The number of senses of a given word, or polysemy, is a very subjective notion, which varies widely across annotators and resources. We propose a novel method to estimate polysemy based on simple geometry in the contextual embedding space. Our approach is fully unsupervised and purely data-driven. Through rigorous experiments, we show that our rankings are well correlated, with strong statistical significance, with 6 different rankings derived from famous human-constructed resources such as WordNet, OntoNotes, Oxford, Wikipedia, etc., for 6 different standard metrics. We also visualize and analyze the correlation between the human rankings and make interesting observations. A valuable by-product of our method is the ability to sample, at no extra cost, sentences containing different senses of a given word. Finally, the fully unsupervised nature of our approach makes it applicable to any language. Code and data are publicly available 1 . * Equal contribution. CX handled the data, generated the rankings, and sampled the examples of section 7. AJPT computed the results, plots, and wrote the paper. Both authors participated in the design of the study.