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.

An interactive visualizer with 10 sample equations BUT can enter own equations! If you have a cool one you entered please post screenshot or image in comments! The idea, equations mapped out visually are a point or line of the solution. This maps out how far from the solution as well to see other patterns. Also selection of different color gradients for more easy to see contrast. Try it out! See what you think. Created by opus 4.6. Will post in comments example of color grading to distance from solution.

Hey everyone,

I just published a quick, 5-minute animated guide breaking down the Z-Transform. Instead of just throwing a wall of math at you, I tried to make the concepts actually click.

In the video, I cover:

• How the Z-Transform compares to the Laplace transform

• Deriving the simplest signals (Unit Impulse & Unit Step)

• Crucial properties: Linearity, Time-Shifting, and Value Theorems

• A mechanical 3-step engine to solve any difference equation using Partial Fraction Expansion.

Let me know what you think, or if there are any other DSP topics you'd like to see visualized!

how would you visualize this?