I contacted Dr. Tiozzo about the similarity of the two fractals and he replied,
From Giulio Tiozzo
"Of course, the similarity between the set of zeros of Littlewood polynomials and Thurston’s entropy bagel is striking.
In fact, I proved a few years ago that the two fractal sets the same inside the unit disk!
The difference is that Baez’s set is symmetric under circle inversion, while Thurston’s set is not: the parts outside of the unit disk are completely different."
The qm article says "A number is totally real if it satisfies a polynomial equation with integer coefficients" etc but I love your spirit and it's totally on quantamags for not stating their inputs up front
Right, but that's not what they're actually plotting - the quanta bagel is algebraic numbers with colour given by the entropy of their corresponding dynamics! Whereas the one I posted is roots of integer polynomials by density.
The fact that they look similar is striking. It means in some sense roots of integer polynomials generate more complex iterative systems, which is surprising (to me).
The quotation from Quanta was incomplete, and the Quanta article doesn't in fact claim that "totally real" means what "algebraic" actually means. Here's what the article actually says (added emphasis mine):
> A number is totally real if it satisfies a polynomial equation with integer coefficients that only has real roots.
No it's narrower than algebraic. If I have this right, the number is totally real if it's a root of a polynomial with integer coefficients, all of whose roots are real. So for example, the (real) cube root of 2 is not totally real, because x^3-2 has one real root and two complex roots. The idea of totally real is that not only is the root real, but all of its images under the polynomial's Galois group are also real.
Added: in fact the Quanta article gets this right. It says "A number is totally real if it satisfies a polynomial equation with integer coefficients that only has real roots." Also, Jordana Cepelowicz is a knowledgeable math writer and I believe she has a PhD in math.
It does seem like a bit of not totally accurate shorthand: totally real isn't a property of a number, but rather of the extension field where it lives. So sqrt(2) is a root of x^2-2 whose roots are +/- sqrt(2), and the splitting field of x^2-2 is Q(sqrt(2)) which is totally real. But sqrt(2) is also a root of x^4-4 whose roots are +/- sqrt(2) and +/- i sqrt(2), so the splitting field doesn't live inside the reals. The wiki article explains it better. I was unfamiliar with this idea before.
That's true but I think it's implicit here we're working with the algebraic extension generated by the number itself (which always contains the Galois conjugates).
I'm not sure if I'm using the jargon properly, but if r is the real cube root of 2, then I thought the extension generated by r is Q(r), which contains only reals. So (x^3 - 2) factors into (x-r) and the irreducible quadratic (x^2 + r*x + r^2). You have to do some extra thing to reach the splitting field, amirite?
> As always, Quanta Magazine reporting is just garbage. The name for such numbers is ‘algebraic’.
Any popularization is inevitably going to run into some inaccuracies—or else it's just re-publishing the technical papers—but the opinions of most mathematicians I know are that, far from being garbage, Quanta's reporting is distinctly better than most.
Absolutely! The bagel 3D-like image and the 3rd image in your link are showing the same thing. Look at the inside of the ring and compare the images side by side. The features align in both images.
Another good one is "Complexity" by Melanie Mitchell. She was a grad student of Douglas Hofstadter's and is with the Sante Fe Institute right now if memory serves.
Is there a reason why the bagel/donut shape seems to appear so much in math/physics? Just yesterday I was watching a kurzgesagt video that explored the idea of the universe being a "hyperdonut", or something. There probably isn't a simple answer but I'd appreciate any attempt to explain it in laymen's terms.
It's the simplest two-dimensional surface[1] with at least one hole in it. In topology - the math branch that deals with properties of geometric objects and how they can be deformed - it's one of the two simplest objects, the other being a sphere. (There's a lot of those, too). Donuts ("tori") have the advantage that they're a bit more complex, so they're more interesting, and have interesting properties. Nobody gets much excited by spheres anymore ;) That also means they get a bit more attention.
That means you'd expect it to be very common. Not having a lot of donuts would be weird. (Now there's a maxim to live by!)
Yes, it's extremely handwavy, and there are probably explanations that are much better and more detailed. Sorry. Best I have.
[1] I lie. It's a compact orientable 2-manifold, not a 2D surface. For laymen's terms, I hope the math folks will let me get away with "surface" though.
Vsauce has an interesting video regarding topology[0]. Probably a bit rudimentary for anyone already familiar with the math, but still entertaining in the typical vsauce way.
Addition looks like a donut with a twist, in both finite and infinite, one-point compactified cases. (A reason we recognize them as “the same” operation.)
Imagine a clock laying on a table, and then placing clocks at each number which are rotated that number of steps. (Equivalently, so the number on the new clock matches the original clock at that spot.) If you trace where the numbers are on these clocks, you get bands that twist around the torus. This shape [0].
So we have:
1. Product of two circles.
2. A simple 2D shape with a hole.
3. Addition.
…all mixed up in one shape. So it shows up a lot of places.
If feel like there is a connection to differential equations here. In those there is quite a lot known about chaos etc... anyone have any insights on that connection? I feel like the mathematicians would have probably covered this.
Chaos theory knows two branches. discrete time and continuous time.
They are surprisingly different branches. differential equations are clearly in the continuous camp, and the basis of most if not all continuous chaos theory.
Self-iteration seems to produce so much complexity from so few rules. The root of so many chaotic things, the collatz conjecture, fractals. And the chemical root of complex life. Funny.
There is quadratic circuits used in Zero Knowledge Proofs, and there is shamir secret sharing, which uses arbitrary degree polynomials.
In general though, cryptography will use polynomials over finite fields. That is very different from these fractals which work on infinite sets that require infinite precision.
I thought of that too. I think at this point most researches don't have much respect for him. If you listen to him or read his book, he sure likes to take credit for a lot stuff and make large claims about revolutionizing science and such. "I discovered this, I did that, etc...". So I can't blame others for avoiding mentioning him and just pretending he doesn't exist.
At the same time this comes from a math background while Wolfram sort of bases his theory on cellular automata with rules, cells, Turing Machines etc.
I think Wolfram's work is quite different in flavour - he focuses on discrete dynamical systems (graphs/cellular automata, stuff like that) whereas OP is about continuous iterated systems like the Mandelbrot set (which afaik have a much older history in mathematics).
His ideas are not mathematically rigorous and would not pass peer review. his ideas are too hand-wavy. I don't think he has ever published anything in peer review or even in a journal format. A doorstop-sized book does not count. No one wants to read all that.
He was a theoretical physics wunderkind before he created Mathematica and became a crank. There are other great physicists like 't Hooft who became cranks in their later years (arguably including Einstein).
(edit: hang on, am I crazy or do these bagels look remarkably similar? despite very different definitions)