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Hyperbolic Fibration Animation 1

Hyperbolic Fibration Animation 1

Owen Maresh asked about making videos of the hyperbolic hopf fibrations, and here is a simple animation rotating the first around.

Cooler stuff to be done:
- Animate a fibration with hyperbolic transformations
- Animate through the 2-parameter family of fibrations
- Show the fibration and quotient surface side by side, and animate pulled back surfaces like Niles Johnson did for S^3

For more details, see my original post:
https://plus.google.com/+RoiceNelson/posts/1w3aoQgj61g

P.S. I'd like to figure out how to make these look better on youtube/vimeo.  I wasn't happy with either, and the video looks much cleaner locally.  It seems video compression doesn't like all those tiny fibers. Suggestions welcome!
https://vimeo.com/146364033

Helicoid in H^3

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Helicoid in H^3

Here are three views of a hyperbolic helicoid, an idea suggested by Saul Schleimer. The first puts the helicoid axis on the vertical line through the origin. The other two apply a hyperbolic isometry to move the axis towards the right. Whenever the axis does not go through the origin, it will be an arc in the ball model.

What curve does a single ideal edge of the surface trace out on the ball boundary? I suspect it is a loxodrome, but I haven't proved this. (Update: It is! see comments below.)

Relevant Links

Euclidean helicoids
en.wikipedia.org/wiki/Helicoid

Loxodromes
en.wikipedia.org/wiki/Rhumb_line


The interactive versions are really well done and fun to play with. My favorites are the wobbly looking ones :D

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The interactive versions are really well done and fun to play with.  My favorites are the wobbly looking ones :D

Originally shared by Malin Christersson

Rolling shapes

I remade some old GeoGebra-examples of rolling hypocycloids and epicycloids to interactive Javascript-versions; and distorted the curves to non-edgy versions.

Interactive versions at:
http://www.malinc.se/m/RollingHypocycloids.php

#math



Hyperbolic Hopf Fibrations

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Hyperbolic Hopf Fibrations

The Hopf Fibration of S^3 is amazing and beautiful.  Rather than describe it here, I'll point you to a lovely online reference with pictures and videos by Niles Johnson.

nilesjohnson.net/hopf.html

To understand it better (and fibrations in general), I recommend this talk by Niles too.

www.youtube.com/watch?v=QXDQsmL-8Us

It turns out there is an analogue of the Hopf fibration for H^3.  In fact, there is not just one "fiberwise homogenous" fibration in the hyperbolic case.  There is a 2-parameter family of them, plus one additional fibration that does not fit the family.  As with S^3, fibers in the H^3 cases are geodesics.  They are ultraparallel in fibrations from the family, and parallel in the exceptional fibration.  I found the following dissertation by Haggai Nuchi a good intro and resource to help think about all this.

www.math.upenn.edu/grad/dissertations/NuchiThesis.pdf

I'm posting three early pictures of H^3 fibrations below, labeled with t…

Nonagonal Firepit

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Nonagonal Firepit

For our 9th anniversary, Sarah Nemec-Nelson and I decided to gift each other a custom firepit with 9 sides. Through her interior design business, Sarah has an excellent contact named Brent Clifton who was excited about the project. He has a computer controlled plasma cutter, and the result is amazing. It's so heavy duty it will likely outlast our short time on this planet.

There are two nonagons in the design, the base and the rim. During design, Brent's wife had a very nice suggestion - scale the rim to the base by the golden ratio φ. But this would be more appropriate for a pentagonal firepit, since φ is the ratio of a pentagon's diagonal to its edge length. The nonagon has its own set of ratios, three of them in fact. Here is a paper all about it!

Peter Steinbach, Golden fields: a case for the heptagon, Mathematics Magazine 70 (Feb., 1997), 22-31. Available at http://www.jstor.org/stable/2691048

So we chose to use the smallest of the three nonagon ratios, …

Visualizing Hyperbolic Honeycombs

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Visualizing Hyperbolic Honeycombs

I'm super excited about this, as it is the first time I've contributed to a submitted paper!

Pick any three integers (larger than 2), and we describe how to draw a picture of a corresponding {p,q,r} honeycomb, up to and including {∞,∞,∞}.

Originally shared by Henry Segerman

New paper with Roice Nelson, "Visualizing Hyperbolic Honeycombs". http://arxiv.org/abs/1511.02851