TheoryOfEverything

Dual 600 Cell (H4+H4/Phi) Petrie Projection to Square,Penta & Hexagon

admin — Mon, 28 Jun 2010 14:10:46 +0000

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:

600 Cell E8 Petrie Projection

On another face it has a pentagonal projection:

Cell 600 dual pentagonal projection face

On the third unique face is the hexagonal projection:

Dual 600 cell Hexagonal face

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.

Cell 600 square projection

A lower quality YouTube is here: 600 Cells E8 to Pentagonal. Download and play the higher quality .avi: cell600E8pent

I also created a 3D perspective spin of that projection (two sets of mutually orthogonal cubic faces). On YouTube:600 Cells 3D Spin. Download the higher quality .avi: cell6003Dspin2

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:

600 Cells H, V, Z Projection Basis Vectors

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).

Animated rotations of the n=4 thru 7 n-cubes projected into 3D

Greg — Fri, 19 Mar 2010 20:52:12 +0000

E8 in the ScienceSim

Greg — Sat, 30 Jan 2010 20:54:01 +0000

New feature in e8Flyer – E8 Polytope edge overlap counts

Greg — Sat, 16 Jan 2010 19:58:30 +0000

A rendering in the OSGrid virtual world of 3 vertices with their Sqrt(2) edges colored based on their overlap count.

New Virtual World for Theory Of Everything

Greg — Wed, 30 Dec 2009 19:04:40 +0000

If you want to see E8 in virtual reality, check out the “Theory Of Everything” region at OSGrid.org

Of course, you will need an SL viewer and an avatar to get “in-world”.

greg

E8 Art Prints now available on t-shirts

Greg — Sun, 06 Dec 2009 23:02:17 +0000

Yes – that’s right, a possible Theory of Everything on a t-shirt

TheoryOfEverything.spreadshirt.com

Another Crystal

Greg — Thu, 26 Nov 2009 00:34:49 +0000

E8ArtPrint0008 Laser Etched in Crystal

Another E8 Jewel Cut Cube (4″x4″x4″)

Greg — Sat, 14 Nov 2009 18:16:53 +0000

Stereo Lithography Models

Greg — Sun, 18 Oct 2009 23:36:27 +0000

Just thought I would show some recent development in generating real objects from the E8 Mathematics.