Geometry—Daughter of Art, Mother of Mathematics

Geometry is “Janus-headed”: Looking backwards it collects, abstracts and systematizes single phenomena and facts of nature, art and mathematics and it provides mathematics with models and structures for these facts. Looking forward it deals with and solves seemingly singular problems partly of practical and applicable nature and partly with pure theoretical problems. For both viewing directions the essential tools are logic reasoning and visualization. The latter ranges from symbolic and metaphoric sketches to images of high precision, where engineers and workmen extract measures and working processes. The ability to interpret such figures is based on quite a lot of unconsciously accepted assumptions, most of them have cultural historic background. To pack a meaning, an idea into a figure, a painting, a sculpture, a temple and to be able to decode this meaning or idea starts with religion and art. The realized paintings, temples, etc. were and still are communication media. Parallel to these symbolic aspects of visualization the communication aspect evolves to making “technical” drawings. Throughout history the correct interpretation of such technical drawings needed and still needs also schooling of the interpreter as well as of the producer of the sketch. Besides the typical geometric disciplines, theory of mappings and differential geometry of curves and surfaces and kinematics, which were necessary for practical technical purposes, geometry became the starting point for the development of mathematical disciplines. The paper aims at discussing historic aspects of the development of geometric visualization.