E126: Hexagonal number spiral structure

size 101: 10,201 numbers and 1,252 primes

size 201: 40,401 numbers and 4,236 primes

size 301: 90,601 numbers and 8,769 primes

size 401: 160,801 numbers and 14,752 primes

size 999: 998,001 numbers and 78,359 primes

Prime counts shown are \(\pi(\text{size}^2)\), i.e., the number of primes \(\le \text{size}^2\).

Tags: number-theory, quantitative-exploration, visualization
See: Valid Tags.

Highlights

• Arranges integers on a hexagonal lattice using a centered spiral enumeration.

• Parameterizable run: vary --size from the command line (default: 301).

• Writes reproducible artifacts (params.json, report.md, and figures) into out/e126/.

Goal

Render a hexagonal number spiral and highlight primes to reveal lattice-aligned structures, then compare how that structure changes as size increases.

Background (quick refresher)

What is a hexagonal number spiral?

A hexagonal number spiral places integers on a hexagonal grid, starting at the center with \(1\) and then spiraling outward. Compared to the square-lattice Ulam spiral, the hex lattice emphasizes different geometric alignments and may reveal different-looking prime structures. See [Kurzweg, 2020].

Optional background pages

• Prime numbers refresher

• Quadratic polynomials (algebraic) refresher

• Euler’s prime-generating polynomial refresher

Research question

Which geometric structures appear when primes are highlighted on a hexagonal spiral, and how stable are they as size increases (i.e., as we include integers up to \(\text{size}^2\))?

Why this qualifies as a mathematical experiment

• Finite procedure: enumerate a finite spiral placement for integers \(1,\dots,\text{size}^2\) and run a deterministic primality test up to \(\text{size}^2\).

• Observable(s): lattice-aligned streaks/clusters and how visibility changes with size.

• Parameter space: size (and, if you extend it, coordinate convention / rendering choices).

• Outcome: figures and a short report capturing the key observation and caveats.

• Reproducibility: outputs saved to out/e126/ with a parameter snapshot.

Experiment design

• Computation: map integers \(1,\dots,\text{size}^2\) to hex-lattice coordinates using a centered spiral enumeration; mark primes.

• Coordinates: integers are mapped to axial hex coordinates \((q,r)\) on a hex grid and embedded into the plane via \(x = q + \tfrac{1}{2}r\) and \(y = \tfrac{\sqrt{3}}{2}r\).

• Prime classification: primes are computed using a sieve up to \(\text{size}^2\).

• Outputs: one or more scatter plots of prime points on the hex lattice (composites omitted).

• Artifacts written:

  • figures/fig_01_hex_spiral*.png (main figure) and figures/e126_hero_<size>.png (published hero)

  • params.json

  • report.md

How to run

make run EXP=e126

Override the experiment size:

make run EXP=e126 ARGS="--size 501"

Direct invocation (always works):

uv run --extra dev python -m mathxlab.experiments.e126 --out out/e126 --size 501

Notes / pitfalls

• Different conventions exist for hex-spiral enumeration; changing the convention can change visible structures.

• Larger size means more numbers (\(\text{size}^2\)) and a longer run time.

• “Looks-true” trap: lattice-aligned streaks do not prove a statement about primes.

Extensions

• Overlay ring boundaries (or distance-from-center contours) to relate features to the spiral geometry.

• Add simple quantitative overlays: density by hex-ring index or by lattice direction.

• Compare hex vs. square spirals at matched N (same number of integers).

Published run snapshot

If this experiment is included in the docs gallery, include the published snapshot (report + params).

Reproduce:

make run EXP=e126

Parameters

• size: 301 (odd); visualizes integers 1..90601.

Notes

• Integers are placed by walking concentric hexagonal rings around the origin.

• Prime-rich lines/curves are expected analogues of the Ulam spiral phenomena.

• This experiment is deterministic; seed does not change the output.

params.json (snapshot)

{
  "size": 301
}

References

• Hex spiral background and construction ideas: [Kurzweg, 2020].

• Exposition connecting multiple prime visualizations: [Brockmann, 2019].

• Recreational-math context and “pattern traps”: [Gardner, 1983], [Hoffman, 1989].

See also References.

Related experiments

• E024: Ulam spiral structure (Ulam spiral structure)

• E124: Klauber triangle structure (Klauber triangle structure)

• E125: Sacks spiral structure (Sacks spiral structure)

• E127: Quadratic prime-run atlas (n^2 + a n + b) (Quadratic prime-run atlas (\(n^2 + a n + b\)))

• E128: Quadratic modular obstructions (Euler-type) (Quadratic modular obstructions (Euler-type))

• E129: Euler lucky constants for n^2 + n + b (Euler lucky constants for \(n^2 + n + b\))

Parameters (example)

• size: 301 (implied total integers: 1..90,601)

Recommended size range

The hex spiral run includes integers \(1,\dots,\text{size}^2\), so runtime and memory scale roughly with \(\text{size}^2\) (because about \(\text{size}^2\) integers are classified as prime/not-prime).

A practical range that works well for most runs:

• Minimum (still meaningful): size = 101

• Default / recommended: size = 301

• Comfortable upper range on typical laptops (varies): size = 999

• Above ~size = 1501: expect noticeably higher runtime (and potentially memory pressure), depending on the machine and the primality implementation.

Rule of thumb: start with 301, then try 501, 701, 901. Increase further only if runtime remains acceptable.

Summary

We enumerate a hex-lattice spiral placement for the first 90,601 positive integers and highlight primes. At this scale, lattice-aligned structures can be visible: some directions show denser prime “streaks” than others.

Key observations

• Prime points show directional structure aligned to the hex lattice.

• Some structures are subtle and become easier to see with larger size and careful marker sizing.

Interpretation

The hex spiral is an alternative geometry for indexing the same integers. As with the Ulam spiral, geometric placement can visually emphasize prime-rich progressions or polynomial-like sequences. This is a strong exploratory tool — but not evidence of a global law of primes.

Caveats

• Different enumeration conventions change the picture.

• Finite window: patterns can look stronger/weaker depending on the chosen size and zoom level.

• Visualization choices matter: marker size, DPI, and axis limits can hide or exaggerate structure.