Most intricate and most beautiful structures in mathematics

Question

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

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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...

Answer 1

The absolute Galois group of $\mathbb{Q}$. It contains the information of all algebraic extensions of the rationals - and is therefore the most important single object of algebraic number theory. Representations of the absolute Galois group are central to many diophantine questions; see for example the Taniyama-Shimura conjecture (aka modularity theorem) which led to a solution of Fermat's last theorem and states in some form that certain Galois representations associated to elliptic curves come from modular forms.

One of the most intricate set of conjectures is dedicated (partly) to the study of representations of the absolute Galois group of $\mathbb{Q}$: the Langlands program.

Answer 2

How about the Leech lattice. This is a 24-dimensional packing of unit spheres where each one touches 196560 others. It is the densest 24-dimensional lattice packing (and very likely the densest 24-dimensional sphere packing, although this has not been proved - EDIT: it has been now). It has a remarkable amount of symmetry, and most of the densest sphere packings known in dimensions < 24 are derived from it (and known sphere packings in dimensions > 24 are nowhere near as dense when normalized for the dimension).

Maybe this is already implicitly included in the list, as it is closely related to the monster vertex algebra.

Answer 3

The (stable or unstable) homotopy groups of spheres are certainly considered intricate and beautiful by topologists.

Here is an interesting (obvious) fact about the stable homotopy groups of spheres that I learned from Vigelik:

In the category of commutative rings (with unit) there is an initial object, $\mathbb{Z}$. This seems to be one reason the integers are important or rather fundamental. But there is something more fundamental! There is a functor from commutative rings to ring spectra, the Eilenberg-MacLane functor. $H\mathbb{Z}$ is no longer initial in this category, the sphere spectrum is! So somehow the stable homotopy groups of spheres are a pretty cool/fundamental ring.

I do not know a lot about the unstable setting, but there is a lot of extra data that it has.

I think that Lennart's point about intricacy and complexity showing up when you start to try to compute the thing sounds like a confusion of what one means by intricate and complex. But it is not, it is not the messiness of the computation that makes it intricate, it is the way of teasing apart the knowledge we do have in meaningful ways that lead me to believe that it is a very intricate object. Especially all the number theory hidden in the chromatic picture, which is part of what Lennart is referring to when he mentions the moduli stack of formal group laws.

Edit: My advisor pointed out another reason that the stable homotopy groups of spheres are cool: $\pi^S_{*}(S^0)=\pi_{*}(B\Sigma_{\infty})$, so the stable homotopy groups of spheres are the homtopy groups of the classifying space of the the category of finite sets and bijections. This is essentially the Barratt-Priddy-Quillen theorem (I am told, I do not know the precise statement). That is pretty cool too! All that information about finite sets sitting there has to be something.

(seems to look fine now, please ignore the bump)

Answer 4

The monster vertex algebra.

It is (to date) the central object in monstrous moonshine, since its character is the $q$-expansion of the modular $J$-function, its automorphism group is the monster simple group, and the graded trace of any element of the monster is the $q$-expansion of a genus zero modular function. The construction of this structure (by Frenkel, Lepowsky, and Meurman) involves ascending a hierarchy of objects that are by themselves quite intricate and beautiful.

1. One begins with the extended binary Golay code of length 24. Up to symmetries, it is the unique copy of $\mathbb{F}_2^{12}$ in $\mathbb{F}_2^{24}$, for which any five basis vectors are contained in a unique codeword (i.e., it forms a Steiner $(5,8,24)$ system). The codewords are separated by Hamming distance at least 8, so even if 3 bits in a code word are changed the error can be corrected. The automorphism group of the Golay code is the sporadic simple group $M_{24}$ of order 244823040.

2. Using the Golay code to produce coordinates of generators, one constructs the Leech lattice $\Lambda$, which is a rather densely packed copy of $\mathbb{Z}^{24}$ in $\mathbb{R}^{24}$. One can also make the Leech lattice as a subquotient of the even unimodular lattice $I\!I_{25,1}$, which has its own exceptional properties. Peter Shor mentioned the Leech lattice in another answer, so I'll just note that its automorphism group is a double cover of Conway's sporadic simple group $Co_1$, which has order 4157776806543360000.

