# Most intricate and most beautiful structures in mathematics

In the December 2010 issue of Scientific American, an article "A Geometric Theory of Everything" by A. G. Lisi and J. O. Weatherall states "... what is arguably the most intricate structure known to mathematics, the exceptional Lie group E8." Elsewhere in the article it says "... what is perhaps the most beautiful structure in all of mathematics, the largest simple exceptional Lie group. E8." Are these sensible statements? What are some other candidates for the most intricate structure and for the most beautiful structure in all of mathematics? I think the discussion should be confined to "single objects," and not such general "structures" as modern algebraic geometry.

Question asked by Richard Stanley

Here are the top candidates so far:

1) The absolute Galois group of the rationals

2) The natural numbers (and variations)

4) Homotopy groups of spheres

5) The Mandelbrot set

6) The Littlewood Richardson coefficients (representations of $S_n$ etc.)

7) The class of ordinals

8) The monster vertex algebra

9) Classical Hopf fibration

10) Exotic Lie groups

11) The Cantor set

12) The 24 dimensional packing of unit spheres with kissing number 196560 (related to 8).

13) The simplicial symmetric sphere spectrum

14) F_un (whatever it is)

15) The Grothendiek-Teichmuller tower.

16) Riemann's zeta function

17) Schwartz space of functions

And there are a few more...

• With great respect for Richard-and without going so far as to call for it's closure,because it is an interesting question-I think if anyone of lesser stature in the mathematical community had posted this question,there would have numerous calls to close it as too general and subjective. – The Mathemagician Dec 12 '10 at 18:16
• The main output of this question will likely be a reinforcement of the ego of each domain/community. Not that good for the unity of mathematics. – Denis Serre Dec 12 '10 at 18:38
• IMO, the statements are sensible because they include the words "arguably" and "perhaps". In other words, I think there is little objective content to them. With all due respect to the OP, I think this question is the epitome of "subjective and argumentative", and I have voted to close for that reason. – Pete L. Clark Dec 12 '10 at 19:04
• I think the question is really good and deserves its place on MO. This said (and this is in no way in reference to the OP, it is just a general rant) I've noticed that often on MO when someone with no points posts a soft question/big list type of question there is always the same bunch of people who rush to close it and to say basically it's lame. When a professor posts a question virtually of the same order, he gets 80 votes up and congratulations on the "amazing question". I've seen many examples of that and it is this same "police" going to every post and deciding what's good, what's bad. – Rachid Atmai Dec 12 '10 at 19:46
• Since this has already attracted a vote to reopen (at time of typing) I've opened a thread on meta MO ( tea.mathoverflow.net/discussion/834/… ); if you feel the question should be reopened please take the discussion there. Also, please vote this comment up for visibility. – Yemon Choi Dec 12 '10 at 20:08

The rotational symmetries within Young's lattice, discovered at the beginning of the 21st century by Ruedi Suter. I think the bilateral symmetry has been known many decades, but rotational symmetries were not, and they seem quite surprising.

The Wikipedia link below clearly explains the idea. The first link is Suter's paper.

http://www.sciencedirect.com/science/article/pii/S0195669801905414

https://en.wikipedia.org/wiki/Young%27s_lattice#Dihedral_symmetry

This is a less-known example. Consider the semigroup $$A_0 = \big\{ \big[\begin{smallmatrix} 0&0\\0&0 \end{smallmatrix}\big], \big[\begin{smallmatrix} 1&0\\0&0 \end{smallmatrix}\big], \big[\begin{smallmatrix} 0&1\\0&0 \end{smallmatrix}\big], \big[\begin{smallmatrix} 0&1\\0&1 \end{smallmatrix}\big] \big\}$$ under usual matrix multiplication. The variety $\mathrm{var}A_0$ generated by $A_0$ is finitely universal in the sense that the lattice of subvarieties of $\mathrm{var}A_0$ embeds all finite lattices. The semigroup $A_0$ is a minimal example since the variety generated by any semigroup of order three or less is not finitely universal.

Brownian walk (or Wiener process, to conform to "single object" stipulation).

Schramm-Loewner evolution. This is a stochastic differential equation which models the scaling limit of many stochastic planar processes.

I think that Stone-Cech compactification has a highly and deeply complexity.

• See my comment to Daniel Geisler's answer. – Deane Yang Dec 13 '10 at 20:08
• I agree with Deane's comment, even though I think a case can be made for $\beta \mathbb N$. (The poster doesn't say what space he is taking the Stone-Cech compactification of, mind you.) – Yemon Choi Dec 13 '10 at 22:10

Many believe (me too!) that finite fields are among the most beautiful mathematical objects (not to mention important).

