When we think of the physical world around us, we invariably deal with abstractions. E.g. a coordinate system is a mathematical abstraction of physical space. Geometric distance is a mathematical abstraction of the actual (i.e. physical) distance in the physical space.

The Walk

But how does an object get from one point in physical space to the next? I daresay it gets there by means of a physical process that I will call ‘the walk’. As such, an abstract geometric distance must be expressible in terms of steps of the physical walk. It is important to note the following key point:

• The physical distance between any two points must correspond to the shortest walk.

Let’s start by considering a 2D Cartesian coordinate system as shown on Fig. 1.

In this model of space, the walk would correspond to movement from one point (which is a vertex on the Cartesian mesh) to the next, each time moving exactly one step up, down, left or right. As such, we can define distance as the number of steps one must take to get from vertex A to vertex B.

The Pythagorean Theorem

Unfortunately, such a Cartesian system coupled with the ‘orthogonal’ walk defined above fails to yield the familiar Pythagorean measure of distance, which we know to be true from our observation of the physical world around us. E.g., we know that on a flat 2D surface, for a right-angle triangle ABC AC² = AB² + BC². However, the orthogonal walk in the Cartesian 2D space of Fig. 1 yields AC = AB + BC, which is a bummer.

Granted, the Pythagorean theorem generally requires irrational numbers to work (save for a limited set of exceptions known as the Pythagorean numbers), and this notion bothered the Greeks greatly. Setting the natural numbers aside, in the quantized space we would expect the Pythagorean theorem to hold at least approximately, with the relationship between the catets and the hypothenuse approaching the known relationship involving the square root as the distances grow and the granularity increases. I find it intuitive and reasonable to expect a quantized space to approximate a continuous space with the increase in distances.

But can we design such a fixed graph in 2D that gives us a realistic representation of physical space where the length of the hypotenuse of a right-angle triangle is calculated using the Pythagorean theorem and the true irrationally transecentenal value of π (defined as a ration between the circumference of a circle to its diameter) is recovered in the limit of infinite distances?

Triangular Tesselation

To see if we can answer this question, let’s consider a different, triangular tessellation of 2D space as shown on Fig. 2.

What do we get? In the example on Fig. 2 AB = 7 steps, BC = 4, and AC = 9. The √(7² + 4²) ≈ 8.1, which gives us the error of (9 – 8.1) / 8.1 = 11.1%.

Does the error improve when we increase the size of the ABC triangle? Let’s examine a larger triangle, Fig. 3.

The larger triangle on Fig. 3 measures AB = 13, BC = 10, AC = 18 steps. The Pythagorean theorem for AC yields √(13² + 10²) ≈ 16.4, which gives us the error of (18 – 16.4) / 16.4 = 9.8%.

Better. We have an improvement, the error has gone down from 11.1% for the smaller triangle on Fig. 2 to 9.8% for a slightly larger one on Fig. 3. I conjecture that the error will continue to decrease with the increase in the size of the triangle, and we will recover a Pythagorean theorem as a limiting case for distances approaching infinity.

A Circle

What about a circle? Using the definition of distance as the shortest walk between two points, we can define a circle as the locus of points equidistant from a center point C. That is, if we walk from the center point C to any point on the circle P, the distance of the walk between R and P must always be R steps.

This concept is illustrated on Fig. 4 for a triangular tessellation of 2D space.

π

What is surprising is that no matter how large the radius R is, the resulting 2D circle in the triangular-tesselated 2D space will always be a hexagon! And the value of π defined as a ratio of the circumference of a circle to its diameter 2R in such space will always be 3.

So, even though we can recover a Pythagorean theorem as a limiting case for infinite distances in the triangular-tessellated space, we will never see the magnitude of π approach the familiar transcendental value of 3.14159… The value of π is fixed in the triangular-tesselated 2D space, whereas the value of π is not fixed in the real world.

A Square Circle

The situation is even funnier in the Cartesian 2D space of Fig. 1. A circle in such space is actually a square, Fig. 5!

The illustration on Fig. 5 tells us that the value of π is a constant 2 no matter the size of the circle in the Cartesian tesselation of 2D space.

Conclusions

What can we conclude from these surprising results?

1. The obvious conclusion is that for a space represented by a graph of statically connected vertices we will never recover the true transcendental and irrational value of π (defined as a ratio of the circle’s circumference to the circle’s diameter) even in the limiting case of R → ∞. I conjecture that any and all such statically connected 2D spaces will yield constant values of π.
2. The less obvious conclusion is that if we are to represent space by a graph, then the connections between the vertices must be dynamic! In other words, vertices must acquire new and lose old edges such that distances between two arbitrary points could be made as finely granular as we experience them to be in the real world. The number π is an attestation to this evolving granularity: We keep discovering more and more digits as the circle grows in size, as such, the granularity of the walk from the circle’s center to its circumference must increase with distance. This is something that does not happen in fixed-form tessellations.
3. Another surprising conclusion is that space cannot be understood without time! This surprising notion underscores Einstein’s deep insight that space and time are one and the same fundamentally. For Einstein, this insight had sprung from the mathematical apparatus of the General Relativity, which is heavily geometrical. For him, it was easier to formulate the mathematical formalism of the General Theory of Relativity by bunching the spatial dimensions and the dimension of time together without giving time any preferential treatment

The Lesson

In my opinion, the exercise with the 2D space tessellations teaches us a different and much more profound lesson. Time at a fundamental level is a part of the structure of space since space vertices must form and destroy connections.

In other words, space is not static. Moreover, we can never hope to understand space by examining it as a ‘frozen snapshot’ or as a ‘static picture’. This approach is fundamentally at odds with reality, and the transcendentally irrational value of π is the proof of it. Static spaces yield constant values of π. Only a dynamic and evolving space can yield an ever-changing value of π that keeps transforming as we refine our measurement.

As such, we must stop looking at space and time separately. This approach is wrong and has nothing to do with the reality we live in. Instead we must think of a dynamic space-time structure, ever evolving and ever changing.

And this is why I think that Stephen Wolfram’s graph model is on the right track. A walk from point/vertex A on the dynamic space-time graph to point/vertex B is a computational process that involves time, not only because time is necessary to make a step as we cannot make all steps instantaneously, but also because time is necessary to allow vertices to create and destroy connections. This requires an algorithm, which means that we do live in a simulation. I.e. the reality we perceive is at a fundamental level an algorithmically-driven computation.