Circle Limit - Wood Engravings by M.C. Escher

Around 1956, M.C. Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter inspired Escher’s interest in hyperbolic tessellations, which are regular tilings of a hyperbolic plane. Escher’s wood engravings Circle Limit I–IV demonstrate this concept. In 1959, Coxeter published his finding that these works were extraordinarily accurate: “Escher got it absolutely right to the millimeter.”

Hyperbolic planes are difficult to explain. In fact, hyperbolic geometry is an extremely huge topic. Many visualizations of hyperbolic planes have been discovered (including the circle limits afore mentioned). Taking Circle Limit III for example (the one with the fishes), here is the gist of what these artworks have to do with hyperbolic geometry:

  • The number of fishes within a distance of n from the center rises exponentially.
  • The fishes have equal hyperbolic area. Yes, the tiny fishes on the very edge of the circle are the same size as the fishes in the center (on an actual hyperbolic plane, anyway).
  • So, the area of a ball of radius n must rise exponentially in n.

Learn more. (x) (x) (x)

theomenroom  asked:

oh, since you're working on a rasterizer, I noticed a related software niche that you might be interested in knowing about: the hyperrogue dev has claimed on his blog that hyperrogue is the state of the art in visualizing hyperbolic geometry. this doesn't surprise me since the other visualizers I've found are really limited java applets. it would be interesting if there were a visualizer that did that (moving the viewpoint around in hyperbolic space) in 3d

Ooh, that sounds like a fun project. It will take quite a bit of research for me to figure out how existing algorithms like raytracing and rasterisation would work with hyperbolic geometry, but yeah, it would be fun to try out. It would be especially cool if I could get it working with WebGL in the browser - in principle I imagine a suitable vertex shader could be written to do the necessary transformation and the rest of the graphics pipeline wouldn’t have to change too much.

Now I’ve basically finished my software rasteriser (there’s going to be one final post on adding textures), I’m going to move to learning real-time graphics APIs like OpenGL, WebGL (basically OpenGL with a JavaScript interface) and Vulkan. A projective space visualiser will need to wait until I’ve gotten decent at OpenGL in plain old Euclidean space, but I appreciate the idea :)

Maryam Mirzakhani

There are moments when we do not agree with what is happening around us… 

Maryam Mirzakhani (born May 3, 1977) is an Iranian-American mathematician and a professor of mathematics at Stanford University.

On 13 August 2014, Mirzakhani became both the first woman and the first Iranian honored with the Fields Medal, the most prestigious award in mathematics. The award committee cited her work in “the dynamics and geometry of Riemann surfaces and their moduli spaces”.

Her research topics include Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry.

In hyperbolic geometry (geometry on the surface of something that looks like a Pringle), the sum of the 3 angles of a triangle is always less than 180 degrees.

Remember in any geometry, lines are length minimizing curves. In regular, flat (Euclidean) geometry these are just straight lines. Making sense of alternative geometries requires one to rethink the idea of a straight line as the shortest path between 2 points. The shape above is made with 3 length minimizing curves and is therefore a triangle.

More on hyperbolic triangles here https://en.wikipedia.org/wiki/Hyperbolic_triangle