A Different Way of Understanding the Number Pi and Our Perceived Reality

The number pi has fascinated us since ancient times. It is intimately linked to the main geometric figure we know: the circle. The myth of pi comes from its simplicity, as it represents the relationship between the circumference C and the diameter D of a circle, two concepts that we can easily grasp. And yet, this ratio C/D leads to an irrational number, whose number of decimal places is infinite and, moreover, these decimals never repeat periodically. It is both difficult to calculate it precisely and even more challenging to memorize a great number of its decimals.

The formulas C=2πR with D=2R have accompanied us since primary school, and no one questions them, whether for practical calculations or for complex equations in cosmology or quantum mechanics.

And yet...

The number pi, irrational and constant, is defined under very specific conditions: a continuous space, also called Euclidean, in which points have no dimensions, or more precisely, have zero dimensions. In this space, lines and geometric figures are perfectly smooth and continuous, and calculations are based on lengths or distances. Thus, to know the length of a wheel's circumference, you simply measure its diameter and multiply it by pi or an approximation of it. This applies to an infinite number of practical and theoretical examples.

The Point of the Compass

However, there is another type of space, called discrete, in which the point has a non-zero dimension (which can be extremely small but never zero), and this point sets the fundamental dimension in this space: it is indivisible and determines the smallest possible dimension. There is therefore no half-point or fraction of a point. What is smaller than the point simply does not exist. To illustrate this, imagine that the point is the tip of my compass, precisely placed on the center of the circle. It is always possible to refine this tip, but once placed, it marks a definite central point for the circumference.

When the point acquires a dimension different from zero, two things change compared to continuous space. First, my diameter D must be represented by an odd number of points. Indeed, the presence of a central point, equidistant from all points of the circle, implies that there are the same number of points on either side of the center. As a result, the diameter will have an odd number of points. Secondly, the circumference must have an even number of points, to respect the principle of symmetry of the points on the circle: each point must have an exactly opposite point, passing through the center. Points thus come in pairs.

The Big Differences

These two rules, stemming from the existence of a fundamental point, bring significant differences to our usual conception of the circle:

The diameter D can no longer be defined as equal to twice the radius R, because multiplying the number of points in the radius by two will always give an even number, whereas the diameter must have an odd number of points. This is due to the presence of a physical central point (with a dimension).
Both the diameter and the circumference will be defined exactly by an integer number of points. There will be no residual space smaller than the point, thus ensuring the completeness of the circle.
Consequently, the ratio between the circumference C and the diameter D will be a fraction of two integers, therefore a rational number (either finite or with periodic decimals). This ratio will no longer be a constant, but a number close to pi. This is confirmed by Dirichlet’s theorem, which states that any irrational number can be approximated by a fraction of two integers with a given precision, although this theorem is not the source of this result.

The Collapse of Continuous Space into Discrete Space

The application of these rules in a discrete space leads to a consequence: only certain circumferences from continuous space can be represented and perceived. These circles will have a diameter with an odd number of points and a circumference with an even number of points. To determine the percentage of circles that can transition from one space to the other, a simple logical calculation must be made:

Take straight line segments with an odd number of points that can qualify as diameters, which reduces the possibilities by half.
Multiply these diameters by pi and round the result to the nearest even integer to select the unique corresponding circumference.
Finally, among all possible circumferences, keep only those that meet these criteria.

This qualification rate is about 15.92%. Only 15.92% of the circumferences in continuous space can be transposed into discrete space. This explains why we speak of a collapse from one space into the other when it comes to circles.

A Quantum Vision of Pi

Here is an explanation that could further enhance the myth of pi. In continuous space, we don’t concern ourselves with the parity or oddness of the circumference and the diameter since there is no point. In fact, there is a point of zero dimension that could become non-zero. Without being observed in discrete space (where points could be counted), the number of points in the circumference and the diameter is both even and odd. This means that, much like in quantum mechanics, states are superimposed. This is the same story as Schrödinger's cat, which is both alive and dead in its box, something that can only be determined by opening the box. Similarly, I can only know if my circle is "viable" or not by entering discrete space and giving the point a non-zero dimension.

Pi has an infinite number of decimals because the precision of points is infinite; they are infinitely small or divided, thus null. But more importantly, the decimals never repeat periodically because there is a permanent uncertainty in the even or odd nature of the points, preventing the validity of the fraction from being fixed. The C/D relationship thus has a quantum nature, represented by pi as long as the parity and oddness of the figures are not observed.

This quantum vision of pi, linked to the fact that it originates from continuous space, raises the issue of the transition from one space to another. Pi is used in formulas that describe both continuous and potentially or openly discrete spaces. The constant that is absolute in continuous space is only an approximation in discrete space, but this leads to a misunderstanding of the discrete nature of things. For example, I am allowed to multiply the even number of points of a straight-line segment to find a result that I will call the circumference, while this segment does not qualify as a diameter and virtually has no center. It works, but it is wrong.

Geometric Collapse and Quantum Collapse

Things get more complicated when one takes an interest in quantum physics. We know that wave functions, like the electromagnetic wave of light, transform into particles (or quanta) when observed. Much has been said about this, notably through the Copenhagen interpretation, which suggests that the observer's consciousness is responsible for the collapse of the wave function.

The geometric collapse of the circle that we have described could provide a parallel to quantum collapse. It would no longer be the conscious observer causing the collapse, but rather the transition from a quantum space, where everything is possible simultaneously and continuously, to our observable space governed by rules that limit these infinite possibilities to a small fraction.

Visible Matter, Dark Matter, Dark Energy

One mystery that intrigues physicists is the presence of Dark Matter and Dark Energy in our observable universe. Only slightly less than 5% of the universe would be observable at our level of precision. These dark entities can only be detected indirectly, and nothing currently allows us to study them through direct observation. If we push the geometric analogy further, Visible Matter could be the collapse of the universe’s total energy into perceptible matter. Dark Matter would be collapsed energy, but not yet perceptible, while Dark Energy would be energy that has not yet collapsed into our observable space and may never collapse because it doesn’t meet the necessary conditions.

If this were true, in this metaphysical speculation, we could say that our reality is unique and that there are no parallel universes due to a lack of energy. If 32% of the available energy has collapsed into visible and dark matter, and 68% remains uncollapsed, there wouldn’t be enough energy to collapse parallel universes unless they came from other sources of energy.

Conclusion

This reflection on the role of Singularity and the point began during the writing of my book of practical philosophy, Singular Life and the Triangle of Illusions (2015, in Spanish only), where I mention on page 15 that the existence of the central point of a circle implies a diameter with an odd number of points. Less than ten years later, this idea has evolved in my metaphysical reflections. The philosophy I call Pointfulness is precisely based on the point, and it is natural to explore its implications in relation to concepts as fascinating as pi, cosmology, and quantum physics. However, these are intuitions, not science in the strict sense, as they fall outside my expertise, training, and experience.

The very idea of a quantum pi, or a witness to a quantum reality, even if it is not strictly scientific, seems elegant to me because it illustrates the thin boundary between the observable and the unobservable, between a point of zero dimension and a point of non-zero dimension, which changes everything and suddenly creates a reality—our reality.

The next step will be the study of square roots, where there are also interesting discoveries to be made about the common point that unites the segments of right angles.