Gerard Westendorp
201 followers -
http://westy31.home.xs4all.nl/
http://westy31.home.xs4all.nl/

201 followers
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The heptagonal case.
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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 mentioned a couple of days ago.
For example a square. (This could also be done with a Schwarz–Christoffel mapping.)
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Crumple 8 milk cartons into a torus.
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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.
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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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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.