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:
- This is a work in progress; everything here is subject to revision.
- 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.
- 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)
- In each stage, the number y is always a positive integer, and is said to be the order of the duad, triad or tetrad.
- The term “duad” has been adopted in preference to “dyad,” which already has a different mathematical meaning.
- 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.
- 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.
- 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.
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.