Equations

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Equations

Postby quickfur » Tue Jun 20, 2017 11:34 pm

Some of you may be wondering what I've been up to lately. Well, many things... but lately, one thing that's somewhat related to this site is my exploration of the graphs of implicit equations in two variables. Here's a sample of an equation I've been considering recently:

Image

The yellow crosshairs represent the coordinate axes. The region represented by the above image is -120 ≤ x ≤ 120, -120 ≤ y ≤ 120.

In case you can't read the tiny font at the top, the equation is:

Code: Select all
12·sin(sin(x - 1) +                                   
       sin(cos(2·π/5)·x + sin(2·π/5)·y - 1) +
       sin(cos(4·π/5)·x + sin(4·π/5)·y - 1) +
       sin(cos(6·π/5)·x + sin(6·π/5)·y - 1) +
       sin(cos(8·π/5)·x + sin(8·π/5)·y - 1)) = 0


The factor of 12 on the left-hand side is not really necessary; it's just a hack to get my graphing program to show the graph lines more clearly. :P

The equation looks a bit scary, but it's actually very simple. It can be rewritten simply as:

sin( sum_{i=0 to 4} sin(cos(i*2π/5)*x + sin(i*2π/5)*y - 1) ) = 0

Or, to put it even more simply:

sin( sum( sin( horizontal line rotated by 0°, 72°, 144°, 216°, 288° ) ) ) = 0

The sine function applied to each line basically replicates it into an infinite set of equally-spaced parallel lines. The outer sine function turns the intersections into a series of blob-like pieces that form pretty patterns.

The pentagonal symmetry is a direct result of the orientations of the generating lines. What's interesting, though, is the resulting patterns in the graph. Due to the fact that the coefficients of x and y in pentagonal orientation are irrational, the pattern of intersections between the 5 sets of parallel lines never repeat, so it produces a kaleidoscopic pattern of intersections that are reminiscient of Penrose pentagonal tilings. But what's fascinating is that the outer sine function creates a pattern of blob-like curves in the graph that exhibit all manners of pentagonal, decagonal, and even 20- and 30-fold symmetries, even though if you look more closely, they are actually not perfectly symmetrical except for the patterns centered on the origin. Yet they show almost-pentagonal / almost-decagonal / etc. patterns in a way that suggests some kind of underlying generating principle that one could use to generate Penrose tilings. If you trace the circle-like arrangements of blobs, you'll notice a lot of 5-membered, 10-membered, and 30-membered rings. They aren't actually real, regular pentagons, decagons, 30-gons, they are somewhat distorted but nevertheless they retain their relationship with the central regular pentagonal symmetry.

As the form of the equation suggests, this is only one member of a class of similar equations that I've been exploring. By summing lines rotated across different angles corresponding to subdivisions of the circle, you can get triangular, square, octagonal, etc. patterns. In fact, I have a graph of a heptagonal version of the equation that shows patterns in 14-gonal and 28-gonal symmetries. Truly fascinating!
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Re: Equations

Postby Klitzing » Wed Jun 21, 2017 11:15 am

Ha, well, Quickfur, your displayed pattern not only looks like the Penrose tiling, it rather is clearly related!
In fact, your sets of parallel lines, used in rotational symmetric overlays, are nothing but the well-known de Bruijn grids, from which the Penrose tiling (and others) can be derived. Cf. e.g. http://www.mathpages.com/home/kmath621/kmath621.htm. (Just as a quick Google result, there are probably better suited citations too.)

--- rk
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Re: Equations

Postby quickfur » Wed Jun 21, 2017 5:13 pm

Haha, I seem to have a knack for rediscovering stuff already known long ago. :lol:

I had never heard of de Bruijn grids before you mentioned it. I "accidentally" stumbled upon this while experimenting with graphs of various nested trigonometric functions. There were various interesting ones like sin√(x^2+y^2)=x*y, but they were all axis-aligned. I wondered if I could get non-axis aligned symmetries, and eventually discovered a way of making graphs with triangular symmetry. One that particularly caught my attention was a pattern based on the periodic triangular tiling of 2-space, that looked like ceramic floor tile patterns. Then I wondered if there were pentagonal or other symmetries analogues as well. That's how I found an early form of the equation with pentagonal symmetry, and noticed that it was remarkably similar to pentagonal Penrose tilings. :D I had no idea that there was an actual relation between the two!

