So I salvaged all my posts.

This is part 2. Here is part 1.

Other stuff by me.

n

You can interpret this geometrically.

Any odd square can be the last term of such a sequence, and any even square the sum of the last 2 terms. Using this, you can construct Pythagorean triples for every number:

1^{2} = 1^{2} -
0^{2}

2^{2} = 2^{2} -
0^{2}

3^{2} = 5^{2} -
4^{2}

4^{2} = 5^{2} -
3^{2}

5^{2} = 13^{2}
- 12^{2}

6^{2} = 10^{2}
- 8^{2}

7^{2} = 25^{2}
- 24^{2}

…

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It has 4 triangles and 3 squares. (Not so wel visible in the picure, it it octahedral in shape ) It is non-orientable: like a Klein bottle, you cannot assign an inner and outer surface to its faces.

If you slice the polyhedron in half, you get the paper model I made. It is is a Mobius strip, that self-intersects itself! You might expect something like Mobius strips to come up in a non-orientble surface like the Tetrahemihexahedron. One way to see it is a Mobius strip, is to look at the cut-out pattern: it is a strip, which has to be glued twisted.

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Derek Wise:

Nice! I'm trying to guess the idea behind this ... look forward to seeing your explanation.

Dmitry Shintyakov:

Have you considered using conformal mappings? Here is a practical recipe:

math.stackexchange.com - Conformal map from Mandelbrot set to Disk

//math.stackexchange.com/questions/2719240/conformal-map-from-mandelbrot-set-to-disk>

Gerard Westendorp:

+Dmitry Shintyakov I did try an appraoch with conformal mappings. I thought in that case the outside of the Mandelbrot would be easier, because the escape to infinity forms rings, and these rings would correspond to the generations of heptagons in a (7,3) hyperbolic tiling. The inside is trickier. Not that I gave up this approach, but I thought of something different. Basically, a circle packing, the middle cirles have to touch their 7 neighbours, and the circles on the edge 'feel' if they are on the edge of the Mandelbrot, and iteratively shrink or expand to get there. (But I hope to get more on that this weekend)

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I can now make 'Indra pearls' style fractals.

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The algorithm I mentioned in my last post seems to be progressing in the right direction.

What I want to do is tile fractals with hyperbolic tilings, using circle packings. So 3 of my favourite subjects come together.

In a circle packing, you create a graph, and then you put a circle (or perhaps an n-sphere) on each vertex. Then you demand that the radii are such that they all touch. This turns out to be possible for a wide range of cases. If you do that with a graph corresponding to a (7,3) hyperbolic tiling, you will obtain a packing from which you can construct the tiling. (still need to do that)

The circles in the middle ultimately get their radii dictated by those on the edge (as in the ‘holographic principle’!). The radii on the edge, you can choose.

So the idea is, for all edge circles, I look at the coordinates, turn them into a complex number, iterate them as a pixel of the Mandelbrot, and note the number of iterations it takes to go to the escape radius of 2 (hmm, maybe I should take a slightly larger number…). This number can be infinite if the pixel is on the inside of the Mandebrot, in which it is truncated to some value. Call this number N. So I want N to a specific target number N_target. If all the edge circles would satisfy that, then they would all be in the escape-time ring near N_target. For large N_target, this approximates the Mandelbrot.

To converge to this situation, I make an edge circle smaller if it escapes too fast to infinity, and larger if it escapes too slow. The middle circles ‘feel’ this, a bit like plant cells, as they try to adapt to touching their neighbours. The way to do that: Make the circle smaller if the triangles corners formed by the ring of neighbours sum to more than 360 degrees, and larger if they sum to less than 360 degrees.

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Right! This is what I was trying to make. I'll make a webpage about it to explain it a bit more in detail, and put in some more results.

Gerard Westendorp:

Nice article about circle packings:

math.utk.edu - www.math.utk.edu/~kens/Notices_article.pdf

//www.math.utk.edu/~kens/Notices_article.pdf>

Harald Hanche-Olsen:

Remember back when the Mandelbrot set was all the rage and we got these movies zooming seemingly forever into the boundary? It would be interesting if such movies could be made with hyperbolic tilings. It would require either a much improved algorithm or a faster computer, though. Preferably both.

Now I want to go back and revisit the basic theory of the Mandelbrot set.

