[Imc-makerspace] Lie Group E8, Double-Covering of the Klein Bottle, etc.

[Stewart Dickson]

Some leftovers from TechKnowLedge this morning:

The Lie Algebraic Group E8 (mathematics) - Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/E8_(mathematics)   E8 X E8 would be a 
functional mapping of E8
onto itself.   Functions which do this can operate within String Theory.

Http://www.endlessuniverse.net
Check out the "Animations" section:
http://www.physics.princeton.edu/~steinh/endlessuniverse/animations.html

How the cover image of "Endless Universe" is topologically a Klein 
bottle which has swallowed itself:
http://www.ifp.illinois.edu/~sdickson/UIUC_CS/LeTopologicon_inside_back_cover001.PNG

The number of interior regions determined by diagonalizing a regular 
polygon:
http://www.math.rutgers.edu/~erowland/polygons-project.html
n^3 < R(n) < n^4  -- it is a phenomenon of super-cubic complexity which 
takes place in the plane.  Who knew?   It's practically fractal!

The topological mesh techniques I came up  to do "thickening 
<http://emsh.calarts.edu/%7Emathart/sw/objView/thicken.html>" of 
mathematical surfaces for sculpture do not cut it for this problem 
(4-coloring the interior regions determined by regular polygons.)   
Andrew Glassner, "Maintaining Winged-Edge Models", Graphics Gems II ; 
James Arvo, ed.; (IV.6 -- pp. 191-201) Academic Press, Inc.; ISBN: 
0-12-064480-0

I think that the Combinatorial Map 
<http://en.wikipedia.org/wiki/Combinatorial_maps> is required to solve 
this.  I have discovered that I need to un-think Glassner's model in 
order to correctly implement the Combinatorial Map in C++.

And "Why 19?"  http://ekadhikena-purvena.tumblr.com
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