primestuff

I was looking at Visualization of Prime Numbers and had a thought.

If you graph the primes in 3d cubes like:

[
    [               # PRIMES
        [1,2,3],    # 2, 3
        [4,5,6],    # 5
        [7,8,9]     # 7
    ],
    [
        [10,11,12], # 11
        [13,14,15], # 13
        [16,17,18]  # 17
    ],
    [
        [19,20,21], # 19
        [22,23,24], # 23
        [25,26,27]  # -
    ]
]

Projected onto a 2D surface and considering only the primes, this looks like.

# Front
[
    [19,2,3],
    [13,5,-],
    [7,17,-]
]


# Side
[
    [2,11,19],
    [5,13,23],
    [7,17,-]
]


# Top
[
    [7,2,3],
    [13,11,-],
    [19,23,-]
]

# The series of primes in the side view are:
2, 5, 7, 11, 13, 17, 19, 23

If the series is new, I will come up with a name.

The longest line of primes in any direction within this graph is length 3 - [3, 5, 7].

Here's some visualisations.

The question becomes, what is the smallest number for which you get no - viewed from the sides or top of the cube.

Ie. which cube is completely filled with numbers when looking from the front, side or top.

The answer is 67.

67x67x67. The largest prime in this series is 300,719. This is not the largest prime in the cube - but from the view of the side.

Under S = 100, where S is the side of the height, width and depth the progression is.

# Cube with sides S x S x S that have a full side below 100

67, 69, 70, 71, 80, 82, 86, 88, 89, 92, 95, 97, 98, 99

I'm a recreational mathematician, so I asked Gemini if it's anything new

"the underlying mathematics is entirely covered by existing number theory. If you brought this to a number theorist, they wouldn’t say you discovered a new theorem, but they would likely say, "That is a really cool visual metaphor to teach prime gaps...You haven't discovered new math, but you have invented a fantastic visualization tool."

It might be flattering me. I didn't have my trusty 'Don't waffle, Don't be sycophantic' system instruction on.

I build the tool with Copilot, mostly GPT-5.4.

I looked briefly at other dimensions ... but 3d visualisation is what we're concerned with for now.

I'd be crazy not to hop on the end of the 67 meme. So here is primestuff.

See 67-side-.png

Details

primestuff renders prime numbers as dot graphs and visualises their structure in 2D grids, interactive 3D cubes, side snapshots, and straight-line prime-run analysis. Numbers are laid out row by row; prime numbers become bright dots, and non-primes remain dark.

Example

Generate a small prime graph image:

uv run python Examples/Primes/generate_img.py

That writes Examples/Primes/primes.png.

Project Layout

• src/primestuff/primes.py contains the reusable package code for rendering prime-dot PNGs, generating interactive 3D prime cubes, and ranking straight-line prime runs.
• Examples/Primes/generate_img.py creates a small prime graph image.
• Examples/Primes/generate_3d_cube.py creates a Plotly-powered 3D prime cube.
• Examples/Primes/generate_3d_cube_side_snapshots.py renders orthographic PNG snapshots of prime cubes.
• Examples/Primes/nD_search/search_nd_projection.py searches 3D, 4D, 5D, and higher-dimensional hypercubes for prime projections with no empty cells.
• Examples/Primes/analyse_line_length.py ranks graph widths by the longest contiguous prime lines.
• Examples/Primes/search_zero_white_box.py searches cube dimensions for projections with no empty cells.

Setup With uv

This project is configured as a uv package project and pins local development to Python 3.13 with .python-version. From the repository root:

uv sync --python 3.13

That creates a Python 3.13 virtual environment and installs the dependencies declared in pyproject.toml:

• numpy
• pandas
• pillow

Run a quick package smoke test:

uv run python -c "from primestuff import estimate_png_dimensions; print(estimate_png_dimensions(30, 100))"

Generate the small example prime graph:

uv run python Examples/Primes/generate_img.py

Generate an interactive 3D prime cube:

uv run python Examples/Primes/generate_3d_cube.py

That writes Examples/Primes/primes_3d_cube.html, a standalone Plotly scene with controls for prime opacity, non-prime opacity, marker size, labels, grid visibility, and camera reset. The default cube is 3 x 3 x 3: 1 sits above 10 and 19, 2 sits above 11 and 20, and 9 sits above 18 and 27. The HTML loads Plotly.js from the CDN, so no extra Python dependency is needed.

Use --square-dots when you want multi-pixel prime cells rendered as filled squares instead of round dots.

Search for the higher-dimensional equivalent of the 3D side length 67:

uv run python Examples/Primes/nD_search/search_nd_projection.py --dimensions 4 5 --max-side 250

By default this looks for the first side length where any orthographic projection axis has no empty cells, matching the 3D 67 result. Add --mode all to require every projection axis to be full.

Current 4D result with the default any projection mode:

uv run python Examples/Primes/nD_search/search_nd_projection.py --dimensions 4 --max-side 1000 --progress 0 --method auto

searching 4D: sides 3..1000, mode=any, method=auto
  found 4D side 295 using line: full projection axis/axes: axis 3
  max number in hypercube: 7,573,350,625
  largest prime in side-on PNG/projection: 7,573,350

API Example

from primestuff import (
  generate_prime_cube_plot_html,
  generate_prime_dot_png,
  rank_widths_by_prime_lines,
)

metadata = generate_prime_dot_png(
    width=30,
    max_number=100,
    output_path="Examples/Primes/primes.png",
    cell_size=1,
)
print(metadata)

best_lines = rank_widths_by_prime_lines(
    range(2, 100),
    max_number=100_000,
    min_length=5,
    workers=1,
)
print(best_lines[:5])

cube = generate_prime_cube_plot_html(
  output_path="Examples/Primes/primes_3d_cube.html",
  plane_width=3,
  plane_height=3,
  layers=3,
)
print(cube)