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...