Comments on: The beauty of roots

The beauty of roots

escabeche — Mon, 04 Jan 2010 18:18:21 -0800

The beauty of roots. From Dan Christensen and Sam Derbyshire via John Baez. If you like algebra: these are plots of the density in the complex plane of roots of polynomials with small integral coefficients. If you don't: these are extravagantly beautiful images produced from the simplest of mathematical procedures. Explore the image interactively here.

By: stavrosthewonderchicken

stavrosthewonderchicken — Mon, 04 Jan 2010 18:36:59 -0800

This kind of thing was exactly how I taught myself to program back in the late '70s on my TRS-80 Model III, in monochrome, at 127x47 resolution. The computer booted to a BASIC command line, and I taught myself to write code (and save it to tape -- no discs back then) to use math functions to create crude (but astonishingly beautiful to me) lightshows from simple math. I think I'd still be doing something like that as a living -- working on game engines or something -- if I hadn't intentionally put computers aside as much as possible for a decade or two starting in my late teens, in favour of girls and booze and motorcycles and travel.

By: closetphilosopher

closetphilosopher — Mon, 04 Jan 2010 18:40:22 -0800

This looks just like a fractal, but the way it's generated is nothing like a fractal. I'm just... my mind has been blown.

By: DU

DU — Mon, 04 Jan 2010 18:43:39 -0800

Well, it looks like the mandelbrot set. And it actually kinda is similar to how you generate a mandelbrot set. Which is not to say this isn't cool.

By: potch

potch — Mon, 04 Jan 2010 18:46:45 -0800

I love the way these images, like fractal plots, display gorgeous organic forms, but in an unnervingly symmetric way- it's like the uncanny valley, but more fascination than horror.

By: FishBike

FishBike — Mon, 04 Jan 2010 18:47:53 -0800

In fact, mandelbrot set similarity is highly correlated with coolness.

By: water bear

water bear — Mon, 04 Jan 2010 18:58:57 -0800

But I think they're the most beautiful near the point (1/2)exp(i/5). This image is almost a metaphor of how, in our study of mathematics, patterns emerge from confusion like sharply defined figures looming from the mist: -John Baez

By: iamkimiam

iamkimiam — Mon, 04 Jan 2010 18:59:47 -0800

Well that's a nice reminder to appreciate beauty, and to not always try to understand it. Stunning.

By: DU

DU — Mon, 04 Jan 2010 19:05:16 -0800

Attempting to understand something is what led to finding this beauty in the first place. Attempting to understand this beauty will lead to future beauty.

By: horsemuth

horsemuth — Mon, 04 Jan 2010 19:17:08 -0800

I read the post, visited the first link, came back here, reread the post, reread it again, and just now figured out that this had nothing to do with Joan Baez.

By: moorooka

moorooka — Mon, 04 Jan 2010 19:34:59 -0800

NICE

By: escabeche

escabeche — Mon, 04 Jan 2010 19:35:07 -0800

Well, it has SOMETHING to do with Joan Baez -- she's John Baez's cousin. Also, it seems quite possible that Sam Derbyshire is the child of National Review columnist and math aficionado John Derbyshire.

By: Mike Buechel

Mike Buechel — Mon, 04 Jan 2010 19:36:02 -0800

Math is the lesbian sister of biology.

By: pwnguin

pwnguin — Mon, 04 Jan 2010 20:02:33 -0800

Interestingly, that last one, is some form of Dragon Curve. I have to admit I haven't studied fractals enough to know if this is an obvious result or not.

By: Lobster Garden

Lobster Garden — Mon, 04 Jan 2010 20:07:52 -0800

Well I don't know shit about math, but this is awesome. Thanks for posting it.

By: jcruelty

jcruelty — Mon, 04 Jan 2010 20:33:36 -0800

Moebius Transformations Revealed

By: jcruelty

jcruelty — Mon, 04 Jan 2010 20:36:25 -0800

Derbyshire wrote one of the 3 (!) pop-math books on the Riemann Hypothesis. I remember thinking the book would have benefited from some interactive Javascript-type visualizations of the Riemann function's zeroes. In my perfect world all math textbooks are 'e-books' with animated diagrams along the lines of the posted link.

By: Joseph Gurl

Joseph Gurl — Mon, 04 Jan 2010 21:04:19 -0800

By: delmoi

delmoi — Mon, 04 Jan 2010 21:25:38 -0800

Well, it looks like the mandelbrot set. And it actually kinda is similar to how you generate a mandelbrot set. -- DU

Huh? Mathematically it's completely different from the mandelbrot set. These are zeros of polynomials and the mandelbrot set has to do with whether an initial parameter of a recursive function will terminate (in C.S. terms). So it's really quite different. Also, the generation is quite different. With a Mandelbrot set, you take each point in your image and calculate whether or not it will 'escape' when it's the constant in the recursive formula z_n+1 = z_n + c where. So for each point on your image x and y, you calculate z_n+1 = z_n + x + yi But for this, you have to step through every single polynomial (within the range you are looking at) and you add one dot where the zero is. With the golden ring image they used polynomials of degree 24 where all the coefficients were either 1 or -1, so you have exactly 2²⁴ polynomials. So really, it seems like the way they are generated is quite different? They're both on the complex plane, though.

