'Entropy bagels' and other complex structures emerge from simple rules | Hacker News Reader

'Entropy bagels' and other complex structures emerge from simple rules

quantamagazine.org

77 pointsrolph47 comments

Awesome! You can get another beautiful bagel by plotting the roots of polynomials with integer coefficients: https://math.ucr.edu/home/baez/roots/

(edit: hang on, am I crazy or do these bagels look remarkably similar? despite very different definitions)

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."

Thanks for contacting him! I'll do some diving into his work, that's super cool =)

It might be worth emailing the two groups of people and putting them in contact with each other. Nobody can know everything and be everywhere at once.

The author of the article linked in the comment above yours has been discussing this here:

https://mathstodon.xyz/@gregeganSF/112007010957379619

This finding is so funny, it's like despite the interconnectedness in the 21st century things like this still happen.

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).

As always, Quanta Magazine reporting is just garbage. The name for such numbers is ‘algebraic’.

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.

Nothing wrong with that.

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.

See: https://en.wikipedia.org/wiki/Totally_real_number_field

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.

Yeah, you've got it right, the Quanta article covered this accurately.

For instance, 'i' is algebraic but not totally real (it isn't a real number!), since it satisfies 'x^2 + 1'.

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?

Pardon me, you're completely right! Thanks for reminding me to refresh my basics before I write =)

> 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.

I highly recommend reading the article, you might be pleasantly surprised =)

I wrote them off a long time ago, but after this comment thread, I’ll give them another shot. Thanks!

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.

Not just you. I knew I'd seen these somewhere before.

I recommend James Gleick's Chaos book. It will change your perspective on the world.

c = -3/2 produces nice "hyperbolic" behavior has been discussed in this video

https://www.youtube.com/watch?v=eC9-1wY7FvU

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.

I read this as a kid and was totally enamored by cellular automata

Reminded me of this recent observation of fractals in nn training https://sohl-dickstein.github.io/2024/02/12/fractal.html

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.

[0] https://youtu.be/egEraZP9yXQ?si=P1BzLzBVW4q1aZ3D

Not exhaustive but a lens , but the 3D rotation groups SO(3) is not simply connected.

The antipodal points Pac-Man and so while a 360 degree rotation is considered 'invariant' that only holds for rigid objects with no connections.

Rolling up a hose, twirling a baton, or the belt trick shows that 720 is the real invariant rotation.

Merge and Delete those two points at infinity and you get a donut.

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.

[0] - https://d2r55xnwy6nx47.cloudfront.net/uploads/2020/05/Knot-S...

Can you elaborate on how addition fits into this? You don't mean vector addition of the "clocks" do you?

I think it’s because they can be modelled “easily” as a 2D plane with no edge in any direction but i am not an expert

The bagel looks like a black hole

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.

https://getyarn.io/yarn-clip/ad99367f-7c77-4566-b160-50efca7...

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.

Is there any value in these types of systems in a cryptography context? I was just thinking about that after reading the article.

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.

A bit like Collatz conjecture

Not a single mention of Stephen Wolfram?

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 has a PhD from CalTech at age 20, under Richard Feynman. Those are published.

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).

Reply on news.ycombinator.com