Hyperbolic plane tilings and polyhedra_

Hyperbolic plane tilings and polyhedra both lie on curved surfaces, with nega...
Tim HuttonTim Hutton - 2014-10-08 21:35:17+0000 - Updated: 2014-10-09 07:31:56+0000
Hyperbolic plane tilings and polyhedra both lie on curved surfaces, with negative and positive curvature respectively. But how can a surface have negative curvature, when the radius of a circle is always positive? It turns out that a circle can have an imaginary radius, and hence a negative squared radius. And what does a circle with an imaginary radius look like? It's a hyperbola. That's why we call it a hyperbolic space.

In my new web toy you can see a sphere morph into a hyperboloid and watch the effect on a plane tiling and its projection as the curvature changes.
http://timhutton.github.io/hyperplay/projection_2D.html

Intriguingly, the hyperboloid model is also how spacetime works in special relativity (Minkowski space), so there's a deep connection between hyperbolic space tessellations and the world you're sitting in right now.

Related demos and source code are here: https://github.com/timhutton/hyperplay
Hyperbolic plane tilings and polyhedra both lie on curved surfaces, with negative and positive curvature respectively. But how can a surface have negative curvature, when the radius of a circle is always positive? It turns out that a circle can have an imaginary radius, and hence a negative squared radius. And what does a circle with an imaginary radius look like? It's a hyperbola. That's why we call it a hyperbolic space. In my new web toy you can see a sphere morph into a hyperboloid and watch the effect on a plane tiling and its projection as the curvature changes. http://timhutton.github.io/hyperplay/projection_2D.html Intriguingly, the hyperboloid model is also how spacetime works in special relativity (Minkowski space), so there's a deep connection between hyperbolic space tessellations and the world you're sitting in right now. Related demos and source code are here: https://github.com/timhutton/hyperplay

Hyperbolic plane tilings and polyhedra both lie on curved surfaces, with negative and positive curvature respectively. But how can a surface have negative curvature, when the radius of a circle is always positive? It turns out that a circle can have an imaginary radius, and hence a negative squared radius. And what does a circle with an imaginary radius look like? It's a hyperbola. That's why we call it a hyperbolic space. In my new web toy you can see a sphere morph into a hyperboloid and watch the effect on a plane tiling and its projection as the curvature changes. http://timhutton.github.io/hyperplay/projection_2D.html Intriguingly, the hyperboloid model is also how spacetime works in special relativity (Minkowski space), so there's a deep connection between hyperbolic space tessellations and the world you're sitting in right now. Related demos and source code are here: https://github.com/timhutton/hyperplay

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Tim Hutton - 2014-10-10 09:37:53+0000 - Updated: 2014-10-10 10:01:31+0000
"For God's sake, please give it up. Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life." -  Wolfgang Bolyai, 1820, urging his son János Bolyai to give up work on hyperbolic geometry.
Tim Hutton - 2014-10-10 09:59:46+0000 - Updated: 2014-10-10 10:00:09+0000
"From this I should almost conclude that the third hypothesis would occur in the case of an imaginary sphere." - Lambert, 1766. A cryptic half-joke half-serious comment that we now see as exactly correct. It illustrates the fascinating mental contortions that accompany the development of new concepts.

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