Having said that, though... now it makes me wonder if there are analogous de Bruijn planes / hyperplanes, from which one may construct aperiodic 3D/4D/etc. tilings?
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Re: Equations

Postby Klitzing » Wed Jun 21, 2017 9:39 pm

quickfur wrote:Haha, I seem to have a knack for rediscovering stuff already known long ago. :lol:

I had never heard of de Bruijn grids before you mentioned it. [...] :D I had no idea that there was an actual relation between the two!

Having said that, though... now it makes me wonder if there are analogous de Bruijn planes / hyperplanes, from which one may construct aperiodic 3D/4D/etc. tilings?


When I recall correctly, there was an equivalence between the de Bruijn grid construction and an according cut-and-project construction for quasiperiodic tilings with zonogonal tiles (pairs of parallel boundaries each, all edges the same size). Or at least wrt. the non-singular cases (rhombs only, resp. just intersections of exactly 2 lines at each crossing), then you'd get a lifting into some Zn.

This then is straight forward to be used with grids of parallel hyperplanes. You'd get according dimensional zonotopal tiles (resp. rhombotopes for the non-singular cases). Again this can be lifted into some Zn. At least wrt. the quasicrystaline symmetries I would wonder when these constructions wouldn't have been explored widely.

Btw. I really like your outer sine function. These curves indeed provide the same visual impression as the atomic position plots provided by several quasicrystallists years ago! Just that yours represent the curves of zero, while theirs were representing the curves of height of the according potential or position probabillity. :)

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Re: Equations

Postby quickfur » Wed Jun 21, 2017 10:52 pm

Haha, maybe they were secretly using my program for generating those plots? ;) :lol:

One thing I've always wondered about aperiodic tilings, though. It seems that the majority of aperiodic tilings known today are of very limited types, like ones with polygonal rotational symmetry, recursive tilings (tiles assemble into larger versions of themselves), and maybe a few other types. But according to the proof of the existence of aperiodic tilings, there is a connection with computability theory, with periodic tiles corresponding with the simplest kinds of computations. That means there ought to exist arbitrarily complex aperiodic tilings with arbitrarily complex, non-repeating patterns, not just polygonal symmetry and recursive tiles. So my question is, where are the rest of these tilings?? From what I understand of the computability hierarchy, which has analogies with the Fast-Growing Hierarchy, these more complex tilings ought to be in the vast majority in the set of all possible tilings. So how come we only know a meager few, and only from relatively simple patterns? Where are the rest of them, and how do we find them?

(One idea I have is to find some kind of mapping from Turing machines to tilings... then we'd be able to generate arbitrarily complex tilings just by constructing a suitably complex Turing machine. Or, for that matter, any Turing-complete language that basically spans the same set of computable functions.)
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Re: Equations

Postby Hugh » Sun Jun 25, 2017 11:53 am

Cool pic! I tried the magic eye (cross-eye) technique and it worked well with it. :)
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Re: Equations

Postby quickfur » Fri Jul 07, 2017 11:13 pm

Here's a fun case I found today:

Image

This is the equation:

y·sin(x - π·sin(x - π·sin(x - π·sin(x - π·sin x)))) = x·cos(y + π·cos(y + π·cos(y + π·cos(y + π·cos y))))

It looks quite complex, but orderly, like a printed circuit board with many connectors. Notice that on the LHS it's all x (except for the first y), and on the RHS it's all y (except for the first x). It's basically a bunch of nested sines and nested cosines.

What happens if we change the last x on the LHS to y, and the last y on the RHS to x?

y·sin(x - π·sin(x - π·sin(x - π·sin(x - π·sin y)))) = x·cos(y + π·cos(y + π·cos(y + π·cos(y + π·cos x))))

Well, you know what happens when you connect two opposite poles of a circuit together... it's called a short-circuit, and it usually has very bad consequences. In this case, it causes a gigantic explosion that completely rips out all the wires and splatters all the components everywhere:

Image

Now that's what I call applied mathematics. :D :lol: :P :nod: :roll: :XD:
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