I really like the Collatz conjecture graph ($3n+1$ problem), the graph showing the evolution of natural numbers under the rule: If $x$ is even, divide by 2; if $x$ is odd, multiply by 3 and add 1. Here's the graph with $100,000$ nodes:

How about true arithmetic. True arithmetic is the set of first order statements that $$\mathbb N$$ satisfies with its usual operations.

One cool thing you can do is construct non-standard models of TA. These will share all first order properties with $$\mathbb N$$, but could be widely different. This is pretty crazy, considering that they since first-order induction is a property of $$\mathbb N$$, it is also a property of these models, even though second-order induction fails. The similarities are great enough that you can even encode the same objects for the most part, leading to infinite structures that act like finite ones. For example, you can have infinite turing machines.

I want to mention the small category $$\bf{\Delta}$$, where its objects are $$[n]$$ for a natural number $$n$$ and its morphism are all maps preserve orders.

I've been told $$\bf{\Delta}$$ is magic for a long time. Topologists use it to say what a space (simplicial set) is, and algebraic-geometors also use it. We even need it to define the higher categories.

I'm still in learning about this category, and I wish I could know some deep reason why it works so well.

The cube model of $$\Bbb R^3.$$ Geodesics are gold strands. I think it's a simple, intricate, and beautiful way to view $$\Bbb R^3.$$

• Never heard of it. You have a link to explain what "the cube model of ${\bf R}^3$" is? – Gerry Myerson Nov 23 '19 at 2:51
• @GerryMyerson It's built by making $\Bbb R^3$ into a compact space, by mapping it inside $S^2,$ and then deforming the structure to a cube. – Jack Zimmerman Nov 23 '19 at 3:16
• Thanks. Surely there's more than one way to map ${\bf R}^3$ to the interior of $S^2$? – Gerry Myerson Nov 23 '19 at 3:18
• Yes I'm sure there is. Do you want me to clarify something? – Jack Zimmerman Nov 23 '19 at 3:22
• Here's an example of a mapping to the inside of $S^2.$ $(x,y,z)\mapsto(x,y,z)/\sqrt{1+r^2}$ so, $\left(\dfrac x{\sqrt{1+x^2+y^2+z^2}},\dfrac y{\sqrt{1+x^2+y^2+z^2}},\dfrac z{\sqrt{1+x^2+y^2+z^2}}\right).$ But you get the same model no matter which mapping you use – Jack Zimmerman Nov 23 '19 at 3:39

The Selberg class, which is conjectured to contain all L-functions of arithmetic interest. It is closed under multiplication, and is a way to formalize the common properties of L-functions for which we believe the analogue of the Riemann hypothesis holds. Many still open conjectures address its structure, like the degree conjecture, that states an invariant of an element of this class called the degree is always an integer, or Selberg's orthonormality conjecture, that says the set of primitive (meaning irreducible) elements form in some sense an orthormal system, suggesting connections with other parts of math such as Hilbert spaces, representation theory, Galois theory...I fell in love with that class many years ago and can't get my eyes off of it !

Shelah's Body of Work. Considering that this list of references is over 100 pages long, I think this a contender.

• No offence, Michael, but you seem to be interpreting the original question rather creatively with this answer. – Yemon Choi Jan 26 '11 at 3:04
• @Choi Indeed, but aren't the best answers creative? If I had said, L, L(A), L[A], the cumulative hierarchy V, the collection of P-names generated by a partial order, or mentioned the PO constructed by essentially 'weaving' together every CCC PO in a ground model to produce a model of MA+'not CH', they would not have elicited the same feelings of respect and amazement, I get when I consider such objects. So I opted for something I could share, which everyone could appreciate, which demands the same sense of respect and amazement. – Not Mike Jan 26 '11 at 4:14
• @Blackmon: "aren't the best answers creative" - it depends on your question, doesn't it? As you may see from some of my previous comment s on some of the answers here, I don't find them particularly good. Moreover, although no one else seems to have mentioned things I find kewl, I haven't taken the opportunity to put "the publication record of the late N. J. Kalton" as an answer... – Yemon Choi Jan 26 '11 at 19:08
• @Blackmon: Because I don't think this is a productive MO question, from the point of view of light rather than heat. I also think "the publication record of the late N. J. Kalton" would be a poor answer to the question; and I am not a fan of several other answers to this question. It doesn't help that the title of this question is not quite the same as what the question seems to actually ask for... – Yemon Choi Jan 27 '11 at 5:01
• @Choi I agree, this question is nothing more than a place to vent about how "awesome you're field is." I made the choice not to do this, and just posted something everyone should be able to appreciate. With that being said, I think this particular question should be closed, for the reasons highlighted in the comments to the original question. – Not Mike Jan 27 '11 at 5:13