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Generative Radial Geometry

Generative Radial Geometry (abbreviated GRG) is a proposed mathematical framework that defines circular structures not through traditional Euclidean parameters, but through an internal, quantized radial mechanism. The theory aims to reconstruct the concept of the circle, its area, and its circumference without relying on π (pi), instead employing discrete summations and projection-based formulas.

Overview

In classical geometry, the area of a circle is computed using the formula A = π·r² and the circumference using C = 2·π·r. Both depend on the prior knowledge of the transcendental number π. GRG proposes that these approaches are externally imposed and introduces an alternative: reconstructing circular measures as the result of internal radial processes.

Core Idea

In GRG, a circle is not seen as a perfect curve enclosing a surface but as the cumulative result of radial activations extending from a central point. Both area and circumference are derived through discrete procedures involving rotation, projection, and quantized accumulation.

For the area, the method uses a discrete summation of vertical projections of radii onto the base axis. The circle is divided into *N* radial steps over a quarter-circle (90°), and each step generates a projection $ h_i $ which contributes to the total area. The formula becomes:

$A(r,N)=4r^{2}\cdot \left({\frac {1}{N}}\sum _{i=1}^{N}\sin \left({\frac {i\cdot \pi }{2N}}\right)\right)$

As N increases, this sum converges to the classical area A = π·r², though π is not used in the construction—it appears as an emergent result.

Similarly, the circumference is determined by connecting adjacent points on the quarter arc obtained by rotating the radius in N steps. Each pair of points defines a chord, and the total length of these chords approximates the arc. The formula is:

$C(r,N)=4r\cdot N\cdot {\sqrt {2-2\cos \left({\frac {\pi }{2N}}\right)}}$

This sum approaches the classical circumference C = 2·π·r as N increases, again without using π in the computation. The result is derived entirely from internal angular subdivision and radial connections.

Comparison to Classical Approaches

Historical approximations of π, such as those by Archimedes (using polygons) and Leibniz (via infinite series), are considered external estimations of a constant. GRG, by contrast, claims to generate both area and circumference intrinsically, using finite and internally defined operations. It reframes π not as a fundamental input, but as a limiting behavior observable from within the system.

Publication

The method was published on [Zenodo](https://zenodo.org/record/16731969) by Stefano Ferragina in 2025, under the Creative Commons Attribution-ShareAlike 4.0 International license.

- DOI:** [10.5281/zenodo.16731969](https://doi.org/10.5281/zenodo.16731969)

Criticism and Limitations

As of 2025, Generative Radial Geometry is a recently introduced concept and has not yet undergone academic peer review or been cited in independent mathematical literature. Its claims of internal reconstruction of area and circumference without π may require further validation and scrutiny. While the method has been formally published in open-access format, its acceptance and notability within the mathematical community remain to be established through third-party analysis and commentary.

References

Ferragina, Stefano. “Generative Radial Geometry and Area Without Pi.” Zenodo, 2025. [DOI:10.5281/zenodo.16761102](https://doi.org/10.5281/zenodo.16761102)