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All posts by jgmoxness
Messier Marathon 2021
The idea is to observe the 110 astronomical Messier objects which are visible to some of the Northern Hemisphere in March/April. The best result is to do it all in one night, which is difficult unless everything works out seamlessly (110 over 11 dark hours leaves only 6 minutes per image, so it isn’t so much about quality but speed).
So I decided to try and set up my astro gear to automate the process of taking 110 Messier objects in one night.
In testing it the evening of March 2, it was going really well so I decided to pull an unplanned all-night’er and got 100 of the 110 (with the missing 10 not being visible to my scope before sunrise due to the obstruction of my house, as well as it being a bit early in the Messier season).
Click here for a video compilation of the shots I took (sorted by their Messier number each using a 2 minute monochrome exposure):
5-color graph solutions to the Hadwiger–Nelson problem
These are a simple animated visualizations I created while studying the problem. They contain a sequence of 2s frames starting at 130 vertices and incrementing 62. The red numbered text are the first few starting vertex numbers showing the hexagonal structure of the construction. On the left is a list of sorted and color coded vertices by the number of edges on a vertex for that frame’s shown vertices (it also lists in the right column the number of vertices with that number of edges & color). The solution data was obtained from: https://github.com/marijnheule/CNP-SAT
Here is a link to a Mathematica Notebook (50Mb) that has interactive ListAnimated graphs with vertex data (vertex number, symbolic and numeric coordinates, number of edges on the vertex in the frame) in Tooltip on mouseover.
3D Visualization of the rays of E6 & E7 in Kochen-specker theory by Ruuge & Waegell/Aravind
In several papers on BKS proofs, Arthur Ruuge’s “Exceptional and Non-Crystallograpic Root Systems and the Kochen-Specker Theorem” https://arxiv.org/abs/0906.2696v1 and Mordecai Waegell & P.K. Aravind’s “Parity proofs of the Kochen-Specker theorem based on the Lie algebra E8” https://arxiv.org/abs/1502.04350v2, in addition to E8, E6 and E7 is studied. Using the visualization developed for my recent paper and prior papers, I present here the related visualizations for E6 and E7 as discussed in those papers.
My Latest paper published on Vixra – 3D Polytope Hulls of E8 4_21, 2_41, and 1_42
https://vixra.org/pdf/2005.0200v1.pdf
or also available directly from this website:
https://theoryofeverything.org/TOE/JGM/3D_Polytope_Hulls_of_E8-421-241-142.pdf
Using rows 2 through 4 of a unimodular 8x8 rotation matrix, the vertices of E8 421, 241, and 142 are projected to 3D and then gathered & tallied into groups by the norm of their projected locations. The resulting Platonic and Archimedean solid 3D structures are then used to study E8’s relationship to other research areas, such as sphere packings in Grassmannian spaces, using E8 Eisenstein Theta Series in recent proofs for optimal 8D and 24D sphere packings, nested lattices, and quantum basis critical parity proofs of the Bell-Kochen-Specker (BKS) theorem.
A few new Figures from the paper.
3rd largest hull of the 74 hulls in 142
Each 3D projection shown lists the projection name, the nu-
meric basis vectors used, and the 421 & 142 overlap color coded
vertex groups, and the projection with vertices (larger) &
6720 edges and the 142 vertices (smaller)
with vertex count in each hull and increasing opacity and
varied surface colors.
a) 24 individual concentric hulls
b) In groups of 8 hulls
with vertex count in each hull and increasing opacity and
varied surface colors.
a) 74 individual concentric hulls
b) In groups of 8 hulls
3D projection with vertex counts color coded by overlaps
a) 54 vertex (42 unique) 421=241 icosidodecahedron (30 yel-
low) & two overlapping icosahedrons (12 red) scaled 1.051
b) 100 vertex (80 unique) 142 non-uniform rhombicosidodeca-
hedron (60 yellow) & two overlapping dodecahedrons (20 red)
scaled 1.0092
c) 154 vertex (122 unique) combination of a & b
d) 208 vertex (122 unique) combination same as c with color
coded vertex counts for both 421 & 241
Note: The internal numbers of the image are the 8 axis (pro-
jection basis vectors).
Now for a few new visualizations that are not in the paper…
3D visualization of E8 1_42 polytope
This is what I expect to be the first ever 3D visualization of the E8 1_42 polytope with 17280 vertices showing concentric hulls of Platonic solid related structures!
of 8 hull listed above. Notice this is a combination of two overlapped dodecahedrons (40) and a Nonuniform Rhombicosidodecahedron (60).
For the sake of completeness in visualization, see below for various projections to 2D. Click these links for a higher resolution PNG or the SVG version.
Nested Lattices of E8 in Complex Projective 4-Space
I read an interesting article about a pattern discovered by Warren D. Smith (discussed at length here):
“The sum of the first three terms in the Eisenstein E_4(q) Series Integers of the Theta series of the E8 lattice is a perfect fourth power: 1 + 240 + 2160 = 2401 = 7^4”
So I decided to visualize the 2401=1+240+2160 vertex patterns of E8 using my Mathematica codebased toolset based on some previous work I put on my Wikipedia talk page.
The image below represents various projections showing 6720 edges of the 240 E8 vertices, plus a black vertex at the origin, and the 2160 Witting Polytope E8 2 _ 41 vertices using the same projection basis (listed at the top of each image along with the color coded vertex overlaps). Click these links for a higher resolution PNG or the SVG version.
