Universe-symmetry-mesh generation, jal
I am always amazed by the depth of information that has been acquired by so many people. What I know is just scratching the surface. Dig into the links and discover for yourself.
In my previous blogs, I start with the minimum number of units, 24 for a bounce, and it agrees with sphere packing in 3d. This produces the hex. or the cubic pattern.
I ask,
“What is the “jiggling” formula to get the densest packing?”
If you played with the model then you saw that as you expand the model from R to 2R to 4R, (hypersurfaces), that there was room for other spheres in between the 60 degrees.
How to make the space large enough to put a sphere into that space is the “jiggling” formula.
Does the mesh generation expand evenly at every step? Does it ever crash? Is the Inverse Square Law violated at any of the steps?
---------
I’ll re phrase by the following example
Imagine that you are at the beach on a hot summer day. The beach is crowded. There are vacant spots but you must find one of them and go to it.
You do not have all day to find and get a spot. You want to enjoy the day before the sun goes down and everyone leaves. (Inflation).
What is the information mechanism for finding the spot?
What are the rules for getting to that spot? Speed? Distances? Time? Path?
What are the possible patterns that could have been created by early arrivers to the beach or would be created by the late arrivals?
=======
Mathematician have a different way of “talking”. I say that the densest sphere packing is 12 spheres in a hex. pattern and then add the minimum scale, this results in having another 12 spheres for a total of 24 spheres, a Leech Lattice. Compare, the radius of a circumference of 12 units, with the radius of Leech Lattice. Does it make a difference if the circumference of the 12 units makes a smaller radius?
Therefore, when I ask, “At what expansion stage does minimum length no longer apply?”, is similar to saying, “ The universe started out as a Leech Lattice but there was a phase transition. Now, we can have spheres touch each other in 3d without minimum length.
The physicist and cosmologists have their own way of “talking” and they calculate that the bounce occurs at 24 units and then they do their calculation on the hypersurfaces, causal sets, after assuming that the “jiggling” was done without causing anything interesting and that all of the connections (symmetries) have been established. (mesh generation)
You cannot have a hypersurface without filling all the “voids”.
As usual, I expect that the “jiggling” formula has already been found by the mathematicians but that it has been overlooked/unrecognized. (by me)
Studying the “jiggling” (mesh generation) formula should/could explain the expansion and the confinement mechanism.
I’ll save a big spot, between R and 2R, for you at the beach.
I’ll see you all in my next blog entry, where we will look at ways of building a universe from what we learned so far.
--------
==========
Math references
http://en.wikipedia.org/wiki/Leech_lattice
Leech lattice
========
----------------
COMPARE
1) http://en.wikipedia.org/wiki/Voronoi_diagram
Voronoi diagram
---
2) http://en.wikipedia.org/wiki/Causal_sets
The causal sets programme is an approach to quantum gravity.
---
3) http://en.wikipedia.org/wiki/Mesh_generation
Mesh generation
---
4) http://en.wikipedia.org/wiki/Causal_..._triangulation
Causal dynamical triangulation
=======
-------------
http://research.microsoft.com/~cohn/
-------
http://front.math.ucdavis.edu/search...00&s=Abstracts
------
http://arxiv.org/abs/math/0408174
The densest lattice in twenty-four dimensions
Authors: Henry Cohn, Abhinav Kumar
(Submitted on 13 Aug 2004)
--------
http://arxiv.org/abs/math.NT/0701080
The Optimal Isodual Lattice Quantizer in Three Dimensions
J. H. Conway
N. J. A. Sloane
Jan 02 2006
The mean-centered cuboidal (or m.c.c.) lattice is known to be the optimal packing and
covering among all isodual three-dimensional lattices. In this note we show that it is also the best
quantizer. It thus joins the isodual lattices , A2 and (presumably) D4, E8 and the Leech lattice
in being simultaneously optimal with respect to all three criteria.
