Diagram Reasoning and Paraconsistent Thinking: Hieromonk Hierotheos, His Ancestry, and Legacy

The article is dedicated to the use of logical diagrams in Byzantine Trinitarian theology. Logical diagrams are a kind of logical computation that is often considered to originate with Euler and Leibniz, but they were, in fact, used by Byzantine theologians since at least the ninth century. Nevertheless, logical diagrams were never so widely accepted as they began to be from the late thirteenth century to the early fifteenth century. The diagrams seem to have been introduced into Trinitarian theology by Eustratius of Nicaea (an authoritative philosopher who did not fare as well as a theologian) in his anti-Latin polemics dating to ca. 1112. From there, the use of diagrams was reclaimed in about the 1140s by the Latinophrone Nicetas “of Maroneia” and rejected in 1256 by the anti-Latin theologian Emperor Theodore II Laskaris. Nevertheless, beginning in the 1270s, their popularity and variability exploded. Eventually, triadological diagrams were “canonized” as the legacy of St. Hierotheos of Athens, the teacher of Dionysius the Areopagite, by Joseph Bryennios in the early fifteenth century. Even the “internal” opponent of Palamite theology, Theophanes of Nicaea, resorted to diagrams in defending his own triadology. The figure who rendered diagrams critical for the “Hesychast” theologians was, in the 1270s, hieromonk Hierotheos. He was able to express with diagrams the inconsistency of the mainstream Byzantine understanding of the Trinity. Nevertheless, his own name would come, in the fourteenth century, under a kind of damnatio memoriae, so that his main ideas circulated rather under the name of Hierotheos of Athens. This article argues that hieromonk Hierotheos passed from the Church of Patriarch Joseph to the Church of Patriarch Arsenius (or the Arsenites). Some of the highly authoritative teachers of the Palamites were in disagreement with the Great Church on the Arsenite issue, refusing to accept the act of 1410, where the Great Church had declared the Arsenites to be on the right side of the conflict. This fact could have affected the memory of hieromonk Hierotheos in the milieu where his works were most in demand. This research was supported by the University of Oxford project ‘New Horizons for Science and Religion in Central and Eastern Europe’ funded by the John Templeton Foundation. The opinions expressed in the publication are those of the author(s) and do not necessarily reflect the view of the John Templeton Foundation

A Method for Graph Drawing Utilising Patterns

This thesis describes a novel method for the layout of undirected graphs. It works by identifying certain patterns within the graph and drawing these in a consistent manner. For graphs to be useful and of benefit to a user, the result must clear and easy to understand. This process attempts to draw graphs in such a manner. Firstly, a background of graph problems and graph drawing is introduced, before the benefits of patterns are explained. Following this, there is an in-depth discussion of a number of existing graph drawing techniques, perceptual theories and methods for subgraph isomorphism. This pattern-based method is then explained in great detail. Firstly, the patterns required are defined and examples given. Then, there is an explanation of the methodology involved in identifying these patterns within a graph. Following on from this, the order in which patterns are drawn based on their connection types to those already drawn is detailed, before a detailed description of each drawing method. Evaluation of this method follows, starting with analysis mainly based on three real world data sources. This is in the form of side-by-side comparisons of graphs drawn with this method and a force-directed method. Following this, a metric based evaluation compares the two methods on edge crossings and occlusion, while also detailing some pattern based metrics. Further evaluation continues in the form of an empirical study. The methodology of this study is detailed before results are displayed. Analysis of these results follows, with conclusions drawn. Finally, potential further work is detailed and possible implementations discussed. All study materials and results are provided in the Appendix for those who wish to repeat the study or analysis

Sugiyama Layouts for Prescribed Drawing Areas

The area of graph drawing is concerned with positioning the elements of a graph on a canvas such that the resulting drawing is well-readable by humans and aids their execution of certain tasks. While known methods are usually well-studied from a theoretical perspective, both their applicability to graphs from practice and their integration into tools from practice are not always satisfactory. This is due to various reasons, for instance, due to known methods usually solving well-defined, self-contained problems that do not cover all of the bits and pieces that must be considered in practice. There, the diagrams the graphs originate from often comprise more than just simple nodes and simple edges, they tend to be messy and complex, and existing methods regularly compute drawings with poor compactness. This thesis is concerned with improving the well-known layer-based layout approach, originally proposed by Sugiyama et al., and devotes special attention to the requirements of dataflow diagrams. It presents new methods for the approach's layer assignment and coordinate assignment steps, and it identifies and illustrates research tasks that are essential to further better the situation in practice