Volume 34 (2015): Issue 4 (December 2015)

We introduce three related mathematical measures of terrain based on a particularly elegant combination of relief and steepness, which we term omnidirectional relief and steepness (ORS). One measure, reduced ORS (RORS), is particularly suited to creating lists of notable features, and another, domain relief and steepness (DRS), measures the ruggedness of a region.

The application of GIS and GPS methods together with qualitative sociological methods is one of the current fields of discussion in studies of human behaviour in space. The authors ask about the justification, manner and value of using the ‘triad’: quantitative GIS measurement methods – qualitative sociological methods – quasi-experiment. A mixed-method approach in an analysis of human movement patterns is introduced. Also, the role of the investigator in such projects is discussed.

This paper describes the previously unknown statistical distribution of adjacency matrix spectra for planar graphs, also known as spatial weights matrices, in terms of the following three readily available eigenvalue properties: extremes, rank orderings, and sums of powers. This distribution is governed by at most six parameters that, once known, allow accurate approximations of eigenvalues to be computed without resorting to numerical matrix methods applied on a case-by-case basis. Parameter estimates for illustrative real-world examples are obtained using nonlinear least squares regression techniques. Three conjectures are proposed, and graphical and trend results are reported for a diverse set of planar graph-based matrices.

Is mathematical category theory a unifying tool for geography? Here we look at a few basic category theoretical ideas and interpret them in geographic example. We also offer links to indicate how category theory has been used as such in other disciplines. Finally, we announce the direction of our research program on this topic as a way to facilitate the learning, and maintenance of learning, of GIS software – and in the spirit of Quaestiones Geographicae, invite debate, comment, and contribution to this program in spatial mathematics.

Entropy has been proposed as a significant tool for an analysis of spatial differences. Using Semple and Gauthier’s (1972) transformation of the Shannon entropy statistic into an entropy measure of inequality and their algorithm, an estimation is made of changes in regional inequality in Poland over the years 2005–2012. The inequality is decomposed into total, inter- and intra-regional types, and an analysis is made of relations holding between them.

Probably, almost everyone has some idea of what is meant by the words ‘spatial mathematics.’ The problem is that 100 people have 100 different ideas, because the concept is not easy to codify. In this paper we suggest a few ways to illustrate differences between ‘spatial’ and ‘non-spatial’ concepts, and ways to introduce spatial approaches where none was present before.

It is possible that Georg Cantor and Walter Christaller may have been aware of one another during their careers, however, there is no indication the two collaborated. Also, there is no documentation that Christaller’s central place theory (CPT) contains any tenets derived from Cantor’s middle third set (CMT). Regardless, CMT and CPT are linked by their constructions as nested hierarchies and the geometry of hexagons. The end points and intervals of CMT may be incorporated, respectively, as anchor points and radii for the hexagonal tessellations of central place theory.

Some geographical phenomena are continuous and exist in whole number dimensions. Topography, for example. Other phenomena, such as population density, depend heavily on the area used in their computation. Some refer to this as existing in 2½ dimensions. Is the difference just because it is a computed, rather than an observed quantity? I argue the case for considering treatment of discrete geographic data as continuous.