3. For any positive definite even lattice $L$, there is a canonical construction of a vertex operator algebra graded by that lattice, called the lattice vertex algebra $V_L$. I think physicists say that it is the algebra of chiral symmetries of a conformal field theory describing a bosonic string propagating in the torus $L \otimes \mathbb{R}/L$ (but I may have mixed up the words). It has an action of the holomorph of the algebraic torus $L \otimes \mathbb{C}^\times$.

4. The "-1" automorphism of the Leech lattice induces an automorphism $\theta$ of $V_\Lambda$, and there is a unique irreducible $\theta$-twisted module $V_\Lambda(\theta)$ that inherits an action of the centralizer $2^{1+24}.Co_1$ of $\theta$ in the automorphism group of $V_\Lambda$. The monster vertex algebra is formed by taking the direct sum of fixed points: $V^\natural = (V_\Lambda)^\theta \oplus (V_\Lambda(\theta))^\theta$.

Apparently, the hard part was proving that the monster acts on $V^\natural$ by automorphisms.

There are some additional conjectural reasons for considering it beautiful:

1. In the same paper where it was constructed, it was conjectured to be the unique vertex operator algebra with central charge 24, character equal to the modular $J$ function, and representation category equivalent to $Vect$. (Naturally, this does not account for higher structure like twisted modules.)

2. Witten suggested that it is dual to pure 3-dimensional quantum gravity with minimal cosmological constant by AdS/CFT correspondence.

Answer 5

Since one of the questions is Are these sensible statements?, allow me to answer that one with a resounding NO, and be on record against the size-ism inherent in the statements of Scientific American. As the largest simple exceptional Lie group, E8 deserves credit for both intricacy and beauty. but the author seems to imply that the size record makes E8 not only more intricate but also more beautiful than the other simple exceptional Lie groups.

Granted I don't know much about Lie groups, but it really bothers me that an aesthetic judgment can be based on size alone. Last I checked, paintings are not judged on their size.

Answer 6

Here an answer in form of a question:

does it strike anyone that many of the candidates for "most intricate and/or beautiful structure in mathematics" proposed here find their natural joint home where they meaningfully relate to each other in... string theory?

• $E_8$ as a GUT group (see the beautiful exposition (Witten 02)), $G_2$ as a structure group, and in fact the whole tower of exceptional generalized geometries inside 11d SuGra;

• the Leech lattice and the monster vertex operator algebra for evident reasons and reasons already mentioned;

• the Grothendieck-Teichmüller tower for obvious reasons ("hence" also the absolute Galois group..);

• geometric Langlands correspondence as but one incarnation of S-duality;

• tmf, being the target of the partition function of the heterotic string (and tmf0(2) as the coefficients for the partition function of the type I superstring; and conjecturally the whole system of $Tmf(\Gamma)$s for F-theory)

These are just the most evident. One could go on about how the motivic Galois group also fits in etc., but I don't want to strain it.

What sometimes makes me wonder is that mathematicians have all that appreciation for all these separate intricate and beautiful phenomena that come out of string theory, one by one, including an impressive list of Fields-awarded work, but that there is little appreciation that closely similar (just vastly "bigger") to how the Moonshine conjecture qualified as a problem in mathematics, the question "What is string theory?" may be one of the deepest open problems possibly not in phyisics, but in mathematics.

Answer 7

The Mandelbrot Set is widely viewed as beautiful and intricate, although I can't give a mathematical definition for those.

The imperfect self-similarities are no accident. Many of the pieces correspond to the behaviors of the critical point $0$ under iteration of $z \to z^2 + c$.To each point in the plane, there is a corresponding Julia set, and the relationship of the point to the Mandelbrot set indicates some of the structure of the Julia set.

Answer 8

One of the most beautiful structures, in my mind, is the classical Hopf fibration, which allows you to visualize the $3$-sphere $S^3$ as a smooth circle bundle over the $2$-sphere. When you view $S^3$ minus a point as $\mathbb R^3$, one can actually draw very nice pictures of this fibration. It's doubly interesting to me because it involves the isomorphism of $S^2$ with $\mathbb C\mathbb P^1$ from complex analysis.

There are actually 4 such Hopf fibrations (spheres which are total spaces of fibre bundles whose base and fibre are also both spheres):

1) $S^1$ is an $S^0$ bundle over $\mathbb R \mathbb P^1 \cong S^1$.

2) $S^3$ is an $S^1$ bundle over $\mathbb C \mathbb P^1 \cong S^2$.

3) $S^7$ is an $S^3$ bundle over $\mathbb H \mathbb P^1 \cong S^4$, the quaternionic projective line.

