The Nature of Pi : Between Euclidean Geometry and Discrete Geometry

The number pi (π) is known as the ratio of the circumference of a circle to its diameter. In Euclidean geometry, π is defined as an irrational number, which represents the circumference C of a circle divided by its diameter D, that is, π=C/D​. Additionally, since D=2R (where R is the radius of the circle), the circumference can be expressed as C=2Rπ.

However, when we consider discrete geometry, in which points have non-zero dimension d, corresponding to the limits of our material world, the relationship becomes more complex. In reality, multiplying the radius R by 2 on a circle is equivalent to counting the central point c, of dimension d, twice. This means that the correct expression for diameter in terms of radius would be D=2R−d.

Consequently, the relation π=C/2R​ is only valid if d is zero, that is to say in a purely theoretical context where the points have no dimension.

In discrete geometry, C/(2R-d)​ is no longer equal to π for any non-zero value of d. On the contrary, this fraction could take a rational value depending on the precision of the measurement of d. This implies that in the framework of discrete geometry, π becomes an abstract approximation, applicable only in an unreal case. As a result, its use may introduce indeterminacy into calculations. To remain faithful to physical reality, it would be better to adopt a reasonable value of d, even a very small one, and to work with rational numbers rather than relying on π, an irrational number. This is all the more important because when using π, the calculated circumference requires infinite precision to close perfectly, which is unattainable in the material world.

In conclusion, the nature of π in the context of discrete geometry reveals the limits of the use of mathematical abstractions in the real world. Adopting an approach based on rational approximations based on d would make it possible to better adhere to our material reality and at the same time to better understand the nature of π. Once again we can appreciate the importance of the point and its definition in our physical world, as well as for philosophical approaches to information and abstractions.