Published March 8, 2026 | Version 1.0.0
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From the Quadratic Sequence to the Conical Helix

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# Zenodo Upload — Metadata

## Title
From the Quadratic Sequence to the Conical Helix

## Creators
Name:        Balban, Dogan
Affiliation: Independent Researcher
ORCID:       0009-0002-5052-6951

## Description

<p>This work develops a complete analytical and geometric framework for the
quadratic integer sequence</p>

<p><em>k</em>(<em>n</em>) = (2<em>n</em>² + 3<em>n</em> + 1) / 6,</p>

<p>which takes integer values precisely when <em>n</em> ≡ 1, 5 (mod 6).</p>

<p>A fine-grained analysis of the first differences reveals an alternating
pattern of <em>minor</em> and <em>major steps</em> whose pairwise sums form — as
proved in this work — the <strong>unique</strong> partition of the difference sequence
into equal-length blocks that yields an arithmetic progression of circle
circumferences. This algebraic uniqueness forces a constant radial
increment ΔR = 12/π and leads, without any further degree of freedom,
to an explicit Archimedean spiral with slope <em>a</em> = 6/π² = 1/ζ(2).</p>

<p>Extending the planar spiral by a linear height function produces a
three-dimensional <strong>conical helix</strong> on a right circular cone with
half-angle α = arctan(2/π) ≈ 32.48°. The algebraic core of the
entire construction is the fundamental identity
(4<em>n</em>+3)² = 48 <em>k</em>(<em>n</em>) + 1, which defines a universal quantity
<em>Q</em>(<em>k</em>) = √(48<em>k</em>+1) from which radius, azimuth, and height of the
helix can be expressed as exact closed functions of a single
parameter.</p>

<p>As a corollary of the congruence-class structure, <strong>all primes
<em>p</em> > 3</strong> appear as a distinguished subset of the integer-indexed
points on the helix.</p>

<p>The repository contains the full LaTeX source code (20 chapters),
all figures, Python verification scripts, and the compiled PDF.</p>

## Keywords
quadratic integer sequence
Archimedean spiral
conical helix
congruence classes
prime numbers
parametric geometry
arc-length parametrisation
Basel problem
zeta(2)

## Language
eng (Main text English)

## License
Creative Commons Attribution 4.0 International (CC BY 4.0)

## Upload Type
Publication

## Publication Date
2026-02-28

## Related Identifiers (GitHub Repository)
https://github.com/dogan1908/quadratichelix_en 

## MSC 2020
11B25 — Arithmetic progressions
53A04 — Curves in Euclidean and related spaces
11A41 — Primes
11B83 — Special sequences and polynomials

## Suggested Citation

Balban, D. (2026). From the Quadratic Sequence to the Conical Helix.
Zenodo. https://doi.org/10.5281/zenodo.18905795

## BibTeX

@misc{Balban2026,
  author       = {Balban, Dogan},
  title        = {From the Quadratic Sequence to the Conical Helix},
  year         = {2026},
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.18905795},
  url          = {https://doi.org/10.5281/zenodo.18905795}
}

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quadratichelix_balban2026.pdf

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Additional details

Software

Repository URL
https://github.com/dogan1908/quadratichelix_en
Development Status
Active

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