4) $S^{15}$ is an $S^7$ bundle over $\mathbb O \mathbb P^1 \cong S^8$, the octonionic projective line.

Answer 9

The Turing degrees are an immensely intricate poset $\mathcal{D}$. Here are some of their remarkable properites:

1. Every countable poset is embeddable in $\mathcal{D}$.
2. $\mathcal{D}$ contains minimal degrees. (a non-zero degree $\mathbf{m}$ with no degree between $\mathbf{0}$ and $\mathbf{m}$)
3. For every non-zero degree $\mathbf{d}$, there is a degree that is incomparible with $\mathbf{d}$.
4. $\mathcal{D}$ contains an antichain of size $2^{\aleph_0}$.
5. No infinite strictly increasing chain in $\mathcal{D}$ has a least upper bound.
6. For every degree $\mathbf{d} \geq \mathbf{0}' $, there is a degree $\mathbf{c} < \mathbf{d}$ such that $\mathbf{c}'=\mathbf{d}$. (Here $\mathbf{c}'$ denotes the set of indices of oracle Turing machines that halt when using $\mathbf{c}$ as an oracle. Note that one must check that this is well-defined on degrees.)
7. For any two recursively enumerable degrees, there is a recursively enumerable degree strictly between them.
8. Any finite distributive lattice can be embedded in the recursively enumerable degrees.

Answer 10

The Littlewood-Richardson coefficients. (Or, if one wants a single object, as per the rules of the game: the representation ring of $S_n$ or $GL_n({\bf C})$. Or the ring of symmetric functions. Or the cohomology ring of the Grassmannian with the Schubert variety basis. Etc., etc.)

On the one hand, the Littlewood-Richardson coefficients have fairly simple geometric descriptions (using such combinatorial gadgets as Young tableaux, honeycombs, or puzzles), but on the other hand obey a number of deep recursive properties. (See for instance my Notices article with Allen Knutson on one aspect of these coefficients.) Last, but not least, they are connected to an amazing number of areas of mathematics (see e.g. Fulton's survey article).

Answer 11

I believe the natural numbers are the most intricate and beautiful structure in all of mathematics. Particularly insofar as all of the other intricate and beautiful structure we actually work with can be encoded via the natural numbers.

Answer 12

Let $g$ and $n$ be positive integers such that $3g-3 + n > 0$ Let $\mathcal{M}_{g,n}$ the moduli stack of genus $g$ nodal curves with $n$ marked points. There are three obvious families of maps

forgetting a point

$$\mathcal{M}_{g,n+1} \rightarrow \mathcal{M}_{g,n}$$

identifying two marked points on two different curves yielding a new one

$$\mathcal{M}_{g_1,n_1+1} \times \mathcal{M}_{g_2,n_2+1} \rightarrow \mathcal{M}_{g_1 + g_2,n_1 + n_2}$$

and identifying two marked points on a single curve yielding a new one with higher genus

$$ \mathcal{M}_{g,n+2} \rightarrow \mathcal{M}_{g+1,n} $$

This system constitutes the so-called Grothendieck-Teichmüller tower. It is indeed intricate and in my opinion, also beautiful. Moreover, it is a conjecture of Grotehndieck that its automorphism group is naturally isomorphic to the absolute Galois group over $\mathbb{Q}$, namely $\mathrm{Gal}(\bar{\mathbb{Q}}|\mathbb{Q})$.

Answer 13

The class of all ordinals. The class of cardinals is embedded within it (if AC holds) since one identifies a cardinal with the smallest ordinal such that the set of all smaller ordinals has that cardinality. ($\aleph_0$ is the cardinality of the set of all finite ordinals, $\aleph_1$ is the cardinality of the set of all countable ordinals, etc. $\aleph_\omega$ is the cardinality of the set of all ordinals whose cardinality is $\aleph_n$ for some finite $n$. ($\omega$ is the ordinal that gets identified with $\aleph_0$ in the aforementioned identification) $\aleph_{\omega+1}$ is the set of all ordinals of cardinality $\aleph_\omega$, and so on. $\aleph_\omega$ is the smallest cardinal greater than $\aleph_0$ that is known not to be equal to $2^{\aleph_0}$.)

But if grading is only based on "intricacy", maybe the class of all sets, conventionally denoted "V" because it looks like the letter V (?) might be in first place. Some people have tried to embed all of mathematics within this thing.

