Tetrahedrons and Pyramids

Monday, January 31, 2011 3:42 pm PST

A brief update as January draws to a close.  I’ve started investigating the structure of the Four-D Froth, which can be considered as filling three dimensions of “space” (in the shape of a regular tetrahedron–i.e. a pyramid with a triangular base) plus one dimension of “time”.  I haven’t quite got to the bottom of things yet, but it seems that at any given “time,” the “space” is subdivided into irregular tetrahedrons and pyramids whose vertices are the centers of the “bubbles”.  The edges of these shapes are the lines joining the centers of touching bubbles.  (The pyramids have four “base” vertices that lie in a plane, forming an irregular quadrilateral, plus an “apex” that lies outside this plane.)  As “time” increases, new bubbles form within these tetrahedrons and pyramids, or within small groups of adjacent tetrahedrons and pyramids, subdividing those spaces into ever smaller pieces.  I’m expecting the “energies” of this system to reside in the triangular and quadrilateral faces that separate these tetrahedrons and pyramids.  Pyramid power?

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Energy Transfers Revisited

Monday, January 17, 2011 8:37 pm PST

I recently revisited the problem of energy transfers in the two-dimensional (Flatrack) model and came up with a new paradigm: instead of residing in the “bubbles”, energy can be considered as residing in the lines which join their centers.  Each line’s energy is equal to one third of the combined area of the two triangles it separates.  (As new bubbles are formed, every line is eventually crossed by another; it thereupon ceases to exist and its energy becomes zero.)

Earlier today I plotted the distribution of energies in the 88,956 lines that exist in the Flatrack model at time 500.  Over 99% of them fell in the range between zero and 3*10^-5, as shown in the graph below.  (Note that area is measured in triangular rather than square units, so the total area in the whole model is 1.)

Distribution of energies at time 500 in the Flatrack

Distribution of energy quanta in the 2-dimensional model at time 500.

I had expected to find spikes in the energy distribution, but to my surprise it resembled this blackbody curve:

Cosmic Microwave Background Spectrum

Cosmic Microwave Background (CMB) spectrum plotted in waves per centimeter vs. intensity

What does this prove?  Well, I don’t claim it proves that my theories are correct, or even that I’m on the right track.  But it does seem to suggest that the same processes are at work in the Flatrack model and in the Universe.