Technological Impact of Modern Abstract Mathematics

Computer technology is requires more advanced mathematics to be used to model quite novel and previously has never experienced world called cyberworlds. As such, this research visits certain nonlinear mathematical domains mostly in algebraic topology. Homotopy, curvature and cellular structured spaces are visited to explore their roles in versatile technological applications. Actually, most of modern high technology manufacturing employs cellular workspaces called work cells. As such the directions we present in this work is technologically quite fundamental and essential. 1 Backgrounds and Motivations 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 [1], [2]. 2 Roles of Invariants in Information Technology At the crucial time to get into the 21 century in a couple of years, information technology is leading our society to unique cyberworlds in cyberspaces spanning on networked computers [3], such as electronically driven international financial worlds controlling the real worlds constantly. For information technology to go along well with human aesthetics for our daily mental health in high-tension modern life, it has to have artistic provisions as well as technological advanced tools. Information broadcasting as interactive high bandwidth communication is expected to highlight information technology. All information will be basically animated because information reflects dynamically changing worlds on one hand and animation artistically better appeal to people as analogue TV has been on the other hand. As are the physical worlds governed consistently by physical laws build on physical invariant preservation such as mass and energy preservation, for animation to be consistent for validity it has to preserve its own invariants. There are a couple of invariants in animation. One most fundamental invariant is the homotopic invariant defined through the homotopic equivalence relationship. Another invariant is defined through curvature. For any object moving through translation and rotation to be identified uniquely, the curvature along the outline of the object is a motion invariant. 3 Homotopic Equivalence The change of cyberworlds in cyberspaces has to be homotopic, meaning continuously deforming, for our animation design to maintain the generality in terms of homotopic equivalence. We explain this briefly here. Let us consider the changes of a mapping function f relating a cyberspace X to another cyberspace Y. After the change, f becomes another mapping function g. In short, we are designing animation, we are designing the continuous deformation of f into g where f, g: X Ø Y. We consider the deformation during the normalized interval I = [0, 1] that can be a time interval or a space interval. X and Y are topological spaces. Let us denote the unchanging part A of the cyberspace X as a subspace A Õ X. Then, what we are designing is a homotopy H, where H: X μ I Ø Y such that ( "x e X ) ( H (x, 0) = f (x) and H (x, 1) = g (x) ), and ( "a e A, "t e I ) ( H (a, t) = f (a) = g (a) ). F is said homotopic to g relative to A, and denoted as f > g (rel A). Now here is a new design problem. That is, how we can design two topological spaces X and Y to be homotopically equivalent X > Y, namely of the same homotopy type. It is done by designing f: X Ø Y and h: Y Ø X such that h Î f > 1X and f Î h > 1Y, where 1X and 1Y are identity maps 1X : X Ø X and 1Y : Y Ø Y. Homotopy equivalence is more general than topology equivalence so that homotopy equivalence can identify a changing cyberspace that is topologically no more equivalent after the change. As an example, let us look at the human body. By the way, the human body is so complex that it is often called a micro cosmos, and it itself is a cyberspace. What we are looking at is a very simple case for an illustration purpose only, but is enough for our purpose. Given a silhouette of the human body of a person dancing. Fig. 1 shows how we can identify it automatically while the silhouette shape is changing during the dance. In manufacturing automation, while a component goes through a manufacturing line, for example to get deformed to a desired shape by a group of automated press machines, the deformation process is specified by a homotopy and validated by homotopy equivalence. In medical imagery, from a set of computed tomographical 2D sliced images, we can reconstruct 3D images homotopically in a following way: considering that a slice is homotopically deformed to the next slice, and then from the algorithm derived from the law of the formation of the organ of which the images are obtained tomographically we can record all the intermediate shapes between the two neighboring slices faithfully without loss [4]. The same applies to the Homotopically equivalent and topologically equivalent Homotopically equivalent but topologically