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.

Here are the candidates so far:

1. The natural numbers (and variations)

2. The absolute Galois group of the rationals

3. The monster vertex algebra

4. The class of ordinals

5. The Cantor set

6. The homotopy groups of spheres

7. The Mandelbrot set

8. Exotic Lie groups

9. The canonical pointed symmetric sequence of simplicial sets

10. Stone Cech compactification (perhaps of the natural numbers)