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*Rotation*

Hyperbolic geometry is an example of a non-Euclidean geometry.

Hyperbolic Pattern 11 by Ross Hilbert on Flickr.

Experiments in Processing: curious eels in hyperbolic sea.

I love hyperbolic geometry, heptagons, and Escher.

The eels move on the sides of a triheptagonal tiling of the hyperbolic plane in the Poincaré disk model. You better appreciate this because getting the coordinates is a real pain in the *** (done with a python script, <a href=“https://github.com/rantonels/poincare“>source</a>).

BONUS: nauseating Moebius translation.

Rotation around the symmetry axis of 4th order in the disk and upper half plane models

another tiling of the hyperbolic plane by regular right-angled pentagons

(source)

“Snakes” - MC Escher (1969)

Math can be beautiful.

These pictures are pieces of art that have been made based off of the Klein and Poincare disk models for hyperbolic geometry (where through any given point there are an infinite number of parallel lines)

This is a tiling of regular right angled dodecahedron in three dimensional hyperbolic space shown through 4 iterations. 12 generators of the tiling are reflections in each of the 12 dodecahedron’s faces.

madore.org

Hyperbolic maze

Have some time to kill? Then try this maze set in the hyperbolic plane, displayed using the Poincaré disk model.

Ok. MIND : BLOWN. **Margaret Wertheim** is officially my newest crush. In this TED talk she explains how hyperbolic geometrical shapes that look very much like coral can be expressed through crochet.

The idea of translating hyperbolic planes into crochet was first thought by **Daina Taimina** (you can watch her TED talk about it **here**) and for the first time, an actual hyperbolic shape could be translated into a real, physical shape.

Margaret Wertheim is one of the creators of *The Institute for Figuring* and has also written many interesting books on the cultural history of physics. Her first book was called *Pythagoras’ Trousers: God, Physics, and the Gender Wars*. The google books summary of it is:

Here is a fresh, astute social and cultural history of physics, from ancient Greece to our own time. From its inception, Margaret Wertheim shows, physics has been an overwhelmingly male-dominated activity; she argues that gender inequity in physics is a result of the religious origins of the enterprise.

Now I’m pumped for reading AND doing some hyperbolic crochet! (iomikron, are you up for some crafts?)

An ideal rotation around the point at infinity in the disc and upper half plane model of the hyperbolic plane

**Escher’s Circle Limit I - IV and Hyperbolic Geometry**

All four of M.C. Escher’s Circle Limit woodcuts are great works of art and superb examples of hyperbolic geometry. Hyperbolic geometry is a form of geometry where the parallel postulate of Euclidean geometry does not hold. This is reflected in the world represented in each of the woodcuts. In each one, the figures are not getting smaller as the reach the edge of the circle as they would in Euclidean geometry. From the figures’ perspective, they are all the same size, regardless of where they are in the circle. Their entire world lies completely within the flat plain of the circle, the edge of which to them remains infinitely far away.

when I read anything about pants

In topology, *pants *are a surface which resemble a pair of pants. You can find out more here and at this article.

Here is a mathoverflow question about turning a pair of pants inside out!

Math Limericks

Gauss was a pretty weird dude,

He said P5 couldn’t be proved.

But instead of admitting,

He neglected submitting,

And left the mathematicians all screwed.

Hyperbolic translation along the horizontal axis in the disc model, and along both horizontal and vertical directions in the band model