# Walter WhiteleyYork University · Department of Mathematics and Statistics

Walter Whiteley

Ph.D. MIT

## About

190

Publications

40,424

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6,040

Citations

Citations since 2016

Introduction

Just submitted Review Article on Rigidity and Projective Geometry:
"Rigidity Through a Projective Lens"
Working on a second review paper focused on parallel drawing, scene analysis and reciprocal diagrams

Additional affiliations

Education

September 1966 - June 1971

## Publications

Publications (190)

In this paper, we offer an overview of a number of results on the static rigidity and infinitesimal rigidity of discrete structures which are embedded in projective geometric reasoning, representations, and transformations. Part I considers the fundamental case of a bar–joint framework in projective d-space and places particular emphasis on the pro...

This is a survey of over 50 years of work by many authors in rigidity for many variations of frameworks and related areas (such as bivariate splines), along with examples of their applications, all selected and presented in the context of projective geometry, both for notation and conceptual terms, including polarity, transfer to spherical and othe...

Knowledge mobilization is becoming increasingly important for research collaborations,
but few methodologies support increased knowledge sharing. This study provides insights,
using a reflective narrative, into a transdisciplinary knowledge-sharing investigation of the
connectivity of educational research to that of other disciplines. As an exempla...

We consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks and point-hyperplane frameworks in \({\mathbb {R}}^d\). In particular, for a graph \(G=(V,E)\) and a framework (G, p), we show that, under forced or incidental symmetry, infinitesimal rigidity for spherical frameworks with vertices in some subset \(X\su...

Geometric Cognition, or the more inclusive Spatial/Visual Reasoning in mathematics and its applications are often overlooked in a focus on mathematical cognition in algebra, calculation and logic. Both school curriculum and pedagogy, and discussions of the cognitive work of the mathematical mind can miss the central role of spatial/visual reasoning...

In this paper we consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks and point-hyperplane frameworks in $\mathbb{R}^d$. In particular we show that, under forced or incidental symmetry, infinitesimal rigidity for spherical frameworks with vertices in $X$ on the equator and point-hyperplane frameworks with th...

Learning and teaching mathematics and statistics lives within the wide cultural context of mathematical and statistical practices in many areas of work and of play across our cultures. We notice that mathematics and statistics are not homogeneous, even as practiced by pure and applied mathematicians and statisticians. The formal logical face (echoe...

This paper finds its origins in a multidisciplinary research group’s efforts to assemble a review of research in order to better appreciate how “spatial reasoning” is understood and investigated across academic disciplines. We first collaborated to create a historical map of the development of spatial reasoning across key disciplines over the last...

Spatial reasoning plays vital roles in choice of and success in STEM careers; yet the topic is scarce in grade-school curricula. We conjecture that this absence may be due to limited knowledge of how spatial reasoning is discussed and engaged across STEM professions. This study aimed to address that gap by asking 19 professionals to comment on a vi...

A one-to-one correspondence between the infinitesimal motions of bar-joint frameworks in $\mathbb{R}^d$ and those in $\mathbb{S}^d$ is a classical observation by Pogorelov, and further connections among different rigidity models in various different spaces have been extensively studied. In this paper, we shall extend this line of research to includ...

The purpose of this poster is to present preservice teachers’ understanding of the symmetry line of a square.

1996 Preprint of second article on Direction-Length Frameworks. Includes Averaging, examples of singular geometric positions, and reciprocal diagrams

Talk on Geometric Themes related to changes of metric
https://www.birs.ca/events/2015/5-day-workshops/15w5114/videos

In this article, we develop a theoretical model for restructuring mathematical tasks, usually considered advanced, with a network of spatial visual representations designed to support geometric reasoning for learners of disparate ages, stages, strengths, and preparation. Through our geometric reworking of the well-known “open box problem”, we sough...

Using multiple representations, and multiple supporting manipulatives, we will present a conceptual network for the classification of quadrilaterals, accompanied by investigations ready for classroom use. We will use paper-folding, sand pouring, hand gestures, MIRA, and manipulation of GSP sketches in these investigations of the pattern or hierarch...

