Building a universe from numbers

Monday, March 19, 2012 4:28 pm PST

This post is written in response to recent requests to explain the mathematical formulations I am investigating, and to show how a physical universe may be created from numbers.  Before I start, here are some warnings:

  1. This is a work in progress; everything here is subject to revision.
  2. I have attempted to define a universe consisting of space and time, with matter and energy governed by physical laws.  I do not yet claim this to be a model of OUR Universe, just of SOME possible universe.
  3. Some heavy mathematics ahead; proceed at your own risk.

The model is built up in stages as follows:

  • Stage 1 describes a universe with 1 dimension of space and 1 dimension of time.
  • Stage 2 describes a universe with 2 dimensions of space and 1 dimension of time.
  • Stage 3 describes a universe with 3 dimensions of space and 1 dimension of time.
  • Subsequent stages are possible but are not considered here, for reasons to be explained later.

At each stage, the model defines a single structure that can be formulated from a numerical, a geometrical or a physical perspective.  It is important to remember that for each stage, all three formulations are equivalent: the elements of the structure are the same from all three perspectives.

THE NUMERICAL FORMULATION

Stage 1: The model consists of all possible duads, where a duad is defined as an ordered pair of coprime non-negative integers.  (“Coprime” means that the two integers do not share a common divisor greater than 1.  All integers are considered to be divisors of zero.)

Notation: a duad is written as a:b>y where a,b ∈ℤ; a,b ≥0; gcd(a,b)=1; and y=m² (writing m=a+b).

Stage 2:  The model consists of all possible triads, where a triad is defined as three ordered non-negative integers a,b,c that have no common divisor greater than 1.  (Note: they need not be pairwise coprime, so for example 6, 10 and 15 would define a triad.)

Notation: a triad is written as a:b:c>y where a,b,c ∈ℤ; a,b,c ≥0; gcd(a,b,c)=1; and (writing m=a+b+c)  y=m²/gcd (a²+b²+ab, a²+c²+ac, b²+c²+bc)

Stage 3: The model consists of all possible tetrads, where a tetrad is defined as four ordered non-negative integers a,b,c,d that have no common divisor greater than 1 (though they need not be pairwise coprime).

Notation: a tetrad is written as a:b:c:d>y where a,b,c,d ∈ℤ; a,b,c,d ≥0; gcd (a,b,c,d)=1; and (writing m=a+b+c+d)
y=m²/gcd(m², a²+b²+c²+ab+ac+bc, a²+b²+d²+ab+ad+bd, a²+c²+d²+ac+ad+cd, b²+c²+d²+bc+bd+cd)

Notes:

  1. In each stage, the number y is always a positive integer, and is said to be the order of the duad, triad or tetrad.
  2. The term “duad” has been adopted in preference to “dyad,” which already has a different mathematical meaning.
  3. The Stage 2 model contains three copies of the Stage 1 model, obtained by setting a=0, b=0 and c=0 respectively.  Likewise, the Stage 3 model contains four copies of the Stage 2 model.
  4. The Stage 1 model also contains two elements (0:1>1 and 1:0>1) each of which can be considered as a copy of a trivial “Stage 0” model that consists of a single monad.
  5. The Stage 1 model is equivalent to the set of “Farey Fractions” (the rational numbers from 0 to 1, expressed in lowest terms).  The duad a:b>y corresponds to the fraction b/a+b.  This is easily seen because if (a, b) are coprime, then so are (b, a+b), and vice versa.

THE GEOMETRICAL FORMULATION

Stage 1 (“Fareyland”): The model consists of a set of Ford Circles that touch a straight line of unit length in a 2-dimensional Euclidean space.  Each duad a:b>y defines a circle of diameter 1/y, that touches the line at barycentric coordinates (a,b).

Note: the barycentric coordinates here are defined by reference to the line: the two endpoints of the line have barycentric coordinates (0,1) and (1,0) and the point (a,b) is located a fraction b/a+b of the way along the line. The Ford Circles all lie on the same side of the line.  One circle touches the line at every rational point along its length.  The circles touch each other but never overlap.  Here is a picture showing the Ford Circles defined by the duads of orders up to 144, together with their corresponding Farey Fractions.  The diameter of each circle is 1 divided by its order.

Ford Circles defined by duads

Click to enlarge: Ford Circles defined by duads of orders up to 144, and their corresponding Farey Fractions. (For space reasons, only the duads of orders up to 25 are labeled.)

Stage 2 (“The Flatrack”): The model consists of a set of Ford Spheres that touch an equilateral unit triangle in a 3-dimensional Euclidean space.  Each triad a:b:c>y defines a sphere of diameter 1/y, that touches the plane of the triangle at barycentric coordinates (a,b,c).  These barycentric coordinates are defined by reference to the triangle, the corners of which have barycentric coordinates (1,0,0), (0,1,0) and (0,0,1).  The Ford Spheres all lie on the same side of the triangle; one sphere touches it at each rational point (in barycentric coordinates); and the spheres touch but never overlap.  It’s rather hard to show a picture of this, but imagine a triangular rack on a pool table, holding three large balls that touch each other.  Then imagine a fourth ball, sitting on the table in the center, that is just large enough to touch all three balls.  Then imagine that you keep on filling the gaps between the balls and the table with more and more balls, each of which is just large enough to fill the gap it occupies.  That is a system of Ford Spheres.

