Tetrahedrons and Pyramids

Monday, January 31, 2011 3:42 pm PDT

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?


Energy Transfers Revisited

Monday, January 17, 2011 8:37 pm PDT

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.

2011 will be a Löschian year

Friday, December 31, 2010 5:56 pm PDT

I’ve been working on other things for a while, but as the year draws to a close I’ve finally found the time to do more work on my PQR Theory.  Specifically, I’ve been looking at the two-dimensional (Ford Spheres) model, which I call the Flatrack because it vaguely resembles a rack of pool balls, but with the bottom surface flattened off against the table.

The “sizes” of the spheres (by which I mean the reciprocal of their diameters) in the Flatrack are the Löschian (or Loeschian) numbers, which are numbers of the form x²+xy+y², where x and y are integers.  The first few such numbers are 1, 3, 4, 7, 9, 12 and 13.   2011 is a Löschian year, as 2011=10² + 10×39 + 39².  (The last Löschian year was 2007 and the next will be 2017.)

When one looks at a group of four Ford Spheres that touch each other in the Flatrack, their sizes are all Löschian numbers that satisfy the Soddy Equation: (a+b+c+d)² = 3(a²+b²+c²+d²).  (Note that this is a special case of the Soddy 5-sphere equation, where one of the spheres is the plane of the pool table, which is geometrically equivalent to a sphere of infinite radius, or of “size” zero.)  I call a set of four positive integers {a,b,c,d} that satisfy this equation a Soddy Quartet.

Interestingly, it seems that all Soddy Quartets consist of 4 Löschian numbers (or of 4 Löschian numbers times a common multiplier).  And every Soddy Quartet that does not have a common factor appears to be represented in the Flatrack by exactly one group of four mutually touching Ford Spheres (together with its symmetrical reflections).  Interesting!  Whatever next?

In the meantime, I would like to wish my reader(s) a Happy New Löschian Year.

Energy transfers – latest update

Tuesday, August 31, 2010 11:25 pm PDT

Well, August has been an exciting month and as it draws to a close I think I finally have a handle on the two-dimensional energy transfer formula.  Looks like each bubble is formed with energy equal to one-third of the area of its surrounding triangle.  It inherits this energy from the three corner bubbles based upon the areas of the three subtriangles formed by joining the center bubble to the corner bubbles.  Specifically, each corner bubble contributes energy equivalent to one sixth of the area of the two subtriangles that meet at that corner.  Still working out the details, including what happens when a bubble is formed within a quadrangle instead of a triangle, but this looks promising.  Then it will be on to the three-dimensional model, which will be based on four-dimensional bubbles (i.e., Ford hyperspheres).  I think I’ll call this model “FourD Froth”.

Energy transfers in the 2-dimensional model–an update

Wednesday, August 18, 2010 10:45 am PDT

For the last week or so I’ve been working on trying to define the formula for energy transfers in the two-dimensional model.  Haven’t quite got to the bottom of it yet, but it looks as if each bubble is created with an energy that depends on the area of the triangle defined by its three parents.  (Some bubbles have 4 parents, which define two adjacent triangles.  For these cases, use the sum of the 2 areas.)  I think the energy is equal to half the area, though it could perhaps be 1/3 or even 100%.  The energy is “inherited” from the parents but it doesn’t seem to be a straight 1/3:1/3:1/3 split.  I got myself tied up in knots (mentally and abdominally) trying to figure this out, and I have some other business to take care of, so I’m giving it a rest until next week.  However, it’s worth mentioning that the 1-dimensional Fareyland model is embedded in the two dimensional model, which should help me figure this out.

I’m beginning to see the light…

Monday, August 9, 2010 3:25 pm PDT

I haven’t defined a metric yet, but when I applied scaling to the two-dimensional model and watched it running on my computer, I noticed something interesting.  I was looking to see how the “sphere of influence” of any particular bubble would spread out over time.  Remember, each bubble in the system touches several larger ones and many smaller ones.  The larger ones (which are formed at earlier “times”) can be considered as its “parents” and the smaller ones (which go on being formed for ever) can be considered as its “children”.  The “descendants” of a bubble are its children, and its children’s children, and so on, and if you consider these  to be “influenced” by it, this defines the law of cause and effect. 

In the two-dimensional model these spheres of influence appear to grow in a mostly circular fashion (with distortions that seem to represent a “gravitational” effect) and I was also interested to notice what seem to be “rays of influence” spreading out from each parent bubble.  I wondered: could these represent photons?

Accordingly I went back to my one-dimensional Fareyland model (which, unlike the two-dimensional model, already has a working definition of energy) to see how much energy is transferred from parents to children when a new bubble is created.  I have marked some of the energy transfers with arrows in this enlarged diagram of part of the system.  (The numbers on the arrows represent the energy transfers before applying scaling.)

Ford Circles (detail)

Energy transfers in Fareyland

The physical interpretation of this is that the bubble at 3/5 is formed at time 5 and receives energy of 1/30 from its left parent (1/2) and 1/20 from its right parent (2/3).  Its total energy is thus 1/30+1/20=1/12.  At time 7 it passes energy of 1/28 to its new child (4/7) on the left; at time 8 it passes energy of 1/48 of energy to its new child (5/8) on the right.  (By this point its energy has been reduced to 3/112).  The children in turn pass their energy on to their children, and so on.  Note that at each point where two bubbles touch, there is an energy transfer from the larger to the smaller, although I have not marked all the arrows.

Now for the really cool bit of today’s message.  When we apply scaling (multiplying everything that happens at time t by a factor of t) all the left-pointing arrows have a value of 1/4 and all the right-pointing arrows have a value of  1/6.  You could regard the arrows as representing the trajectories of two photons (with scaled energies of 1/4 and 1/6) that cross each other.  So it looks as though some things can get past each other in Fareyland, after all.

Meanwhile the bubble at 3/5 continues to give off energy as it spawns children for the remainder of time.  However, due to the effect of scaling, its residual energy grows with the passage of time, so it never runs out: its scaled energy approaches, but always stays above, 1/5.  What does this represent?  Could it be a particle of stable matter, or of vacuum energy?  I guess I’ll have to examine what happens in 2 and 3 dimensions in order to learn more.

Ford Hyperspheres – an update

Thursday, August 5, 2010 4:00 pm PDT

Just a brief update.  The first part of the scaling problem looks simple.  It’s apparent that the smaller bubbles of the quantum froth come later in “time” than bigger ones.  For example, the bubbles which “exist” at “time” t have diameter 1/t in all three spatial dimensions as well as in the temporal one.  (Note that this formulation of “time” is an ordering of events, but it may not be the same as what clocks measure.)

For the universe to be perceived (from within) as at least relatively stable, the bubbles shouldn’t appear to be shrinking or growing. This implies using a measure of space that grows in proportion to t. (That way, Planck’s constant should actually be a constant.) In other words, at “time” t the sides of our tetrahedron (which represent the “circumference” of this universe) will have apparent length t, i.e. t units of the “current fundamental distance” of the universe of Ford hyperspheres (corresponding to the diameter of each individual hypersphere at time t).

My next task: to see if a Lorentzian or Minkowski metric naturally exists in this space.  In the mean time, I’ll leave you with the following thought: Ford Hyperspheres are just like Nissan Cubes, only smaller, with no corners, and with an added dimension.  And, of course, they’re USA-made.