By: that girl

that girl — Mon, 04 Jan 2010 22:16:13 -0800

I was naively hoping for something hair-related. This is probably better, though.

By: abc123xyzinfinity

abc123xyzinfinity — Mon, 04 Jan 2010 22:26:03 -0800

So really, it seems like the way they are generated is quite different? The interesting question then is, are they related?

By: 7-7

7-7 — Mon, 04 Jan 2010 22:32:21 -0800

Inspired by this link I took out my trusty C++ compiler and re-created some of Dan Christiansen's work at greater length to explore some of these beautiful images for myself. Fortunately, my last paying job had left me with some highly unauthorized accounts on computers significantly more powerful than the general public is aware of. I was able to construct a marvelously detailed series of images to zoom in on the mottled, curlicued region of the bottom of the screen I somehow found so intriguing. The swirling forms gave way to a varied squariform, like a metropolis, something built - how absurd, I thought. Something caught my trained eye, a smear barely even a single pixel in size - yet intuitively out of place. I zoomed in further, further - to the limits of my resources. The smudge resolved into reticulated interstices - a net. A single artefact, hidden in the myriad myriad lacunae of this immense mathematical edifice. An object in ten lobes, now becoming apparent as the great silicon engines churned out their unerring calculations, now shockingly, unmistakably clear. Unbelievable. The Sephirot. The Tree of Life. It is said that our mathematicians claim that the universe is in some way built out of mathematical structures. Indeed, the physical laws of the universe being based in mathematics, how could it be any other way? Should it be so surprising then that just as satellites are beginning to have the power to resolve other Earths our most powerful machines for calculation may now be groping towards the deepest question of all? I could feel myself beginning to weep in awe at the profundity. Unfortunately just then Windows crashed and I was unable to repeat the calculation. (The program may be acquired on request.)

By: twoleftfeet

twoleftfeet — Mon, 04 Jan 2010 22:51:12 -0800

Christensen himself thinks that a nicer image comes from doing this for sixth degree polynomials with integer coefficients between -4 and 4, (instead of the fifth degree ones with those coefficients that Baez chose).

By: autopilot

autopilot — Tue, 05 Jan 2010 05:16:22 -0800

delmoi -- I think you lost an exponent somewhere along the way. The Mandelbrot equation is z_n+1 = z_n² + c. I CAN HAS L^AT_EX MATH INPUT? PLEASE?

By: ryanrs

ryanrs — Tue, 05 Jan 2010 06:18:20 -0800

Man, Microsoft ruins everything.

By: madcaptenor

madcaptenor — Tue, 05 Jan 2010 06:40:50 -0800

escabeche: Sam Derbyshire is not John Derbyshire's son. Somewhat fortuitously, John Derbyshire has a webpage of his family history. I couldn't find Sam there; it seems that they're not related and the similarity of names is just a coincidence.

By: Kid Charlemagne

Kid Charlemagne — Tue, 05 Jan 2010 06:52:42 -0800

Is there detailed instructions as to how one generates this sort of thing? I mean I vaguely see whats going on, but I want a gut feel for it and not just a, "that's nice, move to next shiny thing, repeat" understanding.

By: phliar

phliar — Tue, 05 Jan 2010 11:56:42 -0800

Man, that's surprising and wonderful and bautiful! Surprising too that it seems to be less studied. (Anyone want to make a bet on whether or not this will help with the Riemann Hypothesis?) Kid Charlemagne: loop through all the integer coefficients and solve the resulting polynomial, turn on the pixel at the zeroes. (Solving quintic polynomials is left as an exercise to the reader.)

for a in {-4, -4} and a ≠ 0
    for b in {-4, -4} and b ≠ 0
        ...
             for f in {-4, -4} and f ≠ 0
                 solve ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0
                 turn on pixels at each root

... etc. I'm now going to re-read Hardy's Apology.

By: phliar

phliar — Tue, 05 Jan 2010 11:59:19 -0800

Ugh! Those loops should obviously all be {-4, 4}.

By: delmoi

delmoi — Tue, 05 Jan 2010 12:15:44 -0800

I think you lost an exponent somewhere along the way. Heh. I must have been pretty tired when I wrote that comment. there's an extra "where." in there and for some reason the second to last paragraph has a question mark that makes no sense :P But yeah, z_n+1 = z_n² + c (where c is complex) Another big difference, I think is that if you want to 'zoom in' you have to re-generate the image with another degree, multiplying the amount of work by 2*d where d is the degree of the polynomial. So to go one more 'level' you would need 50 times as much work (level 25 vs. level 24). Whereas with a mandelbrot set, you can 'zoom in' by selecting smaller region and picking the real and complex parts of c to cover that smaller area in finer detail.