Stage 3 (“The Four-D Froth”): The model consists of a set of (4-dimensional) Ford Hyperspheres that touch a regular unit tetrahedron in a 4-dimensional Euclidean space.  Each tetrad a:b:c:d>y defines a hypersphere of diameter 1/y, that touches the three-dimensional space of the tetrahedron at barycentric coordinates (a,b,c,d).  Again, these barycentric coordinates are defined by reference to the tetrahedron, the vertices of which have barycentric coordinates (1,0,0,0), (0,1,0,0), (0,0,1,0) and (0,0,0,1).  The Ford Hyperspheres all lie on the same “side” of the three-dimensional space (i.e. all in the same direction in the fourth dimension); one hypersphere touches it at each rational point (in barycentric coordinates); and the hyperspheres touch but never overlap.  Again, this system can be created by filling the gaps with the largest hyperspheres that will fit.

Note: in each stage of the model, the elements are referred to as “bubbles”.

THE PHYSICAL FORMULATION

Stage 1: We now consider Fareyland, instead of being two-dimensional, to represent one dimension of space (measured along the line) plus one dimension of time (measured downward towards the line).

Stage 2: We now consider the Flatrack, instead of being three-dimensional, to represent two dimensions of space (forming a triangle) plus one dimension of time (measured downward towards the triangle).

Stage 3: We now consider the Four-D Froth, instead of being four-dimensional, to represent three dimensions of space (forming a tetrahedron) plus one dimension of time (measured in the fourth dimension towards the tetrahedron).

In general, each stage n of the geometrical model now represents n dimensions of space plus one dimension of time (instead of n+1 dimensions of space).  In this formulation, each bubble represents a point of the n-dimensional space, and all the bubbles of order y are treated as coming into existence at “time” y.  This defines a discrete (rather than continuous) space, shaped as the unit simplex of dimension n, that is constantly subdividing.  In other words, the space only has a finite number of points in it at any given time, and it is important to remember that there is nothing between these points (yet).  This spacetime is therefore said to be granular.

This would give us a universe of a constant size, with a space that was continually becoming ever more finely grained.  However, we now apply a linear scaling factor of y to the space at time y.  This insures that all bubbles have diameter 1 at the time they come into existence, so that the granularity of space does not appear to change over time.  Instead, the space so defined appears to be continually expanding.

We now have a spacetime that exhibits at least some of the properties of a universe like ours.  In this blog post I showed how the Flatrack had quantities that were distributed like the energies in the Cosmic Microwave Background.  I have subsequently been studying how “cause and effect” can travel through the Four-D Froth, on the assumption that each bubble can “influence” only its higher-order (i.e., “later”) neighbors.  My results seem to indicate some sort of “cosmic speed limit,” but I’m not ready to say more at this stage.  And I previously observed some gravitational-type effects, so things are looking interesting.

Why limit the model to 3+1 dimensions?

Apart from the fact that the calculations become more complex and hard to visualize in higher dimensions, I have been reluctant to go beyond 3 dimensions of space plus 1 of time for another reason: embedding.  As I noted above when discussing the numerical formulation, each stage of the model contains embedded copies of the previous, lower-dimensional stages.  This is also true when we consider the geometrical and physical models.  The Four-D Froth is shaped like a tetrahedron, each face of which is an identical triangular copy of the Flatrack.  And each edge of the Flatrack (and of the Four-D Froth) is an identical copy of Fareyland.  If this is true up to three dimensions, we can expect it to be true in higher dimensions too.  This leads to the unsettling idea that our Universe may be just the boundary of a higher-dimensional universe, whose inhabitants could be watching us in the same way as we watch cartoon characters.

Of course, we could always specify that the numbers a,b,c and d in our models must always be non-zero, but this seems to be a bit of an artificial solution.

Another, less worrying problem, is that the models have a symmetry that corresponds to a reordering of the numbers a,b,c and d.  Thus Fareyland has a 2-way symmetry, the Flatrack has a 6-way symmetry, and the Four-D Froth has a 24-way symmetry.  That could mean that as I sit here typing these words, there are 23 other copies of myself (11 right-handed and 12 left-handed ones) doing exactly the same thing in other parts of the Universe.   This could lead to interesting phenomena at the boundary between adjoining mirror-image sections of the model.

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I’m beginning to see the light…

Monday, August 9, 2010 3:25 pm PST

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.


A coincidence of time?

Thursday, August 5, 2010 2:00 pm PST

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….


Ford spheres, reformulated

Saturday, July 31, 2010 9:12 pm PST

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 PST

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 PST

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…


Fareyland?

Wednesday, July 14, 2010 11:36 pm PST

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?