"To most outsiders, modern mathematics is unknown territory. Its borders are protected by dense thickets of technical terms; its landscapes are a mass of indecipherable equations and incomprehensible concepts. Few realize that the world of modern mathematics is rich with vivid images and provocative ideas."
~Ivars Peterson
In mathematics, there is a subject known as Complex Simple Lie Algebra. There are four infinite families and five exceptions. It is a 248-dimensional object (of the Platonic sort), and has a rank 8 root lattice. I can only show 2-dimensional and 3-dimensional projections, as it is somewhat difficult to experience the higher-dimensional cases. The E8 group is the largest of the exceptions. This is difficult stuff and I have barely looked into this subject myself, but it is all very beautiful looking. It has symmetry such that it can be rotated in any direction of a 248-dimensional space and still appear as the same shape.
This has been recently associated with a "Theory of Everything" in physics discovered by the surfer and physicist Garrett Lisi. The validity of this theory has been eliminated, and it has been shown to be incorrect. But that doesn't take away from the mathematical and geometrical beauty of this construct.
Here it is projected on the 2-dimensional plane:
And here projected on the 3-dimensional plane and rotated:
The other exceptional Complex Simple Lie Groups are G2 (rank 2, dimension 14), F4 (rank 4, dimension 52), E6 (rank 6, dimension 78), and E7 (rank 7, dimension 133). Those with smaller ranks can be shown as subgroups of those of higher ranks.
G2 (3-D projection):
To see other 2-D projections, check out Wikipedia.
3-D projection videos were obtained from: http://deferentialgeometry.org/
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