Particular pages for each of these areas are currently under construction. Meanwhile, you can email me for specific information at
Statics and first-order kinematics of bar frameworks, tensegrity frameworks, hinged structures, and their projective polars (woven lines and popsicle stick bombs, sheet structures, molecular models) in any dimension.
Can models of proteins as bar and joint frameworks or as tensegrity frameworks give usefull insights or predictions into their behaviour in folding / unfolding or structural strength? Can we prove that the conjectured counting algorithms for 'molecular models' are exact to the mathematicals models?
A classical geometric construction arising from plane rigidity,
a polarity to polyhedral pictures in scene analysis, the study
of reciprocal diagrams, and Minkowski decompositions of polytopes.
Parallel drawings have also arisen in mathematical studies in areas like the homology of certain symplectic structures on manifolds, and in ergodic theory. We are interested in other such occurances - and in the geometry of other levels of the homology, when parallel drawing occurs as the homology for edges.
Which sets of length, direction and angle constraints produce a locally unique dimensioned drawing in plane CAD?
Questions analogous to the previous one (and to plane rigidity) for lengths and angles on the sphere. Studied, in part, as an analysis of local angles within and between planes passing through the origin (the center of the sphere).
With Franco Salilio, we have refined some transfer principles which convert theorems among these geometries. Given the basic duality of distances and lengths in Hyperbolic and Spherical spaces, this raises important questions about the duality in Euclidean space and the possibilies for further connections of other constraints among these geometries.
Generalizations of first-order kinematics and statics to higher geometric complexes than the graphs of rigidity. Developed to study general projections and liftings of polytopes, and the combinatorics of spherical polytopes (the h-vector and g-vector). Also includes higher dimensional versions of Maxwell's Theorem for projected polyhedra and plane stresses.
Contributions to the theory of multivariate splines, based in large part on the analogy to rigidity (and skeletal rigidity) and a transfer of techniques and results between the two theories. This theory is also studied for its potential transfer back to the theory of rigidity.
The explicit comparison and contrast between the theories of skeletal rigidity (including rigidity of frameworks) and cofactors in multivariate splines.
Specific studies of issues of constructions, constraints, and representations of polyhedra in 3-space. Includes special results taken from other areas of discrete geometry.
We are preparing some presentations of specific types of examples from rigidity and its relatives which are appropriate for direct exploration, using models, dynamic geometry programs, graph theory and simple vector algebra, for use as high school science fair projects, for special presentations, or for undergraduate geometry projects. A goal is to illustrate discrete geometry as an active area of research with important applications and surprising connections.
What is the role of diagrams in formal and informal reasoning?
in research and in pedagogy? in
problem solving by students and researchers?
in recall of mathematics? in texts in mathematics, computer science, etc.?
What 'geometries' (in the sense of Klein's hierarchy of geometries) are used in these diagrams, by the creater, by the expert reader, by the student learning the area?
Here is one bibliography for the general area.
Here is a second site with statements of individual researchers working on
Thinking with Diagrams.
With the development of my course on Information in Visual Form,
I have been doing further reading, reflection, and writing, about how
people learn to change what they see so that their visual thinking is
more effective. This is an important area and people seem to be
quite scattered. I would encourage anyone who shares this interest
to get in touch with me:
Projects funded by NSERC, York University
and personal grants, all related to the above research areas.
Undergraduate students who are interested in doing summer research in geometry (for pay!) should contact me. Information is usually out by late November, with deadlines in early January.