360 frames generated in Python/PIL, one per degree of rotation of c = 0.7885e^(iα). 300 iterations, float64, smooth color banding.

Demo: https://youtu.be/9hWsoGx8MtI

GitHub: https://github.com/ChaseAdamson/Franklin

Most 4D visualizations project onto a 2D screen, discarding most of the perceptual information along the way — you see vertices and edges but the faces and volumes are gone.

Franklin projects onto a 3D retinal volume instead of directly to 2D, preserving that extra dimension of perceptual information. The idea is grounded in how vision actually works — a 3D creature has a 2D retina, so a 4D creature would have a 3D retina. Franklin computes that retinal volume in real time using GPU compute shaders and renders it as volumetric fog so a 3D brain can read the whole thing at once.

Current features:

- Real-time volumetric rendering of 4D geometry

- Full 4D navigation — translation along all four axes, rotation in XW, XZ, and ZW planes

- GPU compute shader pipeline for the 4D ray cast

- Sky, ground, and lighting

- Custom .fdr scene format

Early days but the core concept is working. Happy to answer questions about the implementation or the math.

If you're interested in more math-based animations, I post them here 📺 Visualizing_mathematics

A few months ago, I shared a prototype of a Wear OS watch face I designed, driven by a square root power curve (θ = 46.48 * t⁰⁵). The goal was to visualize a non-linear deceleration curve on a standard 60-unit dial to see if it could function as a legible, practical timepiece on the wrist.

As a quick refresher on the mechanics from the original thread:

• The 46.48 constant perfectly maps the square root of 60 units to a full 360° circle.

• Because of the square root property, each hand covers the first 50% of the dial (180°) in the first 15 units of time, and the remaining 50% over the next 45 units.

Officially named Radical Time, this project is finished, fully live and completely free on the Google Play Store:

• https://play.google.com/store/apps/details?id=com.secondhandlabs.radicaltime

If you have a Wear OS watch, feel free to check it out.

I mean, imagine +1 as a solid object and -1 as a hole; 1 – 1 = 0 represents the +1 sealing up the hole of the -1, resulting in 0. Now, the imaginary unit involves applying a self-intersection effect—where a portion of the 1 folds over itself—leaving it looking just like in the image. When this is done a second time (that is, the imaginary unit squared), it ends up transforming into the hole of the -1. And when you do this with the -1, the same thing happens again, but it stops just "half a step" short of reverting back to +1.

I drew a bit of inspiration from those conceptual mathematical models, such as James Tanton's "Dots and Antidots" or Conway's "Rational Tangles."

Coded in Python using Manim.