Hamilton's first application of the concept of phase space - later so
fruitful in physics - was a prediction in optics: conical refraction in
biaxial crystals. This was one of the first successful predictions of a
qualitatively new phenomenon using mathematics, and created a sensation.
At the heart of conical refraction is a singularity, anticipating the
fermionic sign change underlying the Pauli exclusion principle and the
conical intersections now studied in quantum chemistry. The light emerging
from the crystal contains many subtle diffraction details, whose definitive
understanding and observation have been achieved only recently.
Generalizations of the phenomenon involve radically different mathematical
structures.