I have published my ToE.pdf to vixra.org http://vixra.org/abs/1006.0063 . While looking around vixra, I found a paper by Tony Smith on E8 http://vixra.org/pdf/0907.0006v1.pdf. It has much of the work on his website including references to Lisi and Bathsheba on their visualizations of E8. Some of the references are based on the 600 Cell dual of H4+H4/Phi (conjectured by Richter to be isomorphic to E8). I find the edge counts and vertex math does not work the same as E8 (8D to 4D), but it is interesting to compare visualisations. To that end I recreated here a E8 Petrie projection which is isomorphic to 2D split real even E8 Petrie projection (except it only produces half of the 6720 “shortest edge” counts, of different length). All of these use a single set of 3 projection basis vectors to produce a single 3D projected object with 3 unique faces. On one face, it has the famous E8 Petrie projection:
On another face it has a pentagonal projection:
On the third unique face is the hexagonal projection:
I created a movie (similar to those of Lisi) of the H4+H4/Phi 600 Cells that rotates the 2D H-V E8 face projection to the H-Z Pentagonal face (shown in the middle frames of the movie). This is an interesting rotation because this rotates through a square projection at around Pi/8.
Similar to the work of other 3D artists like Bathsheba and Wizzy in Second Life (SL), this object can be created in Real Life (RL) as 3D laser etched crystal or rezzed as a 3000+ prim object in SL.
The 3 (H,V,Z) basis projection vectors are:
While these are 8D vectors for the 600 cells – they could just as well be 4D (setting the last 4 to zero, as they are simply negatives of the first 4).
Orthogonally, the Zome like model from VisibLie_E8 shows: