
The Beauty of Roots (2011) - goldenkey
http://math.ucr.edu/home/baez/roots/
======
wyager
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.

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

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

~~~
torustic
> 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_.28topo...](https://en.wikipedia.org/wiki/Topos#Elementary_topoi_.28topoi_in_logic.29)

[5]
[https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...](https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence)

[6]
[https://en.wikipedia.org/wiki/Grothendieck%27s_relative_poin...](https://en.wikipedia.org/wiki/Grothendieck%27s_relative_point_of_view)

~~~
Yhippa
Just wanted to say thanks for this amazing answer. My statement wasn't worded
the best but I feel like you captured the essence of what I was trying to say.

I see things like quantum mechanics and think about how instead of say at the
deepest levels of nature that instead of somehow enumerating all the possible
positions of an electron or particle it was more efficient to somehow make it
probabilistic. Likewise for things like fractals instead of having a
ridiculously high-degree polynomial function you have a simple polar function
or the examples in the article that make elegant designs from a simple
formula.

I am rambling at this point but I appreciate your links. I already went down
the rabbit hole on the Wikipedia ones.

As an aside when I took a MSCS class in cryptography I had to refresh on set
theory and remember enjoying that and I guess is what got me thinking about
these types of things again.

~~~
torustic
I've been thinking about something to recommend, since it's a bit ridiculous
to end a relativistic exposition without settling the uncertainty into some
new, interesting starting point.

Minimum description length might be just the right place.

[0]
[https://en.wikipedia.org/wiki/Minimum_description_length](https://en.wikipedia.org/wiki/Minimum_description_length)

[1] [https://www.amazon.com/Description-Principle-Adaptive-
Comput...](https://www.amazon.com/Description-Principle-Adaptive-Computation-
Learning/dp/0262072815/)

Alternately, generating functions.

[2]
[https://en.wikipedia.org/wiki/Generating_function](https://en.wikipedia.org/wiki/Generating_function)

[3] [https://www.amazon.com/generatingfunctionology-Third-
Herbert...](https://www.amazon.com/generatingfunctionology-Third-Herbert-S-
Wilf/dp/1568812795/)

Alternately alternately, various categorical notions.

[4]
[https://en.wikipedia.org/wiki/Realizability](https://en.wikipedia.org/wiki/Realizability)

[5]
[http://stijnvermeeren.be/download/mathematics/essay.pdf](http://stijnvermeeren.be/download/mathematics/essay.pdf)

[6]
[https://ncatlab.org/nlab/show/locally+presentable+categories...](https://ncatlab.org/nlab/show/locally+presentable+categories+-+introduction)

[7]
[https://ncatlab.org/nlab/show/groupoid+cardinality](https://ncatlab.org/nlab/show/groupoid+cardinality)

Alternately alternately alternately, foundational madness of the best kind.

[8] [http://www.forkinganddividing.com/](http://www.forkinganddividing.com/)

[9] [https://www.amazon.com/Blind-Spot-Lectures-
Logic/dp/30371908...](https://www.amazon.com/Blind-Spot-Lectures-
Logic/dp/3037190884/)

------
rootdiver
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)

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

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lugus35
I just see a donut.

~~~
kylek
Really? I found some of these [0][1][2] to be shockingly beautiful.

[0][http://math.ucr.edu/home/baez/roots/polynomialroots05expi02....](http://math.ucr.edu/home/baez/roots/polynomialroots05expi02.png)
[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)

------
ttflee
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.

~~~
ranit
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
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.

