Rigidity of structures;
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.
Modeling proteins as frameworks;
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?
Parallel drawings;
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 occurences - and in the geometry of other levels of the homology, when parallel drawing occurs as the homology for edges.
Directions and angles as constraints in plane CAD;
Which sets of length, direction and angle constraints produce a locally unique dimensioned drawing in plane CAD?
Rigidity and independence for lengths and angles on the sphere;
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).
Transfer of rigidity, independence, and global uniqueness between hyperbolic space, euclidean space, Minkowski and spherical space;
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.
Skeletal rigidity, higher stresses, reciprocal diagrams and projected polytopes;
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.
Multivariate splines via cofactors, projections and homology;
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.
Geometric homology: the analogy between skeletal rigidity and multivariate splines;
The explicit comparison and contrast between the theories of skeletal rigidity (including rigidity of frameworks) and cofactors in multivariate splines.
Describing polyhedra: constructions and constraints;
Specific studies of issues of constructions, constraints, and representations of polyhedra in 3-space. Includes special results taken from other areas of discrete geometry.
Projects for students within Discrete Applied 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.
Diagrammatic reasoning and 'proofs';
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 creator, 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.
Losing mathematical abilities in aging; We spend a lot research time on how we learn mathematics when we are young. As people age - we also lose abilities: spatial reasoning, calculations, These loses have major impacts on how we live - including the loss of a driver's license and difficulty navigating in new environments. GeometricCognition Whiteley. If you share this interest contact walterwhiteley@icloud.com
