Polytwisters

Polytwisters are a class of strange curved four-dimensional shapes related to uniform polyhedra and Hopf fibration. They were discovered by Jonathan Bowers circa 2007, who found 222 of these shapes plus three infinite families, and produced POV-Ray renders of their 3D cross sections.

This repository is a project to clean up and port the original POV-Ray code (unpublished) to a modern toolchain using Blender's Python scripting capabilities. Computing the polytwisters as meshes enables professional-quality animations, as well as digital fabrication for realizing physical cross sections.

This software is in an early stage of development. You will probably have to make some minor changes to get it working on your machine.

Basic usage

I have successfully run the entire pipeline on Linux and Windows on x64. (macOS might work but I had issues installing the CadQuery dependency.)

Requirements:

• Python 3.11 (Python 3.12 does not work due to the CadQuery dependency)
• PDM, a Python package manager
• Blender 4.1
• ffmpeg for video rendering
• git-lfs when checking out the repository. There is a single HDRI used for rendering.

I have tried to keep the Python dependency count small: CadQuery, SciPy/NumPy, and tqdm.

To install:

pdm install

If you want a more minimal installation that does not include CadQuery (e.g. to just render the Blender files):

pdm install --prod

Single render

Compute 10 cross sections of the bloated tetratwister:

mkdir scratch
pdm run compute_hard_section -n 50 tetratwister scratch/tetratwister_sections/

Export .blend file, which has cameras, lights, and render configuration all set up for you:

pdm run export_blend scratch/tetratwister_sections scratch/tetratwister.blend

You can open this file in Blender to see the animation. Drag around the animation frame to view the cross sections.

To render with standard settings, you can use Blender's render feature, or this script:

pdm run render_blend scratch/tetratwister.blend scratch/tetratwister.mp4

Render everything

mkdir scratch
# Compute meshes.
pdm run compute_all_sections -n 50 scratch/sections/
# Export blend files.
pdm run export_blend scratch/sections/ scratch/blends/
# Render blend files.
pdm run render_blend scratch/sections/ scratch/videos/

Other goodies

To generate a SVG vector lineart montage:

python core/make_hard_polytwister.py tetratwister -f svg_montage tetratwister_svg

See python core/make_hard_polytwister.py --help for full options.

Cycloplane research code

This repo also includes code for an ongoing research effort to gather experimental data on how the boundaries of cycloplanes constructed from Hopf fibers intersect. cycloplanes.py implements the algorithms for analysis while test_cycloplane.py runs experiments using both random and fixed cycloplane configurations. All randomness is seeded, so the tests are deterministic.

Install numpy scipy pytest, then run pytest.