Gerard Westendorp:

+Harald Hanche-Olsen Hmm...The algorithm could very probably be much improved. I was quite enthusiastic about fractals in the 80's. I wall papered my flat with a large Mandelbrot, on something like 6X24 A4 papers, each of which took an hour to print on a noisy color matrix printer. After complaints, I inserted some code to check what time of day it was and wait till 10am, until the computer was allowed to continue with printing.

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I made a website on hyperbolic tilings of fractals: https://westy31.home.xs4all.nl/PoincareMeetsMandelbrot/PoincareMeetsMandelbrot.html

Refurio Anachro: Very nice post! I need to try this, too. Btw, you wrote:

> /We have p/7 + p/3 + p/2 = p*41/42, so that is hyperbolic. In fact, it is the closest of all hyperbolic tilings to being non-hyperbolic […]/

I bet that should be /regular/ hyperbolic tilings. What a surprising fact! Maybe it shouldn't be too surprising, as (afair) the 42-gon is the largest polygon appearing in a planar vertex figure.

Edit: Having thought about it a little longer, these two appearances of the number 42 don't seem to be as related as I thought them to be. Irritating. Gerard Westendorp: +Refurio Anachro I think I should have said (7,3,2) is the least hyperbolic triangle of a triangle group. I think this also implies that for example the Klein Quartic has the most symmetries of any genus 3 object. It is tiled by 336 hyperbolic triangles. The angular deficit = 336 * (42-41)/42 * pi = 8*pi. Any other triangle group would uses less tiles. Refurio Anachro: It is related, I just forgot how exactly the story went, and started to doubt when I found myself unable to put it together again. It's not very hard, here's the link to +John Baez ' blog, where I first heard about it:

http://www.math.ucr.edu/home/baez/42.html

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2 post ago I wrote about warping a hyperbolic tiling into the interior or the Mandelbrot. You can use the same algorithm to tile other shapes with (7,3) triangles, as +David Eppstein

The heptagonal case.

Roice Nelson: nice, love it!

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https://westy31.home.xs4all.nl/Hingedpolyhedra/Hingedpolyhedra.html HingedPolyhedra

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Gerard Westendorp:

+Alison Grace Martin

Thanks, didn't know some of that. I discoverd various hinged polyhedra years ago, but later I found that Buckminster Fuller had already discovered the octahedral case, which he called the Jitterbug. His first prototype didn't work, he found out as I did that you need to construct hinges that conserve the normal vectors of the faces. One of your references shows a torus tiled by triangles, in a similar way I did the octahedron.

Jon Eckberg:

+Gerard Westendorp I think I asked this before but I can't find the exchange. What do you use to render the mechanics?

Gerard Westendorp:

+Jon Eckberg I generate '.inc' files for POV-Ray(A freeware raytracer) using VBA. Pov-ray can make a series of .bmp files (the frames). I batch-convert these to .gif, with some cropping and pixel reduction included, using Irfanview, my favorite image viewer. Then make them into an animated gif using the freeware Unfreeze'.

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Here is a Youtube clip of a 6 minute talk I did on Hinged polyhedra:

Hinged Polyhedra & Hinged Tessellations — Gerard Westendorp

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I havn't seen +Jos Leys

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Crumple 8 milk cartons into a torus.

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So armed with the picture I made, I can try to understand the counterintuitive fact that non transitive dice are possible.

Qiaochu Yuan: Here's a fun generalization: math.stackexchange.com - Generalized nontransitive dice

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Roice Nelson: These are awesome!

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For the non-intersecting part of the sequence, a mechanical model is possible. (I have several). But recently, I decided to look at the intersecting part too.

Boris Borcic: the icosahedron being the convex hull of the great dodecahedron, I'd expect their compound to be dull. Gerard Westendorp: +Boris Borcic Well, thats what the hinges form...

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I like this video. Worth watching if you care about the truth on infinite sums and the Riemann zeta funciton.

https://www.youtube.com/watch?v=YuIIjLr6vUA

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The 5X5 version.

Gerard Westendorp:

+Jon Eckberg In a reply of the post linked below, I put photo of an actual working model. It is pretty close to collapsing all the way. Maybe exact is possible, if you turn the hinge connecting 2 heights into a kind of crankshaft that uses the space outside the 3X3 square. (You probably can't follow me here, I would to draw or build it rather than try in words...) plus.google.com - I don’t think anyone noticed yet that if you continue transforming the hinged...

Owen Maresh:

Also, what happens to, say, hyperbolic tilings? Owen Maresh:

And, does the infinite limit work?

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