Sundials and Mathematical Surfaces

The ancient art of sundial designing and making has produced an astonishing variety of designs, many of them ingenious and works of art. The history and a fair sample of various designs can be found in [Ro] or [Pe,Sch,SP]. The principles of working are in many cases forgotten and understood only by a very tedious reverse-engineering, often made much more difficult by wellmeant but incompetent repair work. This is the case e.g. for the famous Schissler bowl sundial, made in 1578, see [S]. Nowadays designing sundials might not be considered an important undertaking. Nevertheless, the art is not dead. Some of the more advanced mean solar time showing sundials are of relatively recent origin, see [Ro,L]. The “missing” latitude-independent sundial was first described in [F]. (Any three of the four variables altitude, declination, azimuth — related to Sun’s position in the sky — and latitude determine the fourth, so in principle only three of them are needed to set up a sundial.) The goal of the present paper is to show that use of mathematics programs can be a tool in invigorating the art of sundial designing. The central principle here is to use throughout the usual equal-hour 24 h clock face. This has the consequence that the shadow must be created by a rather complex surface. Generating pictures of these surfaces or computing files of data describing them requires much numerical processing. This is where mathematics programs show their value. The program MAPLE® is used here throughout, but there are many other programs, equally useful (such as MATLAB® or MATHEMATICA®). As theoretical tool, vector-matrix calculus was used. It is strongly felt that the more usual method of spherical trigonometry has had its time. Separate sections are devoted to the usual solar time sundials, the more complicated mean solar time sundials, and shifted-time sundials. A more mathematically oriented treatment can be found in [Ru]. Earth’s orbital parameters and dynamics, among many other things, are excellently explained in [A] and [Se].