Tuesday, June 25, 2013

Hyperbolic Space

In hyperbolic space, you look ahead, to the left or right, you see the back of your own head, repeated into infinitely.  If you reach out and touch the shoulder in front of you, you feel your own hand on your shoulder.  This is the nature of hyperbolic space.

 Thanks to GenAmbiance for the wonderful ambiance!


 

Hyperbolic tiling from Vladimir Bulatov.

I've always been interested in non-Euclidean geometries.  Recently discovered Vladimir Bulatov wonderful visualizations.  He has set the bar pretty high but the underlying formulas are fairly accessible.



Here is a good theory primer...
http://www.math.brown.edu/~rkenyon/papers/cannon.pdf

A video introduction to fundamental domains.


What do you say?  Ready for some hyperbolic fractals?

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2 comments:

  1. UnknownJune 26, 2013 at 12:58 AM

    On seeing the back of your head: This can happen in hyperbolic, Euclidean or spherical spaces (or better, manifolds) - all it requires is a closed geodesic loop in the manifold. Then looking along this geodesic you will see yourself. The geometry of the space (hyperbolic, Euclidean, spherical or something else) is not important to this feature.

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  2. bJune 26, 2013 at 9:52 PM

    Indeed, I have any affection for the Asymptotic nature of the universe. Convergent or divergent as the specifics may be, I enjoyed the self referential world of numerical modelling. Is this real of the nature of our minds? I cannot say.

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