance between points according to how long it takes light to travel between them. That is, we assume that the paths determined above are the geodesics of the half-plane, and by doing so we learn to “see” hyperbolically. Then we are not troubled by looking at an image like

(image source) and being told that all of the pentagonal shapes are the same size, because we’ve learned to look at things with our hyperbolic geometry glasses on.

M. C. Escher illustrated (or, more accurately, approximated) the hyperbolic geometry of the upper half-plane with his print Regular Division of the Plane VI (1958), shown below (image source).

This design was created during a time Escher was attempting to visually depict infinity. It was shortly before he had encountered the Poincaré disk in a paper by Coxeter, which discovery led to the Circle Limit series. In this print, the geometry of each lizard is Euclidean, structured around an isosceles right triangle. Each horizontal “layer” has two sizes of triangles, one scaled down from the other by a factor of \(\sqrt{2}\). The side lengths of the triangles in one layer are one-half of those in the layer above, so the heights of layers converge geometrically to the horizontal boundary at the bottom. Some of the triangles are outlined in the next image.

Some questions I have about Escher’s print:

• How different would this image look if it were drawn according to proper hyperbolic rules, with each lizard having reflectional symmetry, and each meeting of “elbows” having true threefold symmetry? (This would give the tessellation with Schläfli symbol {3,8}, an order-8 triangular tiling.)
• If we suppose that the right triangles act as prisms, with light moving at a constant speed inside each one, but this speed being proportional to the square root of the triangle’s area, then what will the trajectories of light look like as it moves through the plane? Will they approximately follow circles?
• How many lizards are in the picture?

Coda: Jos Leys has taken some of Escher’s Euclidean tessellations and converted them to hyperbolic ones, in both the disk and the half-plane model.