Shadownomics ⚛️

In 1843, Hamilton realized complex numbers needed not one more dimension but two. One real part, three imaginary - i, j, k. He carved the equation into a Dublin bridge on October 16th. That number system now runs inside every 3D graphics engine, every robotics controller, and every LLM embedding space that maps language into high-dimensional geometry on GPU clusters today. Quaternion multiplication spins two perpendicular circles simultaneously - something that can only happen in four dimensions. i times j is k, but j times i is -k. Rotations don't commute, like a Rubik's cube. That's exactly why game engines and spacecraft use quaternions instead of matrices - no gimbal lock, no numerical drift. The same rotation math that every transformer uses when computing rotary position embeddings to encode token order in the complex plane. 2 minutes. Bookmark & watch today. The number system that powers every 3D engine and nobody taught you about.

The fourth dimension is not impossible to visualize - that claim is at best severely limiting and at worst completely wrong. Every LLM operates in embedding spaces of 4,096 dimensions or more, and the geometric intuitions that make transformer attention work are direct extensions of the same logic this video builds from scratch. The key insight is extrusion. A square is an infinite stack of line segments. A cube is an infinite stack of squares. A hypercube is an infinite stack of cubes - each cross-section a full 3D universe, stacked along a fourth axis perpendicular to all three. The deck of cards model makes this concrete: project 3D space onto a flat surface, free up the perpendicular dimension, and suddenly four dimensions have a home. 18 minutes. Bookmark & watch today. The clearest path into four-dimensional thinking anyone has made.

In 1931 Gödel proved that any sufficiently powerful mathematical system contains true statements it cannot prove. He did it by constructing a sentence that says "this statement is unprovable" - and showing the system chokes on it either way. The same incompleteness that limits formal mathematics limits LLMs. Every reasoning model operates inside a formal system and inherits Gödel's ceiling - there exist true statements about the world that no LLM can ever prove from its training data alone, no matter how large it gets. 25 minutes. Bookmark & watch today. The theorem that set the permanent limits of logic - and AI.

Hamilton spent 10 years trying to build a 3D number system. On October 16, 1843, walking across a bridge in Dublin, he realized he needed 4 dimensions instead - and carved the equation into the stone on the spot. Quaternions are what every robotics system, 3D game engine, and LLM positional encoding falls back on when rotation math gets hard. This video builds the full 4D geometric intuition using stereographic projection - the same trick that maps a sphere onto a flat screen. 30 minutes. Bookmark & watch today. The Alice in Wonderland connection alone is worth it.