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”.
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 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.)
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.
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.
A follow-up to the article that I posted earlier today. The system of hyperspheres I described contains elements that are centered at “times” corresponding to every natural number EXCEPT 2,5,8 and 10. So given today’s date, I couldn’t resist making making this blog entry now. And in doing so, I also noticed that today is the first anniversary of this blog.
Curiouser and curiouser! Of course I shouldn’t be surprised, since PQR Theory holds that everything is connected in ways that we cannot hope to fathom.
UPDATE August 10: Oops! just discovered a programming error. The missing numbers turn out to be 2,5,10 and 14, not as stated above. Oh well….
OK, so here’s an update on where I am with my thinking right now. It looks as though a three-dimensional universe can be defined numerically using quartets of coprime non-negative integers. The physical realization of this is that each quartet defines an element in a system of Ford hyperspheres over the rational points of a unit tetrahedron in a three dimensional space. The network of hyperspheres defines the space-time of this universe, and each hypersphere can be regarded as a “bubble” in its “quantum froth”.
A reminder: the elements in a system of Ford hyperspheres (or spheres, or circles) touch but they don’t overlap; each one’s diameter is a rational number; and the coordinates of its center are also rational numbers in an appropriate coordinate system consisting of three (or two, or one) spatial dimensions, plus one time dimension. Each point of this space with rational coordinates has exactly one hypersphere centered “above” it, at a “time” that corresponds to the inverse of its diameter.
As this spacetime structure is currently defined, it is not expanding but rather, it is becoming more detailed as time progresses. Of course, the inhabitants of such a universe (if they existed) would perceive it differently. Being creatures of this system, they would presumably regard themselves as staying the same size and their universe as expanding. However, the difference is merely one of scale and can be resolved by applying a scaling factor which varies over time.
Note that I’m saying that SOME universe can be defined in this way, not necessarily OUR universe. Many details remain to be worked out. The problems I want to tackle next are those of symmetry and scaling, then it’ll be time to look for evidence of matter and energy in this spacetime froth.