# E125: Sacks spiral structure

**size 101: 10,201 numbers and 1,252 primes**
```{figure} ../_static/experiments/e125_hero_101.png
:width: 80%
:alt: Sacks spiral preview (size 101)
```

**size 201: 40,401 numbers and 4,236 primes**
```{figure} ../_static/experiments/e125_hero_201.png
:width: 80%
:alt: Sacks spiral preview (size 201)
```

**size 301: 90,601 numbers and 8,769 primes**
```{figure} ../_static/experiments/e125_hero_301.png
:width: 80%
:alt: Sacks spiral preview (size 301)
```

**size 401: 160,801 numbers and 14,752 primes**
```{figure} ../_static/experiments/e125_hero_401.png
:width: 80%
:alt: Sacks spiral preview (size 401)
```

**size 999: 998,001 numbers and 78,359 primes**
```{figure} ../_static/experiments/e125_hero_999.png
:width: 80%
:alt: Sacks spiral preview (size 999)
```
*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: {doc}`../tags`.

## Highlights

- Places integers on a spiral using polar coordinates $r=\sqrt{n}$ and $\theta = 2\pi\sqrt{n}$.
- Parameterizable run: vary `--size` from the command line (default: `301`).
- Writes reproducible artifacts (`params.json`, `report.md`, and figures) into `out/e125/`.

## Goal

Render the **Sacks spiral** and highlight primes as a point cloud to reveal spiral-aligned structure,
then compare how the apparent density and “spiral fragments” change as `size` increases.

## Background (quick refresher)

### What is the Sacks spiral?

In the **Sacks spiral**, each integer $n$ is mapped to polar coordinates
$$r_n = \sqrt{n},\qquad \theta_n = 2\pi\sqrt{n},$$
and plotted as $(x,y) = (r\cos\theta,\ r\sin\theta)$.
Highlighting primes yields striking spiral fragments and density variations. See {cite}`Brockmann2019PrimeTime`.

### Optional background pages

- {doc}`../background/prime-numbers`

## Research question

How does the visual “prime spiral density” change as `size` increases (i.e., as we include integers up to $\text{size}^2$),
and which spiral-aligned structures persist across scales?

## Why this qualifies as a mathematical experiment

- **Finite procedure:** compute primes up to $\text{size}^2$ and apply a deterministic mapping $n\mapsto(x,y)$.
- **Observable(s):** spiral fragments, density contrasts, and how visibility changes with `size`.
- **Parameter space:** `size` (and, if you extend it, marker size / rendering choices).
- **Outcome:** figures and a short report capturing the key observation and caveats.
- **Reproducibility:** outputs saved to `out/e125/` with a parameter snapshot.

## Experiment design

- **Computation:** for integers $1,\dots,\text{size}^2$, compute a prime mask; map prime indices to $(x,y)$ via the Sacks rule.
- **Coordinates:** Cartesian coordinates $(x,y)$ are computed from $r=\sqrt{n}$ and $\theta=2\pi\sqrt{n}$ via $(x,y)=(r\cos\theta,\ r\sin\theta)$.
- **Prime classification:** primes are computed deterministically up to $\text{size}^2$ (sieve).

- **Outputs:** one or more scatter plots of prime points (composites omitted).
- **Artifacts written:**
  - `figures/fig_01_sacks_spiral*.png` (main figure) and `figures/e125_hero_<size>.png` (published hero)
  - `params.json`
  - `report.md`

### Axis scaling note

The run includes integers $n\le \text{size}^2$. With the Sacks choice $r=\sqrt{n}$, the maximal radius shown is
$$r_{\max}=\sqrt{\text{size}^2}=\text{size},$$, so the plot naturally fits inside a disk of radius about `size`.

## How to run

```bash
make run EXP=e125
```

Override the experiment size:

```bash
make run EXP=e125 ARGS="--size 501"
```

Direct invocation (always works):

```bash
uv run --extra dev python -m mathxlab.experiments.e125 --out out/e125 --size 501
```

## Notes / pitfalls

- Scatter plots are sensitive to marker size and resolution: the same data can look different depending on rendering.
- Larger `size` means more numbers ($\text{size}^2$) and a longer run time.
- “Looks-true” trap: a visually strong spiral does not by itself imply a theorem about primes.

## Extensions

- Add a faint background of composite points (with transparency) to compare prime vs. composite geometry.
- Plot multiple sizes on identical axes to compare density and visible spiral fragments.
- Add simple quantitative summaries (e.g., radial density bins for primes vs. all numbers).

## Published run snapshot

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

```{include} ../reports/e125.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

::: {dropdown} params.json (snapshot)
:open:

```{literalinclude} ../params/e125.json
:language: json
```

:::

## References

- Exposition connecting multiple prime visualizations (includes Sacks spiral): {cite}`Brockmann2019PrimeTime`.
- Recreational-math context and “pattern traps”: {cite}`gardner1983martin`, {cite}`hoffman1989archimedes`.

See also {doc}`../references`.

## Related experiments

- {doc}`e024` (Ulam spiral structure)
- {doc}`e124` (Klauber triangle structure)
- {doc}`e126` (Hexagonal number spiral structure)
- {doc}`e127` (Quadratic prime-run atlas ($n^2 + a n + b$))
- {doc}`e128` (Quadratic modular obstructions (Euler-type))
- {doc}`e129` (Euler lucky constants for $n^2 + n + b$)

## Parameters (example)

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

### Recommended `size` range

The Sacks spiral run includes integers $1,\dots,\text{size}^2$, so runtime and memory scale roughly with $\text{size}^2$ (prime classification dominates).

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 compute primes among the first 90,601 positive integers and plot prime indices using the Sacks polar mapping.
Even at this moderate scale, visible spiral-aligned prime fragments emerge, with noticeable density variation across radii.

## Key observations

- Prime points form dense spiral segments rather than appearing uniformly “random” in the plane.
- The visibility of segments depends strongly on marker size and figure resolution.

## Interpretation

The Sacks mapping is a deterministic geometric transform of the integer line.
It tends to visually emphasize arithmetic structure (including sequences with prime-rich behavior),
making it a useful exploratory visualization tool — but not a proof technique.

## Caveats

- Rendering matters: marker size, DPI, and axis limits can change perceived structure.
- Finite window: patterns can strengthen or weaken as `size` changes.
- Without quantitative overlays, conclusions remain qualitative.