Non-Euclidean geometry perfectly encapsulates this exciting feature of mathematics. Last week I had the pleasure of attending a lecture by Dr. Evelyn Lamb about hyperbolic geometry--in particular, visualizing its counterintuitive properties. Hyperbolic geometry is a type of Non-Euclidean geometry that was discovered by changing one of the five fundamental axioms chosen by Euclid over two thousand years ago. The axiom in question, called the parallel postulate, states that, given a line and a point off that line, there is a single, unique line that passes through that point and never intersects the first line, no matter how far the two lines are extended. Change this assumption, and the shape of your space is dramatically altered.
Euclidean geometry is the geometry of the flat plane, the geometry I was taught in middle school, the geometry you can draw on a piece of paper. The consequences of choosing Euclid's version of the parallel postulate have been explored for thousands of years. From it we get results like the Pythagorean Theorem and the theorem that the interior angles of a triangle add up to 180 degrees, among others.
So what happens when we change the parallel postulate? Let's rephrase it: given a line and a point off that line, there is NO parallel line that passes through that point that will never intersect the first line. When extended, all lines eventually intersect. This causes the space to close up on itself and gives us spherical geometry, which is familiar to us when we look at a globe. We live on the two-dimensional surface of a sphere, and this axiom system describes it. In this geometry, we get counterintuitive results. For instance, interior angles of a triangle add up to more than 180 degrees.
We can change the parallel postulate again. Here's the third version: given a line and a point off that line, there an infinite number of lines that passes through that point and they never intersect the first line, no matter how far the lines are extended. This is hyperbolic geometry, an even stranger structure that is difficult to describe without the language of mathematics. It is exponentially expansive, and just like a sphere, it is impossible to draw in a Euclidean plane without distortions. This axiom system gives us triangles with interior angle sums that are less that 180 degrees. In fact, in hyperbolic geometry, it is possible to have triangles that have interior angle sums of zero!
Dr. Lamb gave us a few techniques for conceptualizing the hyperbolic plane in the talk, from crochet to 3D printed models. But what captured my imagination most was a pattern for a hyperbolic blanket. So I decided to make it.
How can you translate hyperbolic geometry into the surface of a physical blanket? Using tessellation, or tiling. Just as equilateral hexagons, squares, and triangles can tile the Euclidean plane, other shapes can tile hyperbolic and spherical surfaces.
In the Euclidean plane, we have 360 degrees of space around every point. A tiling of equilateral triangles demonstrates this. At each point, six equilateral triangles meet, each with all interior angles equal to 60 degrees. Multiplying 6 by 60 yields 360, as expected. The picture below demonstrates this tiling, made from toy magnets.
In spherical geometry, there are less the 360 degrees of space around each point. This allows the surface to close into a bounded spherical surface. The tiling below shows five equilateral triangles meeting at each point, each with all interior angles equal to 60 degrees (although in the flat picture, they don't look equilateral due to distortion, you'll have to trust me). Multiplying 5 by 60 gives us 300, much less than the expected Euclidean 360. See below.
Finally, in hyperbolic geometry, there are more that 360 degrees of space around each point. This makes the surface expansive and very floppy. The tiling below shows seven equilateral triangles meeting at each point, each with all interior angles equal to 60 degrees (again, they don't look equilateral due to the pesky distortion). Multiplying 7 by 60 gives us 420, more than the expected Euclidean 360. Here's a picture.
The blanket pattern I used uses a tiling with four pentagons clustered around each point. The interior angles of a pentagon are 108 degrees, so fitting them four-to-a-point gives us 432 degrees to accommodate. The only surface that can do this is hyperbolic. The cozy result is shown below.
There is so much more to explore when it comes to hyperbolic geometry, and much of it is available online. For more art featuring hyperbolic tilings, look to the works of M. C. Escher. For some mathematics-based intuition, I recommend this series of Numberphile videos that explore what it would like to live on a hyperbolic surface.
