Unitary: Language of Topology

Reframing the Dynamics of Engaging with Otherness (Part #3)

[Parts: First | Prev | Next | Last | All | PDF] [Links: To-K | From-K | From-Kx | Refs ]

As a major branch of mathematics, Topology is concerned with properties of forms that are preserved under continuous deformation. These include stretching, but no tearing or gluing. The discipline emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation. It has a notable interest in degrees of connectivity. Related applications include graph theory and knot theory.

The innovative exploration of Topology and catastrophe theory by Rene Thom (Structural Stability and Morphogenesis: an outline of a general theory of models, 1972), further developed by Christopher Zeeman (Catastrophe theory. Selected papers, 1972-1977, 1977), offered a formalization of the intuitive insights of authors from a variety of disciplines (David Aubin, Forms of Explanations in the Catastrophe Theory of Rene Thom: topology, morphogenesis, and structuralism, 2004). As noted by Claude Tannery (Malraux, the absolute agnostic, or, Metamorphosis as universal law, 1991):

Already today the mathematical and abstract models of D'Arcy Thompson and Rene Thom strongly suggest that the creation of new forms and the maintenance of old forms depend on fields of action and chreodes (from chre, necessary, and odos, path), whose subtle combinations bring about not only the crafty bountifulness of nature but also that of our brains. Like magnetic or gravitational fields, the fields that create forms are immaterial, abstract... Rene Thom wondered whether a form of forms might exist, an archetypal chreode which extended into families of forms, and whether, on the tree of these families, those [various] tools and organs might occupy hologous positions. Malraux, too, at the most advanced stage of his meditation on forms created by artists, had to ask himself whether there might exist "a language of forms which transcends civilizations" and whether there might exist forms that he preferred to call primordial rather than archetypal.... Rene Thom's rigorous equations give a phenomenal foundation to the intuitions of Goethe and Malraux. (p. 261-262)

Topology has a variety of applications of considerable significance at this time, whereas catastrophe theory has fallen out of fashion in recent decades.

Design and operation of computer processors: For example, as noted by Shih Kuo (Intel® 64 Architecture Processor Topology Enumeration, Intel® Software Network, 27 September 2010):

Processor topology information is important for a number of processor-resource management practices, ranging from task/thread scheduling, licensing policy enforcement, affinity control/migration, etc. Topology information of the cache hierarchy can be important to optimizing software performance.

Crystallography: Topology offers a valuable approach to the study and description of of complex crystal structures, notably with respect to the design of new organo-metallic materials with valuable properties. (M. O'Keefe, Periodic Structures and Crystal Chemistry aka the Topological Approach to Crystal Chemistry. 2010; M. M. J. Treacy, K. H. Randall, S. Rao, J. A. Perry and D. J. Chadi (1997). Enumeration of periodic tetrahedral frameworks. Zeitschrift für Kristallographie: 212, 1997, 11, pp. 768-791). O'Keeffe has for example categorized and published details of 64 of the 128 optimized uninodal tetrahedral frameworks.

Protein structure: As noted by Minh N. Nguyen and M. S. Madhusudhan (Biological insights from topology independent comparison of protein 3D structures. Nucleic Acids Research, 19 May 2011):

Comparing and classifying the three-dimensional (3D) structures of proteins is of crucial importance to molecular biology, from helping to determine the function of a protein to determining its evolutionary relationships. Traditionally, 3D structures are classified into groups of families that closely resemble the grouping according to their primary sequence. However, significant structural similarities exist at multiple levels between proteins that belong to these different structural families.... Structural comparison is effected by matching cliques of points...: (i) 9537 pair-wise alignments between two structures with the same topology; (ii) 64 alignments from set (i) that were considered to constitute difficult alignment cases; (iii) 199 pair-wise alignments between proteins with similar structure but different topology; and (iv) 1275 pair-wise alignments of RNA structures. This second data set is used to quantitatively assess the performance... when structure similarity is low, as in the case of distant homologues. They include 64 pair-wise alignments

Security and surveillance: Extensive use if made of graph theory in the analysis of complex networks of transactions (telephone communications, financial transfers, etc) in an effort to identify potential threats to security or fraudulent activity. Notable examples of such applications include that of Netmap Analytics.

Social networking: Some use is made of graph theory in applications to analyze or represent social networks, notably using the extensive data sets now offered by social media.

Knowledge representation: An increasing variety of uses is made in the analysis and representation of networks of concepts, as networks, but with relatively little effort to extent the analysis into the higher forms of order which are so characteristic of the preoccupations of Topology (Knowledge-Representation in a Computer-Supported Environment, 1977; Preliminary NetMap Studies of Databases on Questions, World Problems, Global Strategies, and Values, 2006)

Simulation: Extensive use of Topology, knot theory and graph theory has been variously considered in the simulation of international relations and global modelling (Simulating a Global Brain: using networks of international organizations, world problems, strategies, and values, 2001; Computer-aided Visualization of Psycho-social Structures Peace as an evolving balance of conceptual and organizational relationships, 1971; Norman Schofield, A topological model of international relations. Paper presented to Peace Research International meeting, London, 1971).

Questionable applications: The application to the elicitation of intelligence for targetted assassination, from surreptitious surveillance, is of special concern (From ECHELON to NOLEHCE: enabling a strategic conversion to a faith-based global brain, 2007).

[Parts: First | Prev | Next | Last | All | PDF] [Links: To-K | From-K | From-Kx | Refs ]