The Beautiful Geometry of Language

Grant Sanderson, creator of the beloved mathematics YouTube channel 3Blue1Brown, brings his signature visual clarity to explaining one of the most important concepts in modern AI: word embeddings. He shows how we can represent words not as discrete symbols but as points in high-dimensional space, where proximity indicates semantic similarity. This geometric representation of meaning allows machines to understand that “king” is related to “queen” in the same way that “man” is related to “woman”—a relationship captured as vectors in meaning-space.

Sanderson’s gift is making abstract mathematical concepts visually intuitive. He walks through how these embeddings are learned from text, how they capture analogies and relationships, and why they’re so powerful for everything from translation to search to language generation. The talk touches on both the elegance of the mathematics and the philosophical questions it raises about the nature of meaning itself. If meaning can be captured geometrically, what does that suggest about how language works and how our own brains might represent concepts?

For anyone seeking to understand the mathematical foundations of modern language AI, from word2vec to the transformers underlying GPT models, this talk provides essential conceptual grounding. Sanderson’s explanations are accessible to non-mathematicians while remaining rigorous enough to satisfy those with technical backgrounds. The talk exemplifies how mathematical thinking can reveal hidden structure in domains that seem purely symbolic or cultural, showing that the meaning humans create through language has a geometry we can visualize and compute with.