Later edit: The "\aleph"s and the "\omega"s are failing to get rendered when I view this thing. Look at the code and you'll see them.

Answer 14

Ok, I'll throw my hat in the ring: I like the classical Cantor set.

Not only does it demonstrate the complexity that relatively simple subsets of the real line have, it illustrates an important property of measures on the real line - namely, that measurability has nothing to do with cardinality of the set (i.e. this is an uncountable set with measure zero!)

It also gives an example of a completely disconnected subset of $\mathbb{R}$ that literally has no components - it contains no open intervals of $\mathbb{R}$ in its power set.

There are many, many more observations one can make about the Cantor set, but I think the obvious ones make my point very nicely. When I teach real analysis, this is an example I think I'll be using a great deal to illustrate properties of the real line.

Answer 15

This has been forgotten so far: http://en.wikipedia.org/wiki/Riemann_zeta_function

Answer 16

$\operatorname{Spec}(\mathbb{Z})$.

It can also be thought as the set of prime numbers. I don't know if it can really be considered "intricate"...

Answer 17

In my view it is difficult to come up with an alternative to any of the exotic Lie groups, which are unquestionably quite intricate but are also beautiful because they express the properties of certain geometric spaces using both fundamental algebra (i.e., groups) and geometric structures of their own (i.e., Riemannian geometry). I don't know $E_8$ particularly well, but I still have vivid memories of Robert Bryant's lectures describing the structure of $G_2$.

Answer 18

The generalized cohomology theory known under the name Topological modular forms:

Here's a picture of the graded ring $\mathit{tmf}\;^*(pt)$:
http://www.staff.science.uu.nl/~henri105/PDF/TmfRing.pdf

And here's the spectral sequence used to compute it:
http://math.mit.edu/conferences/talbot/2007/tmfproc/henriques-tmfSS.pdf
Its $E_2$ page is $Ext_{A(2)}(\mathbb F_2,\mathbb F_2)$, where $A(2)$ is this beast:
http://www.staff.science.uu.nl/~henri105/PDF/A2.pdf

Here's another spectral sequence, that computes the closely related graded ring $\mathit{Tmf}\;^*(pt)$:
http://math.mit.edu/conferences/talbot/2007/tmfproc/EllipticSpectralSequence.pdf

Answer 19

The outer automorphisms of the group $S_6$ of all permutations of a set of six objects.

$6$ is the only number for which $S_n$ has any outer automorphisms.

The group inner automorphisms of $S_6$ is a subgroup of index $2$ in the group of all automorphisms of $S_6$, and --- here's a (probably) unexpected fact: It's one of exactly three subgroups of index $2$, no two of which are isomorphic to each other.

Answer 20

The Surreal numbers as constructed by Conway.

They contain a copy of many objects already on the list ($\mathbb{N},O_n$) and so surpass them in complexity, and they are recursively defined from the empty set using $O_n$ length recursions which is incredibly beautiful.

I would also propose the absolute Galois group of the field of fractions of the Grothendieck ring of the ordinals, however I have very little understanding of this object (nor does anyone else to my knowledge).

Answer 21

I have voted for ${\mathbb N}$; but let me nevertheless propose an object living in the analytical realm, namely the Schwartz space ${\cal S}$ of infinitely differentiable functions $f:{\mathbb R}\to{\mathbb C}$ that for $|x|\to\infty$ together with their derivatives go to zero faster than any power $1/|x|^n$. The "intricateness" of this space stems from the many operations you can perform in it and from the fact that these operations are intertwined with each other in miraculous ways. $$ $$ Responding to a comment: You have (a) ordinary multiplication and convolution, (b) "multiplication" with arbitrary polynomials $p(x)$ and operations $p(D)$, (c) multiplication with functions of the form $x\mapsto e^{iax}$ and the translation operator $T_a: f(\cdot)\mapsto f(\cdot-a)$ and (d) scaling of the variable $x$ resp. $\xi$. The Fourier transform $\Phi$ interchanges in each of these three cases the respective operations; and at heart of it all is Gauss' normal distribution $x\mapsto {1\over \sqrt{2\pi}}\int e^{-x^2/2} dx$ which stays fixed under $\Phi$. And, last not least, there is a scalar product which is preserved by $\Phi$.

Answer 22

I heard good things about F_un!

Answer 23

Another one: Chaitin's Omega constant.

Original answer by: none

Answer 24

I don't know if it's the most beautiful or the most intricate, but I certainly think the random graph $G(n,p)$ deserves consideration, if only for philosophical reasons.