To effectively use spatial reasoning, children need to learn the conventions and interpretations of the 2D representations of the 3D objects and be able to move fluently between each dimension. 2D representations are commonly uncritically allowed to stand for the 3D object. This chapter explores the im- plications of this oversight.

A rigidity theory is developed for bar-joint frameworks in $\mathbb{R}^{d+1}$
whose vertices are constrained to lie on concentric $d$-spheres with
independently variable radii. In particular, combinatorial characterisations
are established for the rigidity of generic frameworks for $d=1$ with an
arbitrary number of independently variable radii, and...

Video at http://www.fields.utoronto.ca/video-archive/event/283/2014

A survey of geometric themes in rigidity theory, with an overview of 40 years of work.
PDF uploaded and link to video
http://www.fields.utoronto.ca/video-archive/2014/08/283-3572

Assur graphs are a tool originally developed by mechanical engineers to
decompose mechanisms for simpler analysis and synthesis. Recent work has
connected these graphs to strongly directed graphs, and decompositions of the
pinned rigidity matrix. Many mechanisms have initial configurations which are
symmetric, and other recent work has exploited th...

Between the study of small finite frameworks and infinite incidentally periodic frameworks, we find the real materials which are large, but finite, fragments that fit into the infinite periodic frameworks. To understand these materials, we seek insights from both (i) their analysis as large frameworks with associated geometric and combinatorial pro...

It is well known that (i) the flexibility and rigidity of proteins are central to their function, (ii) a number of oligomers with several copies of individual protein chains assemble with symmetry in the native state and (iii) added symmetry sometimes leads to added flexibility in structures. We observe that the most common symmetry classes of prot...

Volumes of Polytopes in Spaces of Constant Curvature (N. Abrosimov, A. Mednykh).- Cubic Cayley Graphs and Snarks (H. Ademir, K. Kutnar, D. Marusic).- Local, Dimensional and Universal Rigidities: A unified Gram Matrix Approach (A. Alfakih).- Geometric Constructions for Symmetric 6-Configurations (L.W. Berman).- On External Symmetry Groups of Regular...

A basic geometric question is to determine when a given framework G(p)G(p) is globally rigid in Euclidean space RdRd, where G is a finite graph and p is a configuration of points corresponding to the vertices of G. G(p)G(p) is globally rigid inRdRd if for any other configuration q for G such that the edge lengths of G(q)G(q) are the same as the cor...

A molecular linkage consists of a set of rigid bodies pairwise connected by revolute hinges where all hinge lines of each body are concurrent. It is an important problem in biochemistry, as well as in robotics, to efficiently analyze the motions of such linkages. The theory of generic rigidity of body-bar frameworks addresses this problem via fast...

Plane pictures of three-dimensional convex polyhedra, plane sections of three-dimensional Dirichlet tessellations, and flat spider webs with tension in all the threads are essentially the same geometric object. At the root of this remarkable coincidence is a single geometric diagram that permits us to offer a unified image of the connections among...

Executive Summary of Recommendations 1. Future models of pre-service teacher education in Ontario should ensure that candidates receive instruction in (a) mathematics content knowledge, (b) mathematics pedagogy knowledge, and (c) the integration of both mathematics content knowledge and mathematics pedagogy. Candidates in the Primary/Junior/Interme...

Investigating rates of change in volume without calculation leads to an enriched sense of the optimization process and encourages reflection and connection among different approaches.

This article focuses on the development and problematization of a task designed to foster spatial visual sense in prospective
and practicing elementary and middle school teachers. We describe and analyse the cyclical stages of developing, testing,
and modifying several “task drafts” related to ideas around dilation and proportion. Challenged by par...

How a protein functions depends both on having basic stable forms (tertiary structure) and having some residual flexibility supported within that structure. The modeling of protein flexibility and rigidity in terms imported from physics and engineering has been developed through the theory of generically rigid frameworks and a fast combinatorial al...

