An Excursion in High-Dimensional Geometry
Pick two random directions in a thousand-dimensional space. They are almost perfectly perpendicular — not approximately, not usually, but with mathematical near-certainty. This is concentration of measure, and it quietly underpins both the AI systems you use every day and the thermodynamics you learned in school.
Inevitably Orthogonal
A Monte Carlo experiment reveals something strange: random vectors in high dimensions are nearly orthogonal, always. One symmetry argument — five lines — explains why.
How a Polynomial Becomes a Bell Curve
A single exponent controls the entire shape story, from U-shaped at d=2 through flat at d=3 to Gaussian. We derive it by slicing a sphere and counting what's left.
Why Your AI Search Works
The noise floor of cosine similarity is a geometric guarantee, not an engineering achievement. The same mathematics — applied to velocity space — derives the Maxwell-Boltzmann distribution as a theorem.