4 ms·

The Beauty of Roots (2011)

204 points by goldenkey 10y ago

ttflee 10y ago
The title is not very accurate. The article mostly talks about roots of polynomials in the field of complex numbers and shows some beautiful fractal images derived from the roots.
- stephencanon 10y ago
  Plotting the roots is about as reasonable of a way to talk about what a polynomial "looks like" as any.
- ranit 10y ago
  You are right and the problem is that the HN rule to keep the original title was not followed. The title is "The Beauty of Roots".
  - pbhjpbhj 10y ago
    This is why I favour original titles with editorialised sub-titles. You could even allow the user to choose which to show if you feared that having a sub-title would make for too much clutter.
  - sctb 10y ago
    That's right, we just changed the submission title to the original from “What do polynomials really look like?”.
- thomasahle 10y ago
  And a really interesting explanation about why roots inside the unit circle seems to land on the dragon curve!
Yhippa 10y ago
Someone smarter than I will likely know this but are probability and functions nature's compression techniques?
- GFK_of_xmaspast 10y ago
  What does that even mean.
- torustic 10y ago
  > Someone smarter than I will likely know this but are probability and functions nature's compression techniques? One of the nice things about mathematics is that we don't need to say anything like that. Probabilities are nature's probability techniques. Functions are nature's function techniques. And compression is nature's set of compression techniques. You might find category theory useful here.[0][1][2] Take the category Set, which is the category of sets.[3] Just imagine an n-dimensional structure of dots and arrows, where the dots and arrows can each represent some other n-dimensional structure of dots and arrows. Set is the structure of that form which is constrained precisely so that it represents the idea of sets, and functions between them. You can consider other structures like Set[4] that are constrained (or freed) to represent all manner of properties, structures and other stuff. Some of this stuff resembles branches of mathematics. Some resembles physics. Some resembles philosophy. Science takes observation and uses mathematical models to transform that observation into prediction. So long as a model is sufficiently general, in a computational sense, then you will find that you _can_ rewrite the whole of science in terms of that model. But this doesn't consider the question of elegance -- and the related question of what you want to consider fundamental. And the product question of those two: how do you measure/compute most effectively to maximize elegance-of-stuff upon that fundamental-structure? As someone who cares deeply about constructive mathematics, I consider things like topoi, the Curry-Howard(-Lambek) isomorphism[5], and Grothendieck's relative perspective[6] to be fundamental, and a beautiful foundation for the modelling of any and all information dynamics (≃ 'nature', 'thought', 'computing'). You may have some other perspective. There is a vast space of ways to approach the problem of modelling experience. To be scientific, the thing to avoid is appending some _intention_ to the processes that your model describes. Causality is real and everywhere, but there is no way to say "nature compresses by using probability and functions" that doesn't beg the question. Either