An explanation of the number or points and symmetries of starbursts

Abstract: Starbursts are the light intensity patterns seen when small bright sources are looked at night, typically stars. Starburst shapes are produced when the presence of the eye's wave aberrations generates caustics (light concentration) at the retina. A fascinating, but never explained fact about starbursts is that they usually present a $p$-fold symmetry pattern. We provide a theoretical explanation of the number of points and symmetries of starbursts, based on the geometric and algebraic properties of the wave aberration function expressed as a Zernike polynomial expansion. Specifically, we investigate the number and distribution of saddle cusps of Gauss of the Hessian of the wave aberration function. We also establish the connections between those points with the symmetries and the number of starburst points. We found that starbursts are likely generated by axially symmetric dominated wave aberrations with some amount of non-axially symmetric terms. For instance, whereas a wave aberration with a dominant spherical aberration (Zernike polynomial $Z_4^{0}$) plus $Z_3^{3}$ may induce a $3$ points starburst with a $3$-fold symmetry, a wave aberration combining $Z_4^{0}$ and $Z_4^{4}$ may induce a $4$-fold symmetry starburst with $4$ or $8$ points.