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][1]
________________________

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...


  [1]: http://mathoverflow.net/users/2807/richard-stanley