compression, probability and functions are part of your singular model of nature, or they are competing models. You do math -- or coming from different directions you do physics, info theory, etc. -- by finding a general method that reworks them as a coherent idea. You're doing the first step of that when you form a surface analogy between (probability,functions) ≃ (nature,compression). If this kind of question bothers you enough, you can make it your life's work to follow the rabbit hole all the way down. It's not so much a question of smartness (though rigor is a huge part of truth-seeking) -- just how inexorably bothered you are by the idea of competing models of reality. Cryptography, machine learning, and complexity theory are immediate implications of analogies similar to your own (p,f)≃(n,c). Should you want to study further (and don't particularly identify with the constructivist tone of this comment), those are probably the most appropriate fields in which to look for elegant solutions to your questions. [0] https://en.wikipedia.org/wiki/Category_theory https://en.wikipedia.org/wiki/Category_theory [1] https://ncatlab.org/nlab/show/nPOV https://ncatlab.org/nlab/show/nPOV [2] https://arxiv.org/abs/math/0004133 https://arxiv.org/abs/math/0004133 [3] https://en.wikipedia.org/wiki/Category_of_sets https://en.wikipedia.org/wiki/Category_of_sets [4] https://en.wikipedia.org/wiki/Topos#Elementary_topoi_.28topoi_in_logic.29 https://en.wikipedia.org/wiki/Topos#Elementary_topoi_.28topo... [5] https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon... [6] https://en.wikipedia.org/wiki/Grothendieck%27s_relative_point_of_view https://en.wikipedia.org/wiki/Grothendieck%27s_relative_poin...
lugus35 10y ago
I just see a donut.
- catnaroek 10y ago
  I was going to say something about orbitals, but donuts are obviously awesomer. :-p
- kylek 10y ago
  Really? I found some of these [0][1][2] to be shockingly beautiful. [0]http://math.ucr.edu/home/baez/roots/polynomialroots05expi02.png http://math.ucr.edu/home/baez/roots/polynomialroots05expi02.... [1]http://math.ucr.edu/home/baez/roots/polynomialroots08i.png http://math.ucr.edu/home/baez/roots/polynomialroots08i.png [2]http://math.ucr.edu/home/baez/roots/polynomialrootssmall.png http://math.ucr.edu/home/baez/roots/polynomialrootssmall.png
wyager 10y ago
Interesting to see Greg Egan's name come up! I am a huge fan of his books. I guess I shouldn't be surprised to see him involved in the math/physics research community. Egan's stories are often of the "one big lie" variety. He makes up some fact (e.g. fundamental particles are composed of 12-dimensional wormholes, we live in an uncountably infinite multiverse that can be traversed, you can build a machine that combines a space and time dimension, etc.) and then follows the made-up fact to some fascinating conclusion. He is clearly very intelligent and has substantial background in many scientific fields, which makes his sci-fi books quite mind-bending.
peter-slovak 10y ago
The title got my hopes up for some biology stuff, but this was so much better.
rootdiver 10y ago
I have made a python implementation of this fractal if anyone is interested : https://github.com/Alexander-0x80/Beauty-of-roots https://github.com/Alexander-0x80/Beauty-of-roots
chmaynard 10y ago
It's interesting that Dr. Baez calls himself a mathematical physicist, not an applied mathematician or a theoretical physicist. Can someone explain why?