------------
http://arxiv.org/abs/0804.0637v1
A Complete Classification of Ternary Self-Dual Codes of Length 24
Authors: Masaaki Harada, Akihiro Munemasa
(Submitted on 4 Apr 2008)
----------
http://arxiv.org/abs/0805.2205
Mass formula for self-orthogonal codes over Z_{p^2}
Authors: Rowena A. L. Betty, Akihiro Munemasa
(Submitted on 15 May 2008)
A quaternary code is said to be even if the Euclidean weight of every codeword
is divisible by 8. Every quaternary even code is self-orthogonal.
----------
http://arxiv.org/abs/math/0405441
Local Covering Optimality of Lattices: Leech Lattice versus Root Lattice E8
Authors: Achill Schuermann, Frank Vallentin
(Submitted on 23 May 2004 (v1), last revised 10 Nov 2004 (this version, v4))
-------
http://arxiv.org/abs/0804.0036
Complexity and algorithms for computing Voronoi cells of lattices
Authors: Mathieu Dutour Sikiric, Achill Schuermann, Frank Vallentin
(Submitted on 31 Mar 2008 (v1), last revised 16 May 2008 (this version, v2))
---------
http://arxiv.org/abs/0805.2705
Three-dimensional Random Voronoi Tessellations: From Cubic Crystal Lattices to Poisson Point Processes
Authors: Valerio Lucarini
(Submitted on 18 May 2008)
---------
http://arxiv.org/abs/math/0506200
On packing spheres into containers (about Kepler's finite sphere packing problem)
Authors: Achill Schuermann
(Submitted on 10 Jun 2005 (v1), last revised 9 Sep 2006 (this version, v2))
--------
Cool Dynamic Sphere!
http://www.bugman123.com/Engineering/Unstructured.m1v
---
http://search.arxiv.org:8081/?query=...20670&byDate=1
mesh generation
Displaying hits 1 to 10 of 124.
--------
http://www-users.informatik.rwth-aac.../software.html
meshing programs
--------
A good intro to mesh generation
http://www.mit.edu/~persson/publications.html
http://www-math.mit.edu/~persson/mesh/
http://www-math.mit.edu/~persson/the...esis-color.pdf
Mesh Generation for Implicit Geometries
by
Per-Olof Persson
----------
========
physic references in the next post
In my previous blogs, I start with the minimum number of units, 24 for a bounce, and it agrees with sphere packing in 3d. This produces the hex. or the cubic pattern.
I ask,
“What is the “jiggling” formula to get the densest packing?”
If you played with the model then you saw that as you expand the model from R to 2R to 4R, (hypersurfaces), that there was room for other spheres in between the 60 degrees.
How to make the space large enough to put a sphere into that space is the “jiggling” formula.
Does the mesh generation expand evenly at every step? Does it ever crash? Is the Inverse Square Law violated at any of the steps?
---------
I’ll re phrase by the following example
Imagine that you are at the beach on a hot summer day. The beach is crowded. There are vacant spots but you must find one of them and go to it.
You do not have all day to find and get a spot. You want to enjoy the day before the sun goes down and everyone leaves. (Inflation).
What is the information mechanism for finding the spot?
What are the rules for getting to that spot? Speed? Distances? Time? Path?
What are the possible patterns that could have been created by early arrivers to the beach or would be created by the late arrivals?
=======
Mathematician have a different way of “talking”. I say that the densest sphere packing is 12 spheres in a hex. pattern and then add the minimum scale, this results in having another 12 spheres for a total of 24 spheres, a Leech Lattice. Compare, the radius of a circumference of 12 units, with the radius of Leech Lattice. Does it make a difference if the circumference of the 12 units makes a smaller radius?
Therefore, when I ask, “At what expansion stage does minimum length no longer apply?”, is similar to saying, “ The universe started out as a Leech Lattice but there was a phase transition. Now, we can have spheres touch each other in 3d without minimum length.
The physicist and cosmologists have their own way of “talking” and they calculate that the bounce occurs at 24 units and then they do their calculation on the hypersurfaces, causal sets, after assuming that the “jiggling” was done without causing anything interesting and that all of the connections (symmetries) have been established. (mesh generation)
You cannot have a hypersurface without filling all the “voids”.
