The beauty of roots

January 4, 2010 6:18 PM

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.
posted by escabeche (29 comments total)

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.
posted by stavrosthewonderchicken at 6:36 PM on January 4, 2010

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.
posted by closetphilosopher at 6:40 PM on January 4, 2010

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.
posted by DU at 6:43 PM on January 4, 2010 [3 favorites]

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.
posted by potch at 6:46 PM on January 4, 2010

In fact, mandelbrot set similarity is highly correlated with coolness.
posted by FishBike at 6:47 PM on January 4, 2010 [1 favorite]

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
posted by water bear at 6:58 PM on January 4, 2010

Well that's a nice reminder to appreciate beauty, and to not always try to understand it. Stunning.
posted by iamkimiam at 6:59 PM on January 4, 2010

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.
posted by DU at 7:05 PM on January 4, 2010 [7 favorites]

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.
posted by horsemuth at 7:17 PM on January 4, 2010 [5 favorites]

NICE
posted by moorooka at 7:34 PM on January 4, 2010

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.
posted by escabeche at 7:35 PM on January 4, 2010 [3 favorites]

Math is the lesbian sister of biology.
posted by Mike Buechel at 7:36 PM on January 4, 2010 [1 favorite]

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.
posted by pwnguin at 8:02 PM on January 4, 2010 [1 favorite]

Well I don't know shit about math, but this is awesome. Thanks for posting it.
posted by Lobster Garden at 8:07 PM on January 4, 2010

Moebius Transformations Revealed
posted by jcruelty at 8:33 PM on January 4, 2010

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.
posted by jcruelty at 8:36 PM on January 4, 2010 [1 favorite]

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.

Or a relation of electronic music pioneer Delia Derbyshire?
posted by Joseph Gurl at 9:04 PM on January 4, 2010

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.
posted by delmoi at 9:25 PM on January 4, 2010 [1 favorite]

I was naively hoping for something hair-related. This is probably better, though.
posted by that girl at 10:16 PM on January 4, 2010

So really, it seems like the way they are generated is quite different?

The interesting question then is, are they related?
posted by abc123xyzinfinity at 10:26 PM on January 4, 2010

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.)
posted by 7-7 at 10:32 PM on January 4, 2010 [12 favorites]

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).
posted by twoleftfeet at 10:51 PM on January 4, 2010

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?
posted by autopilot at 5:16 AM on January 5, 2010

Man, Microsoft ruins everything.
posted by ryanrs at 6:18 AM on January 5, 2010

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.
posted by madcaptenor at 6:40 AM on January 5, 2010

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.
posted by Kid Charlemagne at 6:52 AM on January 5, 2010

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.
posted by phliar at 11:56 AM on January 5, 2010 [1 favorite]

Ugh! Those loops should obviously all be {-4, 4}.
posted by phliar at 11:59 AM on January 5, 2010

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.
posted by delmoi at 12:15 PM on January 5, 2010