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.

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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 simplicial symmetric sphere spectrum

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

11) Schwartz space of functions