Technological Impact of Modern Abstract Mathematics
Tosiyasu L. Kunii
tosi@kunii.com
Computational Science Research Center
Hosei University
3-7-2 Kajino-cho, Koganei City
Tokyo 184-8584, Japan
Abstract
Recent advances of modern abstract mathematics are opening up new doors in technology and applications. Often the human common sense bound by the traditional mathematical framework limits our technological capability.
For example, computer display technology is basically relying on projective geometry that has been serving as the framework of human view representation. Hence, a displayed image is usually a result of the projection of objets as seen from a single viewpoint. Cellular spatial structures allow more generalized computer display technology with multiple and concurrent viewpoints. For automobile assembly, concurrent assembly is common and various parts are assembled from multiple directions simultaneously. There is no common display technology appropriate for such applications. Rapid prototyping is also benefited from cellular spatial structures to realize complex shapes as seen from various directions. Dental care, for example, requires this type of rapid prototyping.
Another example is free form generation. Free forms are widely used in industrial product design. Traditionally free forms have been generated from polygons (e. g. triangles) by modifying their surfaces by splines or some free form functions. They can be directly generated homotopically. Cellular spatial structures with a homotopic framework are a typical example of modern abstract mathematics in algebraic topology where versatile technological applications are abundantly discovered.
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