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.
I was
reading about Kolmogorov.
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:
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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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.
5 comments
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.
9 comments
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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Comments:
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.
Comments:
Roice Nelson: nice, love it!
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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.
9 comments
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:
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.
Comments:
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.
Comments:
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.
Comments:
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
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.
12 comments
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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