Ford spheres, reformulated

Saturday, July 31, 2010 9:12 pm PDT

As I mentioned in my post about Ford Spheres, the system can also be defined by partitioning the unit interval into three rational intervals. On looking into this formulation, I’ve found a way to simplify it. Rather than choosing three rational numbers that add to 1, we just need to choose three coprime positive integers a, b and c, say. We can then define the partition by taking a/(a+b+c), b/(a+b+c) and c/(a+b+c) as our three rational numbers that add to 1.  These can be envisaged as the coordinates of a point in an equilateral unit triangle.  The three coordinates represent the distance of the point from each of the three sides of the triangle (measuring each coordinate along a line parallel to one of the other two sides).  And if the time dimension is represented by a+b+c, the sum of our three integers, then we would be deriving space as a partition of time.  The most exciting thing about this definition is that it naturally extends to higher dimensions of space.  So watch this space (even if right now it is still only an area).


Ford Spheres

Thursday, July 22, 2010 11:18 am PDT

OK, so here’s the latest update. I’ve been working on extending Fareyland into higher dimensions. I’ve now found a two-dimensional version of Fareyland, based on a unit triangle (i.e. an equilateral triangle of side 1).  At each rational point on this triangle it is possible to place a sphere that touches the triangle at that point, with a radius of 1/2d, where d is the denominator of the square of the point’s distance from any corner of the triangle. (If different corners give different values of d for a particular point, then take their lowest common multiple.) Amazingly, none of these spheres overlap but they just touch their neighbors nicely.  (This system is known as the Ford Spheres.)

This arrangement can also be generated by placing three spheres of unit diameter, one on each corner of the triangle, so that they touch each other, and then filling in the gaps between the spheres by placing smaller spheres between every three touching spheres. Keep doing this over and over and you eventually generate every sphere described above. 

I haven’t worked out the details yet, but I’m fairly confident that this system can also be defined by partitioning the unit interval into three sections of rational length, or equivalently, by choosing three rational numbers that add to 1, which in turn is equivalent to choosing two rational numbers that add to not more than 1 (since they will define the third).  So here we have a relationship between integers (OK, rational numbers, which are just pairs of integers) and a two-dimensional space.  I’m still working out the details of the time dimension but it looks as though each sphere represents a “bubble” of spacetime.  If this works, it will give me a system with two dimensions of space and one dimension of time, and I can start looking for the physical laws that govern this system.  One interesting facet of this approach is that the space it defines does not expand, but instead becomes more detailed over time.  (Of course, if you were inside the system, these phenomena would both look pretty much the same.)  And of course, as we had in Fareyland, the edges wrap around so that each corner of the triangle is actually the same point.  This gives a symmetrical space like the one I was speculating about here. (This particular space has a 6-way symmetry.)

A couple of notes here: I haven’t proven this mathematically yet, but I have verified it by computer for spheres down to a radius of 1/800 (I found 19,072 of them).  And to clarify what I mean by “rational point”, I should add that this is a point which has rational coordinates in a system where the x- and y-axes are at 60° to each other, instead of the more usual 90°.  (I may find it convenient to change the way that the coordinates are measured when I start to consider partitions.) 

The most encouraging thing is that as I investigate this system, I expect it will become fairly clear how to extend it into three spatial dimensions.  Stay posted for more details as I find them.

Could our Universe be symmetric?

Wednesday, July 21, 2010 2:52 pm PDT

One cube or six?

This Rubik's cube, reflected between two angled mirrors, shows six different aspects.

Here’s a thought that I’m examining. If we were living in a symmetric universe, would we know it?

My idea is that perhaps the Universe is not only deterministic, but symmetric like the line-land I described in my post about Fareyland. Now Fareyland had only one dimension and a two-way symmetry, but when we extend this idea from one up to three dimensions, it gives us a universe with 24-way symmetry. In other words, what if our Universe is composed of 24 identical sectors (implying that there are 23 other copies of ourselves “out there”)? Would we be able to detect this?

Well, unless we were near to the middle of our sector, there would probably be vastly different distances between ourselves and each of the other 23 sectors, so that when we looked out into space we would observe them at greatly varying stages in their development (as well as from different angles). Would we notice the symmetry?

Have you ever eaten in a restaurant which has a wall of mirrors? At first glance (especially if you are not near the mirrors) the room looks to be twice as big as it really is. Or have you ever been in one of those mirror-mazes that seem to be much bigger than they really are? You may not notice the mirrors until you see yourself reflected in one. And if the mirrors were to reflect, not what is happening now, but what happened at some time in the past (as they would if they were zillions of light years away) would we notice them at all?

Food for thought…


Wednesday, July 14, 2010 11:36 pm PDT

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.

Ford Circles

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?