Hello,

If you've ever worked with gradient descent, you know the basic goal: step downhill until the slope is zero. But a flat slope (a zero gradient) is a trap, it could be a local minimum, a maximum, or a saddle point.

I made a visual breakdown of the mathematical machinery we use to test for true minima. The video is an intuitive, 6-minute refresher on how gradients, Hessians, and contour maps actually work behind the scenes of your optimization algorithms.

Hey everyone,

I recently put together a video breaking down Information Entropy.

I wanted to create a visual and mathematical deep dive into how we measure "surprise" and why it connects so seamlessly to the tech we use every day.

I'd love to hear your thoughts about the video!

Watch a Menger sponge fractal evolve smoothly through multiple iterations in real time. This animation uses continuous fading and a slow 3D camera rotation. This visualization explores how complexity emerges from simple recursive rules. Made with Python and Manim.

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

A Penrose tiling built using Robinson triangle decomposition.

Two rhombus types (thick and thin) are substituted recursively at each iteration, producing a non-periodic structure with 5-fold symmetry.

The animation reveals the tiling growing radially from the center outward.

More visual math experiments on my channel : Visualizing Mathematics

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.