As usual, I expect that the “jiggling” formula has already been found by the mathematicians but that it has been overlooked/unrecognized. (by me)
Studying the “jiggling” (mesh generation) formula should/could explain the expansion and the confinement mechanism.
I’ll save a big spot, between R and 2R, for you at the beach.
I’ll see you all in my next blog entry, where we will look at ways of building a universe from what we learned so far.
--------
==========
Math references
http://en.wikipedia.org/wiki/Leech_lattice
Leech lattice
========
----------------
COMPARE
1) http://en.wikipedia.org/wiki/Voronoi_diagram
Voronoi diagram
---
2) http://en.wikipedia.org/wiki/Causal_sets
The causal sets programme is an approach to quantum gravity.
---
3) http://en.wikipedia.org/wiki/Mesh_generation
Mesh generation
---
4) http://en.wikipedia.org/wiki/Causal_..._triangulation
Causal dynamical triangulation
=======
-------------
http://research.microsoft.com/~cohn/
-------
http://front.math.ucdavis.edu/search...00&s=Abstracts
------
http://arxiv.org/abs/math/0408174
The densest lattice in twenty-four dimensions
Authors: Henry Cohn, Abhinav Kumar
(Submitted on 13 Aug 2004)
--------
http://arxiv.org/abs/math.NT/0701080
The Optimal Isodual Lattice Quantizer in Three Dimensions
J. H. Conway
N. J. A. Sloane
Jan 02 2006
The mean-centered cuboidal (or m.c.c.) lattice is known to be the optimal packing and
covering among all isodual three-dimensional lattices. In this note we show that it is also the best
quantizer. It thus joins the isodual lattices , A2 and (presumably) D4, E8 and the Leech lattice
in being simultaneously optimal with respect to all three criteria.
------------
http://arxiv.org/abs/0804.0637v1
A Complete Classification of Ternary Self-Dual Codes of Length 24
Authors: Masaaki Harada, Akihiro Munemasa
(Submitted on 4 Apr 2008)
----------
http://arxiv.org/abs/0805.2205
Mass formula for self-orthogonal codes over Z_{p^2}
Authors: Rowena A. L. Betty, Akihiro Munemasa
(Submitted on 15 May 2008)
A quaternary code is said to be even if the Euclidean weight of every codeword
is divisible by 8. Every quaternary even code is self-orthogonal.
----------
http://arxiv.org/abs/math/0405441
Local Covering Optimality of Lattices: Leech Lattice versus Root Lattice E8
Authors: Achill Schuermann, Frank Vallentin
(Submitted on 23 May 2004 (v1), last revised 10 Nov 2004 (this version, v4))
-------
http://arxiv.org/abs/0804.0036
Complexity and algorithms for computing Voronoi cells of lattices
Authors: Mathieu Dutour Sikiric, Achill Schuermann, Frank Vallentin
(Submitted on 31 Mar 2008 (v1), last revised 16 May 2008 (this version, v2))
---------
http://arxiv.org/abs/0805.2705
Three-dimensional Random Voronoi Tessellations: From Cubic Crystal Lattices to Poisson Point Processes
Authors: Valerio Lucarini
(Submitted on 18 May 2008)
---------
http://arxiv.org/abs/math/0506200
On packing spheres into containers (about Kepler's finite sphere packing problem)
Authors: Achill Schuermann
(Submitted on 10 Jun 2005 (v1), last revised 9 Sep 2006 (this version, v2))
--------
Cool Dynamic Sphere!
http://www.bugman123.com/Engineering/Unstructured.m1v
---
http://search.arxiv.org:8081/?query=...20670&byDate=1
mesh generation
Displaying hits 1 to 10 of 124.
--------
http://www-users.informatik.rwth-aac.../software.html
meshing programs
--------
A good intro to mesh generation
http://www.mit.edu/~persson/publications.html
http://www-math.mit.edu/~persson/mesh/
http://www-math.mit.edu/~persson/the...esis-color.pdf
Mesh Generation for Implicit Geometries
by
Per-Olof Persson
----------
========
physic references in the next post
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