Tag Archives: 4D
Universal Time’s Arrow as a Broken Symmetry of an 8D Crystallographic E8 folding to self dual H4+H4φ 4D QuasiCrystal spacetime
Animations of Wolfram’s Cellular Automaton rule 224 in a 3D version of Conway’s Game of Life.
While this does not yet incorporate the E8 folding to 4D H4+H4φ construct, the notion of applying it, as the title suggests, to a fundamental particle physics simulation of a “Quantum Computational Universe” or an emergent (non)crystallographic genetic DNA (aka. Life) is an interesting thought…
Watch a few notional movies, some in left-right stereo 3D. Best viewed in HD mode.
These are snapshots of all the different initial conditions on the rule 224. Each set of 25 has a different object style and/or color gradient selection.
Playing around visualizing non-crystallographic DNA/RNA
I’ve got interaction between the DNA/RNA protein visualizations (e.g. “1F8V- PARIACOTO VIRUS REVEALS A DODECAHEDRAL CAGE OF DUPLEX RNA) with the VisibLie_E8 projections of crystallographic E8 to non-crystallographic H4 (and to dodecahedral H3 in 3 dimensions, of course).
These pics are a simple (naive) merge of the D6 projected using the E8 to H4 folding matrix and the Protein DB at http://www.rcsb.org/ for 1F8V).
3D particle interactions on H4 and H4φ 24-cell and snub-24-cells
D6 projected to 3D using the E8 to H4 Folding Matrix
This blogs.ams.org/visualinsight article is an American Mathematical Society post based on some of my work. I’ve made an animation of the D6 polytope with 60 vertices and 480 edges of 6D length Sqrt(2). It is similar to the one in the article by Greg Egan. This version was made from my VisibLie E8 demonstration software.
Best viewed in HD mode.
The process for making this using the .nb version of VisibLie E8 ToE demonstration:
1) Side menu selections:
- slide “inches” to 4 (lowers the px resolution)
- select “3D”
- set bckGrnd “b” (black)
- uncheck the shwAxes box
- check the physics box (to give it interesting vertex shapes/colors)
- change the fileExt to “.avi” and check the fileOut box (to save the .avi)
- check the artPrint box (to give it cylinder-like edges)
2) Go to the #9) HMI nD Human Interface pane:
2a) Top menu selections:
- set pScale to .08 (to increase the vertex shape sizes)
2b) Inside the pane selections:
- set “eRadius” to .01 (for thicker edge lines)
3) Go to the #2) Dynkin pane and interactively create (or select from the drop down menu) the D6 diagram.
BTW – this actually selects the subset of vertices from the E8 polytope that are in D6.
4) Go to the #3) E8 Lie Algebra pane:
4a) Top menu selections:
- E8->H4 projection
- check the edges box
- “spin” animation
- 30 animation “steps”
or simply upload this file or copy / paste the following into a file with extension of “.m” and select it from the “input file” button on the #9 HMI nD Human Interface pane.
(* This is an auto generated list from e8Flyer.nb *)
In the next version I will just make this another “MetaFavorite”, so it will be in the dropdown menu in all versions.
E8 to H4 Folding Matrix work cited in the American Mathematical Society Blog
This blogs.ams.org/visualinsight article is an American Mathematical Society post based on some of my work.
This animated Gif from the article is from Greg Egan
Chaotic Attractor Animations
Fun with 8D rotations of E8
This is a little clip containing three different 8 dimensional rotations of E8. While it may seem like a simple rotation of an object in 3-space, if you look carefully (especially the first rotation), you can see sets of vertices moving in different directions.
The object (the 240 vertices of E8 along with 1220 of 6720 edges of 8D length Sqrt) is not “moving”. It is the camera that is moving in an 8 dimensional space.
Best when viewed in HD!
The previous 3D animations here and here are created by projecting the 8 dimensional object of E8 into 3 and simply rotating (spinning) that object around in 3-space. I like to call those animations 4D, since animation visualizes changes over time – a 4D space-time object if you will…
Math only version of E8 to H4 projection
E8 to 3D projection using my H4 folding matrix
Best viewed in HD mode…
Last year I showed that the Dynkin diagram of the Lie Algebra, Group and Lattice of E8 is related to the Coxeter-Dynkin of the 600 Cell of H4 through a folding of their diagrams:
I had found the E8 to H4 folding matrix several years ago after reading several papers by Koca and another paper by Dechant, on the topic. The matrix I found is:
Notice that C600=Transpose[C600] and the Quaternion–Octonion like structure (ala. Cayley-Dickson) within the folding matrix. Only the first 4 rows are needed for folding, but I use the 8×8 matrix to rotate 8D vectors easily.
The following x,y,z vectors project E8 to its Petrie projection on one face (or 2 of 6 cubic faces, which are the same).
On another face are the orthonormal H4 600 cell and the concentric H4/Phi. There are 6720 edges with 8D Norm’d length of Sqrt, but I only show 1220 in order to prevent obscuring the nodes.
These values were calculated using the dot product of the Inverse[4*C600] with the following H4 Petrie projection vertices (aka. the Van Oss projection), where I’ve added the z vector for the proper 3D projection. Notice I am still using 8D basis vectors with the last 4 zero, as this maintains scaling due to the left-right symmetries in C600).
The Split Real Even (SRE) E8 vertices are generated from the root system for the E8 Dynkin diagram and its resulting Cartan matrix:
This is input from my Mathematica “VisibLie” application. It is shown with the assigned physics particles that make up the simple roots matrix entries:
Please note, the Cartan matrix can also be generated by srE8.Transpose[srE8]
It takes Transpose[srE8] and applies the dot product against the 120 positive and 120 negative algebra roots generated by the Mathematica “SuperLie” package (shown along with its Hasse diagram, which I generate in the full version of my Mathematica notebook):
Interestingly, E8 in addition to containing the 8D structures D8 and BC8 and the 4D Polychora, contains the 7D E7, as well as the 6D structures of the Hexeract or 6 Cube. I also showed several years ago that this projects down to the 3D as the Rhombic Triacontahedron. The Rhombic Triacontahedron (of Quasicrystal fame) contains the Platonic Solids including the Icosahedron and Dodecahedron! This is shown to be done through folding of D6 to H3 using rows 2 through 4 of the E8 to H4 folding matrix!!