Wednesday, April 21, 2010

Numbers for Names, Part 3

I think it's time to finish this run on numbers and names, but first, I found a company called E8.com.

E8's ownership includes 1% owned by Moover Toys. Their 1% of E8 cost them \$88,888.88, thus 100% of E8 is \$8,888,888.88. Do you think the people in this company have a thing about the number 8?

The company website says E8.com's mission is to provide the most advanced, intuitive, user-interfacing platform in the world based upon the unifying supersymmetry of the exceptional Lie group E8 and its corresponding 248 dimensions.

I don't understand any of this. It's not elementary math. But I like this artwork. Click for a larger version:

Here's a description of what it might mean (condensed from the original here):

The Lie group E8 consists of 240 points that are tightly and symmetrically packed in 8-dimensional space. This object has 696,729,600 symmetries. In comparison, the 8 points at the corners of a 3-dimensional cube have only 48 symmetries.

We can't visualize objects in 8 dimensions, but we can draw 2-dimensional projections. For example, if you shine a flashlight on a cube, its shadow could look like a hexagon. If you orient the cube correctly, the shadow looks like a regular hexagon (6-sided figure; all sides equal and all angles equal).

This diagram does the same for the E8 root system. The light is "shining" on these 240 points so the 2-dimensional shadow they cast is as symmetric as possible. The E8 has 60 symmetries: 30 rotations and 30 reflections. The 240 points wind up in 8 concentric rings of 30 points each. Those are the black dots.

The lines are "shadows" of the lines that frame this shape, back in the imaginary 8-dimensional space. Each line connects a point to its nearest neighbors among the other 239. Each of these 240 points has 56 nearest neighbors - they are very tightly packed!

Here's how the picture looks on my giant monitor, zoomed to 1200% of its original size! Click for the larger version.
PS - I was unable to confirm the 60 symmetries ... I started to count, but ZZZzzzzzzzzzzz