## Salvaged GooglePlus Account of Gerard Westendorp - Part2

In october 2018, it was rumored that Google plus is intending to shut down.
So I salvaged all my posts.
This is part 2. Here is part 1.
Other stuff by me.

When he was 5 years old, he noticed that
n2 = 1 + 2 + 3 +… +(2n-1).
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:

12 = 12 - 02
22 = 22 - 02
32 = 52 - 42
42 = 52 - 32
5
2 = 132 - 122
62 = 102 - 82
7
2 = 252 - 242

### I just saw this documentary movie on Escher.

I hope it will become available outside the Netherlands. I've seen a lot of Esher stuff, but this is perhaps the best. He saw himself as an untalented artist/mathematician, but he just had to do what he did, because he was so captivated by it. https://www.youtube.com/watch?v=ul7gtJWM2YU

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### The Tetrahemihexahedron

The *Tetrahemihexahedron* is the only uniform polyhedron with an odd Euler characteristic: It has Euler characteristic 1. The same as a flat piece of paper, but this is a closed surface, which kind of has genus ½.
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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### Geodesic loops on the Cube

I was surprised that geodesic loops on the cube are quite complex! I was having some difficulty with geodesics on a non-convex polyhedron. So I tried looking at the cube first. On a sphere, all geodesics are great circles, which wrap around the sphere exactly once. The cube is kind of a deformed sphere, but its geodesic loops are quite different: They wrap around the cube in seemingly complex ways. You can make a geodesic over the cube’s surface by wrapping a ribbon over it. It would be nice if the ribbon forms a closed loop. How do we fix that? Suppose the edges of the cube have length 1, and the ribbon displaces sideways by distance delta every time it crosses a whole face. If it crosses N faces, then N*delta needs to be a whole number. In this case, N = 12 and delta = 1/4.

Maris Ozols: American football provides an even more counter-intuitive example where a geodesic can intersect with itself. See the picture here: en.wikipedia.org - Systolic geometry - Wikipedia Gerard Westendorp: +Maris Ozols Sorry for the slow reply, I was on a vacation. It looks like the subject of geodesics on a surface is much more interesting and complex than you might think if you just look at the spherical case!
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### RSA and Shor's algorithm

I worked a while on trying to understand RSA and Shor's algorithm. I think I get the idea now, and I made a web site on it: https://westy31.home.xs4all.nl/GerardQuantum/MyExplanationOfRSAandShor.html

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### I am trying to tile the Mandelbrot with a hyperbolic (7,3) tiling

This is work in progress, but I got excited by the first results. I am trying to tile the Mandelbrot with a hyperbolic (7,3) tiling. The idea I had seems to work! I'll be back with more, including explanation.

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 mentioned a couple of days ago. For example a square. (This could also be done with a Schwarz–Christoffel mapping.)

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The heptagonal case.

Roice Nelson: nice, love it!
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### Hnged Polyhedra

I made a new website on hinged polyhedra:
https://westy31.home.xs4all.nl/Hingedpolyhedra/Hingedpolyhedra.html HingedPolyhedra HingedPolyhedra westy31.home.xs4all.nl
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### Another hinged construction.

This one shows that you could have almost unlimited expansion ratio in ‘auxetic’ materials. I used 3X3 sub-squares, which leads to an expansion factor of 5 for each square but if you go to nxn, the expansion ration becomes (2n-1). (I chopped of the corners of squares that are not a hinge) The red triangles can be shrunk to zero. Now, they actually form an additional degree of freedom, you can hinge them independently. I left them in for aesthetic reasons.

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### A new hinged polyhedron.

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 on Google plus for a while, but he is still making cool video's: http://www.josleys.com/show_gallery.php?galid=376 Gallery : Connected sum of two real projective planes This sites features mathematical images and animations made by Jos Leys. josleys.com
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Crumple 8 milk cartons into a torus.

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### Non transitive dice.

The 3 dice shown are ‘non transitive dice’: On average, A will throw higher than B, B will throw higher than C, *but* C will throw higher than A. (NB average over the number of wins, not over the amount thrown) The dice were an ‘exchange gift’ from Numberphile blogger James Grime in the Gathering for Gardner conference. One way to check the non transitivity claim is to make 3 6X6 tables of possible outcomes. Each of these squares has an equal probably, so we can just count squares. Result: All 3 dice throw on average 3.5, like normal ones. [A beats B beats C beats A] each with a probability of 17/36. All have 4/36 probability for a draw.
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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### Laser cut Jitterbug.

The Jitterbug is a hinged polyhedron discovered by Buckminster fuller.

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### A laser-cut hinged polyhedron.

The hinges can be 'snapped' to the faces, allwing for fast assembly.

Roice Nelson: These are awesome!
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### Hinged polyhedra

Hinged polyhedra have made a come-back on my priority list. In this animation, the ‘hinged icosadodecahedron’ morphs between: icosadodecahedron snub dodecahedron rhombic dodecahedron compound of icosahedron and great dodecahedron great ditrigonal icosidodecahedron And some strange “retrosnubs”...
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.

Numberphile v. Math: the truth about 1+2+3+...=-1/12 Niles Johnson: I liked this too. And I liked the Numberphile video! I'm amazed at how much this issue has roiled the mathematics corners of YouTube. Gerard Westendorp: +Niles Johnson Remarkable that the Numberphile clip got 6 million views. This is stuff that is really worth understanding, the Riemann zet function, infinity, Ramanujan sums, ...
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The 5X5 version.