Beyond the Shadow

Abstract

Far more than mere dark recesses, shadows have long served as toolsl to aid scientific communication, explanation, and calculation. Herodotus noted that Thales of Miletus systematically forecast an eclipse in 585 B. C. Kepler used the shadows of protruberances on the moon to calculate their elevation above base level (http://www.depauw.edu/sfs/backissues/8/christianson8art.htm). Eratosthenes of Alexandria used the shadow of an obelisk to apply a theorem of Euclid to measure, with remarkable accuracy, the circumference of the Earth (http://www.imagenet.org/, Ebook on Spatial Synthesis).

More generally, a shadow is a projection of a 3-dimensional object into a 2-dimensional space (and even more generally, of an n dimensional space into and n-1 dimensional space). Sometimes one focuses only on the shadows, as in the case of the eclipse. Sometimes one focuses only on the object itself. When the system is taken together, however, both shadow and what casts the shadow, it is then that understanding arises. As Minkowski noted: "Henceforth Space by itself, and Time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality" (http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_spacetime.html).