Some of the particular projections of the Witting Polytope may need 8D rotations applied to the basis vectors to find better symmetries with the Gosset, but this is a start using my standard set of projections.
The 240 vertices of the Gosset Polytope are generated using various permutations:
(* E8 4_21 vertices *)
e8421 = Union@Join[
Eperms8@{1, 1, 1, 1, 1, 1, 1, 1}/2,
perms8@{1, 1, 0, 0, 0, 0, 0, 0}];
The 2160 vertices of the Witting Polytope are generated using various permutations:
(* E8 2_41 vertices *)
e8241=Union@Join[
perms8[{1,0,0,0,0,0,0,0}4],
perms8[{1,1,1,1,0,0,0,0}2],
Eperms8[({2,0,0,0,0,0,0,0}+1)]]/4;
Another view shows just the 2160 Witting Polytope vertices. Click these links for a higher resolution PNG or the SVG version.
Another great source of visualizations on E8 and this Witting Polytope is here.
Now visualizing in 3D the structure in 3D using rows 2-4 of the E8->H4 folding matrix, we get:
Visualizing the concentric hulls of E8 with the 24-cells of H4 & H4Φ
I’ve had a number of related posts on this topic, but I wanted to present a few new pictures and PDF documents that combine the 7 concentric hulls forming Platonic solid related sets of E8 vertices projected to 3D with vertices of five pairs of 24-cell objects.
These break down the E8 structure into familiar 3D objects, such as the icosahedron with its dual the dodecahedron, the icosidodecahdron, and the 16-cell with its dual the 8-cell (aka. Tesseract) combined to create the self-dual 24-cell.
You can see that 6 of the yellow vertices connect to the 24-cell. Rotating that 24-cell four times in 3-space by π/5 gives the connections to the rest of the vertices in H4Φ and completes the 30 vertex icosidodecahedron . The same is true for the 120 vertices of H4 using the corresponding 24-cell in H4.
b) Snub 24-cell highlighting four π/5 rotations of the 24-cell (black) in red, green, blue, yellow.
The following paper hulls-24cells-combined.pdf (13MB) & interactive Mathematica Notebook hulls-24cells-combined.nb (34MB) contains a comprehensive set of images that show the contents of the 7 concentric hulls of Platonic solid related shapes as well as their integration with one of the rotations of 24-cells in H4 and H4Φ in E8 (the same one used in the image above).
In the E8 Petrie projection, every hull (each with 2 or 4 overlapping vertices) pair into left/right patterns. See the set of four icosahedrons that occupy the 3rd hull in both 2D and 3D (the yellow edge sets belongs to the H4Φ and the blue sets belong to H4) :
The pair of dodecahedrons that occupy the 5th hull in both 2D and 3D (the yellow edge set belongs to the H4Φ and the blue belong to H4) :
Below is an animation that cycles through the sequence of 2D Petrie Projections of the pairs of hulls. One cycle shows each frame individually and the other builds the E8 Petrie from the previous 2D hull.
Below are images of the left (H4) and right (H4Φ) 2D Petrie projections of the hulls (which are defined by the Norm’s of 3D projected vertices using the E8->H4 folding matrix rows 2-4 as basis vectors). The two 30 vertex 4 & 7 Icosidodecahedron hulls and four 0th hull (points) are omitted leaving the gaps in the diagram. When these gap vertices are combined with the 48 3rd hull Icosahedrons (above), they make up the 112 integer D8 group assigned to the Bosons (48) and 2nd generation Fermions (64) in the physics model described below.
The combined set of hulls projected using the rows of the folding matrix as basis vectors is shown below from previous work:
This paper hull-list-3b.pdf (4MB) & interactive Mathematica Notebook hull-list-3b.nb (20MB) contains the visualizations and detail E8 vertices associated with each Platonic solid related concentric hulls. This paper 24cell-list-3b.pdf (6MB) & interactive Mathematica Notebook 24cell-list-3b.nb (20MB) contains the visualizations and detail for the five 24-cells in each H4 & H4Φ. Both of these papers also have detail information about its assigned physics particle based on a modified A.G. Lisi model (shown below). This paper describes these particle assignment symmetries in more detail.
Latest Paper – Unimodular rotation of E8 to H4 600-cells
Please see my latest paper that describes some advances in understanding the E8 to H4 rotation matrix
https://theoryofeverything.org/TOE/JGM/Unimodular-Rotation-of-E8-to-H4.pdf
Abstract: We introduce a unimodular Determinant=1 8×8 rotation matrix to produce four 4 dimensional copies of H4 600-cells from the 240 vertices of the Split Real Even E8 Lie group. Unimodularity in the rotation matrix provides for the preservation of the 8 dimensional volume after rotation, which is useful in the application of the matrix in various fields, from theoretical particle physics to 3D visualization algorithm optimization.
Stereoscopic 3D Interactive Solar system simulation (shoemaker-Levy-9)
Please try: SolarSystem.cdf
It is a Mathematica 11/12 Computable Document Format (CDF) web interactive app that has been purpose built for visualizing Solar System Orbital Mechanics
It requires local installation of the free Mathematica CDF plugin.
This now includes the 1998 OR2 “Planet Killer” that will pass very near Earth in April 2020 (when selecting the Asteroid vs. Comet).
Now with improved UI, better planet scaling, and Anaglyph viz.