Uncle Arnold’s talk about mathematical trinities is a very fun read.

Here a summary:

Real-complex dualities

  • free will vs predestination
  • pi0 vs pi1
  • Z2 vs Z
  • modes and quasimodes vs Berry phase and integer quantum Hall effect
  • S1 (projective variety) vs S2
  • S1 (group) vs S3
  • real vs complex trigonometric polynomial stratification

Real-complex-quaternion trinities:

  • R, C, H
  • E6, E7, E8
  • tetrahedron, octahedron, icosahedron
  • A3, B3, H3
  • D4, F4, H4
  • first three Hopf bundles
  • polynomials, Laurent polynomials and modular polynomials, aka with poles at 0, 1, infinity
  • numbers, trigonometric numbers, elliptic numbers
  • quadratic, hermitian, hyperhermitian forms
  • flat connection monodromy, vector bundle curvature, ?
  • hydrodynamic helicity, Chern-Simons functional, ?
  • Whitney, Chern, Pontryagin classes
  • homology, K-theory, elliptic homology
  • 60-60-60, 45-45-90, 30-60-90
  • the 27 straight lines on a cubical surface, the 28 tangents of a quartic plane curve and the 120 tritangent planes of a canonic sextic curve of genus 4
  • classical simple groups, ?, sporadic groups