Three-dimensional drawing of lattice-like structures

Abstract

Hasse Diagrams are a way to draw lattices in an easy-to-understand way. For complicated lattices with lots of edges, the conventional 2D approach usually is not adequate as the diagram is too cluttered. Too many edges and edge crossings make the picture unpleasing and hard to comprehend. In this thesis, we describe a way to model lattices in 3D and a Java program which implements this approach. Furthermore, this program acts as a wrapper for the cgol program proposed in the master thesis of A. Zuga j. The program cgol is a theorem prover for ortholattices. A proof search with cgol results either in a proof in a sequent-style calculus or an ortholattice which acts as a counter-example for the given formula to prove. This "counter lattice" is taken and transformed into a 3D Hasse Diagram. Such a Hasse Diagram depicts the transitive reduct of a lattice. As it cannot be guaranteed that all input represent a transitively reduced lattice, an approach which automati- cally calculates the transitive reduct of the input lattice is chosen. This thesis describes some of the graph theoretic and algorithmic founda- tions of graph drawing. Besides the fundamental aspects, psychological and physiological implications of graph understanding are also taken into ac- count. It describes a program which is devised according to the theoretic guidelines. This program acts as a wrapper for the program cgol to sim- plify user-interaction. Our implementation is then compared to two other prominent state-of-the-art lattice drawing programs.