| _version_ | 1866901311681724416 |
|---|---|
| author | pallemans, gregory |
| author_facet | pallemans, gregory |
| contents | <h2>Introduction</h2> <p><em>What if the circle were not the fundamental structure of rotation, but an unattainable ideal a boundary endlessly approached by discrete forms?</em></p> <p>This work proposes a polygonal reformulation of Euler’s formula by replacing the continuous unit circle with a regular polygon inspired by natural honeycomb geometry. Rather than assuming perfect circularity as the underlying structure of rotation, we explore the idea that the circle itself may be understood as a mathematical limit a membrane between the discrete and the continuous.</p> <p>Instead of modelling rotation using the classical complex exponential eⁱˣ, we introduce a discrete exponential<br><strong>E(k) = ωᵏ</strong>,</p> <p>based on the <em>n</em>-th roots of unity. In this formulation, rotation becomes fundamentally discrete rather than continuous, emerging from finite symmetries rather than infinitesimal motion.</p> <p>The approach further defines a family of polygonal circularity constants<br><strong>πₙ = n sin(π / n)</strong>,</p> <p>including the elegant hexagonal value <strong>π₆ = 3</strong>, which aligns directly with the geometry of honeycombs and lattice-based natural structures.</p> <p>This framework bridges classical mathematics, bio-inspired geometry, discrete physics, and lattice models. It invites an interdisciplinary reconsideration of the role of the circle not as the primitive object of nature, but as an ideal limit toward which discrete structures converge without ever fully reaching.</p> <p>This upload contains the English PDF version of the work:<br><strong>“From the Sphere to the Hive: A Polygonal Reformulation of Euler’s Formula.”</strong></p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17904974 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | From the Sphere to the Hive: A Polygonal Reformulation of Euler's Formula pallemans, gregory <h2>Introduction</h2> <p><em>What if the circle were not the fundamental structure of rotation, but an unattainable ideal a boundary endlessly approached by discrete forms?</em></p> <p>This work proposes a polygonal reformulation of Euler’s formula by replacing the continuous unit circle with a regular polygon inspired by natural honeycomb geometry. Rather than assuming perfect circularity as the underlying structure of rotation, we explore the idea that the circle itself may be understood as a mathematical limit a membrane between the discrete and the continuous.</p> <p>Instead of modelling rotation using the classical complex exponential eⁱˣ, we introduce a discrete exponential<br><strong>E(k) = ωᵏ</strong>,</p> <p>based on the <em>n</em>-th roots of unity. In this formulation, rotation becomes fundamentally discrete rather than continuous, emerging from finite symmetries rather than infinitesimal motion.</p> <p>The approach further defines a family of polygonal circularity constants<br><strong>πₙ = n sin(π / n)</strong>,</p> <p>including the elegant hexagonal value <strong>π₆ = 3</strong>, which aligns directly with the geometry of honeycombs and lattice-based natural structures.</p> <p>This framework bridges classical mathematics, bio-inspired geometry, discrete physics, and lattice models. It invites an interdisciplinary reconsideration of the role of the circle not as the primitive object of nature, but as an ideal limit toward which discrete structures converge without ever fully reaching.</p> <p>This upload contains the English PDF version of the work:<br><strong>“From the Sphere to the Hive: A Polygonal Reformulation of Euler’s Formula.”</strong></p> |
| title | From the Sphere to the Hive: A Polygonal Reformulation of Euler's Formula |
| url | https://doi.org/10.5281/zenodo.17904974 |