Answer 25

Maybe this wouldn't be my first choice, but I still think it's worth being on the list: Gödel's constructible universe $L$.

I would argue that it is intricate because it can serve as a model for "all of mathematics" (i.e., ZFC), furthermore answering many combinatorial questions left open by ZFC alone. Even though most(?) set theorists will probably argue that it is not "the" true model giving the right answer to these questions, it is still undoubtedly a rich and complex structure, moreover one in which the axiom of choice and the continuum hypothesis are not only true but "explained".

But it is also beautiful because of its connections with higher computability theory (e.g., the sets of integers constructed at the level $\omega_1^{\mathrm{CK}}$ of the constructible hierarchy, where $\omega_1^{\mathrm{CK}}$ is the smallest nonrecursive ordinal, are exactly the hyperarithmetical sets, i.e., the (lightface) $\Delta^1_1$ sets of the analytic hierarchy), and, in a related manner, because of Jensen's results on the "fine structure" of $L$. In a very intuitive way, I'd say that $L$ consists of sets that are ultimately "computable" (iterating the Turing jump as far as it can be), a perfectly regular construction that prohibits any randomness.

So even if set theorists are unhappy with $L$ because it forbids really large cardinals, and even if they try to construct something better (the core model), I argue that Gödel's original $L$ is still something immensely intricate and beautiful.

Answer 26

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:

Answer 27

I like the hyperbolic plane where Escher's "circle limits" live.

In the hyperbolic plane you can turn your car (i.e. constant aceleration perpendicular to the constant speed) and not manage to close its trajectory (there are equidistant curves and horocycles).

Also, the symmetry group of the tiling by right angled hexagons contains all but a finite number of closed surface groups. So you can build almost all closed surfaces gluing these hexagons.

People in the hyperbolic plane wont agree on the angle between two stars (i.e. boundary points) but if you average the measurements of other people around you the result will agree with your own measurement (hence everyone thinks they are right).

The modular group (and its congruence subgroups) are important in number theory (which I know next to nothing about) and also in complex analysis (where, for example, the congruence subgroup $\Gamma(2)$ is the covering group of the plane minus two points and allows one to prove that if an entire function omits two values in its image it must be constant).

The list could go on (the Gauss-Bonnet theorem, Brownian motion escapes with positive speed to infinity, Anosov property the geodesic flow, quasi-geodesics are at bounded distance from geodesics, the area of a convex hull is bounded by a constant times the number of points, the strong isoperimetric inequality holds i.e. perimeter is greater then volume for all sets, etc, ...).

Answer 28

The relationship between the discrete order and the multiplication on the natural numbers leads to, among other things, the study of gaps between primes. I would nominate a class of structures S(n), which are the sets of integers relatively prime to the nth primorial (p_1p_2...p_n) as a collection worthy of the labels beautiful and intricate. The symmetry and self-similar nature appeal to many, and while the construction is simple, there are many simple facts remaining to be established about the S(n). For one, the largest gap between consecutive members of S(n) seems to be unknown. (Cf Erik Westzynthius's cool upper bound argument: update? for a weak upper bound; I hope to post an improvement soon.)

Gerhard "Ask Me About System Design" Paseman, 2010.12.13

Answer 29

How about $\mathbb{R}^n$? I hope people don't consider this example too simplistic. After all, the structure of $\mathbb{R}^n$ gives rise to all of the theory of topological and differentiable manifolds. Specific important highlights include the theory of "algebraic" equations (inverse and implicit function theorems) and the local theory of differential equations (jets, forms, integral submanifolds).

Answer 30

Hard to pin down the object I find the most intricate and beautiful mathematical structure, but in my opinion this structure has yet to emerge from the growing body of our knowledge. I believe a structure mathematicians are gradually understanding is manifesting itself in various areas of algebra, number theory, analysis, probability theory, theoretical physics, geometry, topology, set theory and several others, and one might thus say that the most intricate and beautiful structure in mathematics is mathematics itself. More precisely it is some unknown object that all mathematicians study from different sides, angles and perspectives without yet knowing the name for the thing that their discoveries actually have in common. Some evidence to this is given by repeated patterns occurring in several answers here, as well as by some striking instances of partial unification of mathematical concepts, such as (to name very few) Connes' noncommutative geometry, Arakelov theory, Segal's modular functors, topos theory and homotopy type theory.