Focusing between Mathematics and Physics. A Longterm History of Caustics from the 17th to the 20th Century
Mathematics
Optics, Quantum Optics and Physics of Atoms, Molecules and Plasmas
Final Report Abstract
The project gives a long-term historical overview of the study of the phenomenon and geometry of caustics. Caustics are special kinds of effects of bright light caused by reflection or refraction. If light rays from the sun get reflected by the inner concave side of a cylindrical cup, a bright so-called cuspidal curve becomes visible on the bottom of the cup, with an even brighter point where its two branches meet, the so-called singularity. The bright light is caused there from many reflected light rays coinciding. In mathematical terms, the caustic is here the surface enveloping all reflected rays, and the bright curve we see is a section of this surface by a horizontal plane. Caustics, therefore, are a mathematical generalization of the notion of a focal point, like we have in burning lenses or mirrors. It is a notion at the core of geometrical optics, a field of scientific study at the interface between mathematics and physics. But as a peculiar light phenomenon associated with mirrors and lenses, it is also a notion that plays into the history of technology and of the arts. A long-term history of caustics traces precursor lines of development in geometrical optics, in the empirical study of light phenomena, in technological developments of transparent and reflecting materials, as well as in the transmission of knowledge, that culminate into a notion caustics. In the 17th century, the notion of caustics is discussed explicitly and transparently in various contexts by many authors. Its mathematical treatments reflect, in particular, the implications of the momentous invention of the differential and integral calculus. After a period of relative stagnation in the 18th century, the mathematical study of caustics is revived in the 19th century in the context of algebraic surfaces, the wave theory of light, and generalized theories of systems of rays. The twentieth century witnesses a transformation of these earlier concepts, again in different contexts, like astrophysics, catastrophe theory, as well as computer rendering and ray tracing software. The study uses modern tools like dynamic geometry software, simulation of light effects using modern rendering software, and replication of optical experiments, in addition to the analysis of published and unpublished primary sources, to investigate the historical development in its mathematical, technological, and epistemological implications.
