Designing visually rich mathmatical investing tools for repetitive geometric artifacts

Facilitating understanding and learning is the ultimate goal for designing visually rich mathematical investigation tools, but there are many intermediate steps that should be addressed before working on this goal. This thesis addresses one particular step: the design of visual mathematical interaction. This step lies in the intersection of geometry (low-dimensional Euclidean geometry), abstract algebra, theoretical computer science, and human-computer interaction (HCI). Geometry provides examples of rich visual artifacts that have many subtle properties. Abstract algebra provides a means and a framework for understanding and describing the artifacts. Theoretical computer science provides a means for operationalizing the algebra. And finally, HCI provides the insight to design software that links geometric artifacts to abstract algebra. The linking relies on creating visualizations of abstract algebra that allow for meaningful interactions that are meant to embody mathematics yet reduce the prerequisite knowledge associated with the theory. The visualizations are integrated into investigation tools and provide the focus for interaction. This research relies on the design principle of conversation with materials (back and forth feedback from interacting with new creations) to gain insight into the issues surrounding the design of investigation tools. The choice to explore repetitive geometric artifacts, such as polytopes (polyhedra of any dimension), tilings, and crystal lattices, with investigation tools was directed by the combination of their potential for simple algorithmic description and their highly visual nature. There are a number of results that precipitated from the process of design and construction of the investigation tools: the creation of a platform for specialized user-studies, the articulation and development of interaction techniques, the formation of a number of connections between computation and geometry, and the realization of software approaches for introductions to subtle mathematics. In particular, the importance of automata (and in one case 2D Turing Macliines) was demonstrated through the versatile and flexible use in each of the prototypes. The automata’s computation power was harnessed to describe the geometric artifacts and in some cases the automata’s expressive power was harnessed as a processing metaphor in graphical interfaces for navigation and construction.

Name thesis:degree_name

Ph D

Discipline thesis:degree_discipline

Computer Science

Grantor dc:publisher

The University of Western Ontario

Year dc:date.issued

2004

Author dc:creator

• Morey, Jim

Advisors dc:contributor.advisor

• Sedig, Kamran
• Mercer, Bob

• interface design
• tiling
• polyhedra
• polytopes
• crystal lattices
• abstract algebra
• Turing machines
• automata
• microworlds

Language dc:language.iso

en

Handle dc:identifier.uri

https://hdl.handle.net/20.500.14721/38904

OAI identifier oai:identifier

oai:uwo.scholaris.ca:20.500.14721/38904