Walter Whiteley is a mathematician with a strong and enduring interest in geometry that has conducted important research on tensegrities along with his other interests. He is on the faculty of York University, Toronto, Canada.

Whiteley's Current Tensegrity Research

Some of Whiteley's current research into tensegrity, sometimes in collaboration with Connelly:

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?

Whiteley's has a broad interest in rigidity as well; rigidity is an important issue in tensegrity research. Some of his rigidity concerns include

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.

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, eulcidean space and sperical 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.

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

Some Articles by Whiteley

Some selected articles below.

Second-Order Rigidity and Pre-Stress Stability for Tensegrity

## Whiteley, Walter

## Table of Contents

Walter Whiteleyis a mathematician with a strong and enduring interest in geometry that has conducted important research on tensegrities along with his other interests. He is on the faculty of York University, Toronto, Canada.## Whiteley's Current Tensegrity Research

Some of Whiteley's current research into tensegrity, sometimes in collaboration with Connelly:

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?Whiteley's has a broad interest in rigidity as well; rigidity is an important issue in tensegrity research. Some of his rigidity concerns include

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.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, eulcidean space and sperical 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.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## Some Articles by Whiteley

Some selected articles below.

## Second-Order Rigidity and Pre-Stress Stability for Tensegrity

Link: http://www.scribd.com/doc/35313047/Second-Order-Rigidity-and-Pre-Stress-Stability-for-Tensegrity-by-Connelly-Whiteley

AMS (MOS) classification: Primary 52C25, Secondary 70B15, 70C20, 73K99.

## Links and References

Whitely's page at York University: http://www.math.yorku.ca/Who/Faculty/Whiteley/menu.html

York University: http://www.yorku.ca/web/index.htm

Portal To MathematicsPeople: Burkhardt, Connelly, Kenner, Hart, Whiteley