Shapes and Their Approximate Descriptive Proximities

Abstract

This chapter introduces a relaxed form of descriptive proximity, an approximation approach to determining the closeness of descriptions of nerve shapes which is highly application-oriented. It is seldom the case that a pair of cell complexes have matching descriptions, even though the particular cell complexes are close with the exception of one or more of the feature values in the descriptions of the complexes. This anomaly in descriptive proximities between cell complexes is prevalent in physical systems in spacetime, where cell complexes with matching descriptions are usually not found. For example, the wavelength of the reflected light from one triangulated surface shape $\text{ sh }A$ may be very close to the wavelength of the reflected light from triangulated surface shape $\text{ sh }B$. If we choose wavelength the reflected light from a surface shape as a feature to consider, then the description $\varPhi (\text{ sh }A)$ would usually not equal $\varPhi (\text{ sh }B)$. To circumvent this problem, approximate descriptive proximities are introduced in this chapter.