Geometry has always treated π as a given — a fundamental constant embedded in the formulas of circles. But what if π is not a primitive, but a product? What if it can emerge naturally from a more fundamental geometric structure?My name is Muhammad Gashi, an independent researcher driven by conceptual inquiry. I recently completed and published a paper that proposes a new formulation of circular geometry: one that does not use π at all.
Instead, this methodology — which I call the “Gashi Methodology” — derives the essential equations of the circle using only the sine function and a triangle-based geometric model.
Because it views the circle as the limit of triangular construction, π is not an input, but rather an output — an emergent numerical result of internal structure.
This insight leads to a more profound philosophical observation:
“The circle is never constructed — only approached.” I propose: The circle is a concept, not a constructed reality.
“Gashi Methodology”: π as a Conclusion, Not a Postulate — A Triangle-Based Approach to Circular Geometry
Geometry has always treated π as a given — a fundamental constant embedded in the formulas of circles. But what if π is not a primitive, but a product? What if it can emerge naturally from a more fundamental geometric structure?My name is Muhammad Gashi, an independent researcher driven by conceptual inquiry. I recently completed and published a paper that proposes a new formulation of circular geometry: one that does not use π at all.
Instead, this methodology — which I call the “Gashi Methodology” — derives the essential equations of the circle using only the sine function and a triangle-based geometric model.
Because it views the circle as the limit of triangular construction, π is not an input, but rather an output — an emergent numerical result of internal structure.
This insight leads to a more profound philosophical observation:
“The circle is never constructed — only approached.”
I propose: The circle is a concept, not a constructed reality.
From the laws of “Gashi Methodology”:
C = R × sin(θ) × (360° / θ)
π = (sin(θ) × (360° / θ)) / 2
Published Paper:
DOI: https://doi.org/10.5281/zenodo.15665567
Copyright Registration:
SafeCreative ID: 2506172165045
ORCID: https://orcid.org/0009-0008-4185-1887
Email: gashi.mhd@gmail.com
If these ideas resonate with your mathematical or conceptual curiosity, I would be honored to engage in further discussion or collaboration.