Systems Theory/Isomorphic Systems

Isomorphism
Isomorphism is the formal mapping between complex structures where the two structures contain equal parts. This formal mapping is a fundamental premise used in mathematics and is derived from the Greek words Isos, meaning equal, and morphe, meaning shape. Identifying isomorphic structures in science is a powerful analytical tool used to gain deeper knowledge of complex objects. Isomorphic mapping aids biological and mathematical studies where the structural mapping of complex cells and sub-graphs is used to understand equally related objects.

Isomorphic Mapping
Isomorphic mapping is applied in systems theory to gain advanced knowledge of the behavior of phenomena in our world. Finding isomorphism between systems opens up a wealth of knowledge that can be shared between the analyzed systems. Systems theorists further define isomorphism to include equal behavior between two objects. Thus, isomorphic systems behave similarly when the same set of input elements is presented. As in scientific analysis, systems theorists seek out isomorphism in systems so to create a synergetic understanding of the intrinsic behavior of systems. Mastering the knowledge of how one system works and successfully mapping that system’s intrinsic structure to another releases a flow of knowledge between two critical knowledge domains. Discovering isomorphism between a well understood and a lesser known, newly defined system can create a powerful impact in science, medicine or business since future, complex behaviors of the lesser understood system will become revealed.

Methods
General systems theorists strive to find concepts, principles and patterns between differing systems so that they can be readily applied and transferred from one system to another. Systems are mathematically modeled so that the level of isomorphism can be determined. Event graphs and data flow graphs are created to represent the behavior of a system. Identical vertices and edges within the graphs are discovered to identify equal structure between systems. Identifying this isomorphism between modeled systems allows for shared abstract patterns and principles to be discovered and applied to both systems. Thus, isomorphism is a powerful element of systems theory which propagates knowledge and understanding between different groups. The archive of knowledge obtained for each system is increased. This empowers decision makers and leaders to make critical choices concerning the system in which they participate. As future behavior of a system is more well understood, good decision making concerning the potential balance and operation of a system is facilitated.

Uses
Isomorphism has been used extensively in information technology as computers have evolved from simple low level circuitry with a minimal external interface to highly distributed clusters of dedicated application servers. All computer scientific concepts are derived from fundamental mathematical theory. Thus, isomorphic theory is easily applied within the computer science domain. Finding isomorphism between lesser undeveloped and current existing technologies is a powerful goal within the IT industry as scientists determine the proper path in implementing new technologies. Modeling an abstract dedicated computer or large application on paper is much less costly than building the actual instance with hardware components. Finding isomorphism within these modeled, potential computer technologies allows scientists to gain an understanding of the potential performance, drawbacks and behavior of emerging technologies. Isomorphic theory is also critical in discovering “design patterns” within applications. Computer scientists recognized similar abstract data structures and architecture types within software as programs migrated from low level assembler language to the currently used higher level languages. Patterns of equivalent technical solution architectures have been documented in detail. Modularization, functionality, interfacing, optimization, and platform related issues are identified for each common architecture so to further assist developers implementing today’s applications. Examples of common patterns include the “proxy” and “adapter” patterns. The proxy design pattern defines the best way to implement a remote object’s interface, while the adapter pattern defines how to build interface wrappers around frequently instantiated objects. Current research into powerful, new abstract solutions to industry specific applications and the protection of user security and privacy will further benefit from implementing isomorphic principles.

Comparing real vs model
The most powerful use for isomorphic research occurs when comparing a synthetic model of a natural system and the real existence of that system in nature. System theorists build models to potentially solve business, engineering and scientific problems and to gain a valid representation of the natural world. These models facilitate understanding of our natural phenomena. Theorists work to build these powerful isomorphic properties between the synthetic models they create and real world phenomena. Discovering significant isomorphism between the modeled and real world facilitates our understanding of the our own world. Equal structure must exist between the man-made model and the natural system so to ensure an isomorphic link between the two systems. The defined behavior and principles built inside the synthetic model must directly parallel the natural world. Success in this analytical and philosophical drive leads man to gain a deeper understanding of himself and the natural world he lives in.