Here’s what I’ve been working on lately. I’ve been trying to show how a universe can be defined by numbers. I started by attempting to define a universe with one dimension of space and one of time. I did this using the so-called “Farey Fractions”, which are simply the rational numbers between 0 and 1.
We can generate these numbers in stages as follows:
The first generation is just the numbers 0/1 and 1/1, our endpoints. There is nothing between them yet.
The second generation adds all the fractions with 2 on the bottom, so we now include ½.
Next, the third generation includes the fractions with 3 on the bottom, which gives us 1/3 and 2/3.
The fourth generation also includes 1/4 and 3/4. (We don’t list 2/4 because that’s the same as ½ which we already got in the 2nd generation.)
We keep on doing this and by the 6th generation we have
0/1, 1/6, 1/5, 1/4, 1/3, 2/5, ½, 3/5, 2/3, 3/4, 4/5, 5/6, 1/1.
Note that this series displays a two-way symmetry with the number ½ in the center. Everything that happens on the left of the center is reflected on the right. Note also that each fraction first appears in the generation that corresponds to its denominator (the bottom number in the fraction). More interestingly, it is always “generated” by the fractions on either side, in the sense that if at some generation a/b and c/d are adjacent fractions in the series, then there will appear between them (at generation b+d) the fraction (a+c)/(b+d).
Does this system of generation have a physical interpretation? Well, you could say that each fraction a/b represents a “particle” (of matter or energy) that comes into existence at “time” (i.e., generation) b. We consider the two endpoints to represent a single particle, so making a circular universe that “wraps around” from 1/1 to 0/1. (Let’s call this particle “G,” after the “GO” square on a Monopoly board.)
We define the mass (or energy content) of a particle at any given time as equal to half the difference between the fractions that define its left and right hand neighbors. This definition makes the total mass/energy of the system equal to 1 at all times, so mass/energy is conserved. Thus, for example, at time 6 the particle 1/4 will have mass/energy of 1/15th (=1/2*(1/3-1/5)), while at time 7 its mass/energy would be 3/70ths (=1/2*(2/7-1/5)). (The particle G has mass 1/t at time t.)
If at some stage two particles a/b and c/d are neighbors, then these particles will interact at time b+d to create a new particle between them. The mass/energy content of this new particle will be ½|c/d-a/b| at the time it is created. Of course its “parent” particles will between them have lost an equivalent amount of mass/energy.
We can also define a distance function by setting the distance between two particles at time t as the number of steps from one to the other (going via G if this is the shorter way round). This gives us an expanding universe where the distance between G and ½ represents the diameter of the universe. Of course this is only a one-dimensional universe so nothing can get past anything else, and as a result it’s not very interesting.
But here’s something that I think is really cool: if you imagine these fractions strung out along the rational line between 0 and 1, you can draw circles touching the line at every rational point. At each point (represented by the fraction p/q) draw a circle of radius 1/2q². The amazing thing is that these “Ford Circles” never overlap, but each point’s circle just touches its neighbors’ circles. For example, the circle of radius 1/50 that belongs to the point 2/5 will touch the circle of radius 1/18 that belongs to the point 1/3, as well as the circle of radius 1/8 that belongs to the point ½, the circle of radius 1/98 belonging to 3/7, the circle of radius 1/128 belonging to 3/8, the circle of radius 1/288 belonging to 5/12 and so on–see the diagram below.
Right now I’m working on extending this idea into two dimensions, then (hopefully) I’ll be able to take it to the third dimension and see how similar a universe this gives to our own. There are some exciting things emerging–in two dimensions I have already noticed the emergence of structures that look and behave like black holes. Watch this space for more details.
In the meantime I will leave you to ponder the philosophical problem associated with Fareyland: did it exist before I defined it?
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