204 points by goldenkey 10y ago

The title is not very accurate. The article mostly talks about roots of polynomials in the field of complex numbers and shows some beautiful fractal images derived from the roots.

stephencanon 10y ago
Plotting the roots is about as reasonable of a way to talk about what a polynomial "looks like" as any.
ranit 10y ago
You are right and the problem is that the HN rule to keep the original title was not followed. The title is "The Beauty of Roots".
- pbhjpbhj 10y ago
  This is why I favour original titles with editorialised sub-titles. You could even allow the user to choose which to show if you feared that having a sub-title would make for too much clutter.
- sctb 10y ago
  That's right, we just changed the submission title to the original from “What do polynomials really look like?”.
thomasahle 10y ago
And a really interesting explanation about why roots inside the unit circle seems to land on the dragon curve!

Yhippa 10y ago

Someone smarter than I will likely know this but are probability and functions nature's compression techniques?

GFK_of_xmaspast 10y ago
What does that even mean.
torustic 10y ago
> Someone smarter than I will likely know this but are probability and functions nature's compression techniques? One of the nice things about mathematics is that we don't need to say anything like that. Probabilities are nature's probability techniques. Functions are nature's function techniques. And compression is nature's set of compression techniques. You might find category theory useful here.[0][1][2] Take the category Set, which is the category of sets.[3] Just imagine an n-dimensional structure of dots and arrows, where the dots and arrows can each represent some other n-dimensional structure of dots and arrows. Set is the structure of that form which is constrained precisely so that it represents the idea of sets, and functions between them. You can consider other structures like Set[4] that are constrained (or freed) to represent all manner of properties, structures and other stuff. Some of this stuff resembles branches of mathematics. Some resembles physics. Some resembles philosophy. Science takes observation and uses mathematical models to transform that observation into prediction. So long as a model is sufficiently general, in a computational sense, then you will find that you _can_ rewrite the whole of science in terms of that model. But this doesn't consider the question of elegance -- and the related question of what you want to consider fundamental. And the product question of those two: how do you measure/compute most effectively to maximize elegance-of-stuff upon that fundamental-structure? As someone who cares deeply about constructive mathematics, I consider things like topoi, the Curry-Howard(-Lambek) isomorphism[5], and Grothendieck's relative perspective[6] to be fundamental, and a beautiful foundation for the modelling of any and all information dynamics (≃ 'nature', 'thought', 'computing'). You may have some other perspective. There is a vast space of ways to approach the problem of modelling experience. To be scientific, the thing to avoid is appending some _intention_ to the processes that your model describes. Causality is real and everywhere, but there is no way to say "nature compresses by using probability and functions" that doesn't beg the question. Either compression, probability and functions are part of your singular model of nature, or they are competing models. You do math -- or coming from different directions you do physics, info theory, etc. -- by finding a general method that reworks them as a coherent idea. You're doing the first step of that when you form a surface analogy between (probability,functions) ≃ (nature,compression). If this kind of question bothers you enough, you can make it your life's work to follow the rabbit hole all the way down. It's not so much a question of smartness (though rigor is a huge part of truth-seeking) -- just how inexorably bothered you are by the idea of competing models of reality. Cryptography, machine learning, and complexity theory are immediate implications of analogies similar to your own (p,f)≃(n,c). Should you want to study further (and don't particularly identify with the constructivist tone of this comment), those are probably the most appropriate fields in which to look for elegant solutions to your questions. [0] https://en.wikipedia.org/wiki/Category_theory https://en.wikipedia.org/wiki/Category_theory [1] https://ncatlab.org/nlab/show/nPOV https://ncatlab.org/nlab/show/nPOV [2] https://arxiv.org/abs/math/0004133 https://arxiv.org/abs/math/0004133 [3] https://en.wikipedia.org/wiki/Category_of_sets https://en.wikipedia.org/wiki/Category_of_sets [4] https://en.wikipedia.org/wiki/Topos#Elementary_topoi_.28topoi_in_logic.29 https://en.wikipedia.org/wiki/Topos#Elementary_topoi_.28topo... [5] https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon... [6] https://en.wikipedia.org/wiki/Grothendieck%27s_relative_point_of_view https://en.wikipedia.org/wiki/Grothendieck%27s_relative_poin...

lugus35 10y ago

I just see a donut.

catnaroek 10y ago
I was going to say something about orbitals, but donuts are obviously awesomer. :-p
kylek 10y ago
Really? I found some of these [0][1][2] to be shockingly beautiful. [0]http://math.ucr.edu/home/baez/roots/polynomialroots05expi02.png http://math.ucr.edu/home/baez/roots/polynomialroots05expi02.... [1]http://math.ucr.edu/home/baez/roots/polynomialroots08i.png http://math.ucr.edu/home/baez/roots/polynomialroots08i.png [2]http://math.ucr.edu/home/baez/roots/polynomialrootssmall.png http://math.ucr.edu/home/baez/roots/polynomialrootssmall.png

wyager 10y ago

Interesting to see Greg Egan's name come up! I am a huge fan of his books. I guess I shouldn't be surprised to see him involved in the math/physics research community. Egan's stories are often of the "one big lie" variety. He makes up some fact (e.g. fundamental particles are composed of 12-dimensional wormholes, we live in an uncountably infinite multiverse that can be traversed, you can build a machine that combines a space and time dimension, etc.) and then follows the made-up fact to some fascinating conclusion. He is clearly very intelligent and has substantial background in many scientific fields, which makes his sci-fi books quite mind-bending.

peter-slovak 10y ago

The title got my hopes up for some biology stuff, but this was so much better.

rootdiver 10y ago

I have made a python implementation of this fractal if anyone is interested : https://github.com/Alexander-0x80/Beauty-of-roots https://github.com/Alexander-0x80/Beauty-of-roots

chmaynard 10y ago

It's interesting that Dr. Baez calls himself a mathematical physicist, not an applied mathematician or a theoretical physicist. Can someone explain why?