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.)
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.