A number of recent papers have studied when symmetry causes frameworks on a graph to become infinitesimally flexible, or stressed, and when it has no impact. A number of other recent papers have studied special classes of frameworks on generically rigid graphs which are finite mechanisms. Here we introduce a new tool, the orbit matrix, which connec...

In this paper, we combine separate works on (a) the transfer of infinitesimal rigidity results from an Euclidean space to the next higher dimension by coning (Whiteley in Topol. Struct. 8:53–70, 1983), (b) the further transfer of these results to spherical space via associated rigidity matrices (Saliola and Whiteley in arXiv:0709. 3354, 2007), and...

Recent work from authors across disciplines has made substantial contributions to counting rules (Maxwell type theorems) which predict when an infinite periodic structure would be rigid or flexible while preserving the periodic pattern, as an engineering type framework, or equivalently, as an idealized molecular framework. Other work has shown that...

A three-dimensional model and geometry software can help develop students' spatial reasoning and visualization skills.

This chapter traces the long-term cognitive development of mathematical proof from the young child to the frontiers of research. It uses a framework building from perception and action, through proof by embodied actions and classifications, geometric proof and operational proof in arithmetic and algebra, to the formal set-theoretic definition and f...

The decomposition of a linkage into minimal components is a central tool of
analysis and synthesis of linkages. In this paper we prove that every pinned
d-isostatic (minimally rigid) graph (grounded linkage) has a unique
decomposition into minimal strongly connected components (in the sense of
directed graphs), or equivalently into minimal pinned i...

Body-bar frameworks provide a special class of frameworks which are well understood generically, with a full combinatorial theory for rigidity. Given a symmetric body-bar framework, this paper exploits group representation theory to provide necessary conditions for rigidity in the form of very simply stated restrictions on the numbers of those stru...

A longstanding problem in rigidity theory is to characterize the graphs which are minimally generically rigid in 3-space. The results of Cauchy, Dehn, and Alexandrov give one important class: the triangulated convex spheres, but there is an ongoing desire for further classes. We provide such a class, along with methods to verify generic rigidity th...

Recent results have confirmed that the global rigidity of bar-and-joint frameworks on a graph G is a generic property in Euclidean spaces of all dimensions. Although it is not known if there is a deterministic algorithm that runs in polynomial time and space, to decide if a graph is generically globally rigid, there is an algorithm (Gortler et al....

In our previous paper, we presented the combinatorial theory for minimal isostatic pinned frameworks–Assur graphs–which arise in the analysis of mechanical linkages. In this paper we further explore the geometric properties of Assur graphs, with a focus on singular realizations which have static self-stresses. We provide a new geometric characteriz...

A significant range of geometric structures whose rigidity is explored for both practical and theoretical purposes are formed by modifying generically isostatic triangulated spheres. In the block and hole structures (P, p), some edges are removed to make holes, and other edges are added to create rigid sub-structures called blocks. Previous work no...

The sensor network localization problem is one of determining the Euclidean positions of all sensors in a network given knowledge of the Euclidean positions of some, and knowledge of a number of inter-sensor distances. This paper identifies graphical properties which can ensure unique localizability, and further sets of properties which can ensure...

We present some conjectures about the transfer of results for generic global rigidity in IR n to IR n+1 under some natural graph operations. These operations include: power graphs G k to G k+1 and coning G with a new vertex to form G * {v}. We also discuss transfer of generic global rigidity results among Euclidean, spherical, and hyperbolic spaces...

Maxwell's rule from 1864 gives a necessary condition for a framework to be isostatic in 2D or in 3D. Given a framework with point group symmetry, group representation theory is exploited to provide further necessary conditions. This paper shows how, for an isostatic framework, these conditions imply very simply stated restrictions on the numbers of...

We introduce the idea of Assur graphs, a concept originally developed and exclusively employed in the literature of the kinematics community. The paper translates the terminology, questions, methods and conjectures from the kinematics terminology for one degree of freedom linkages to the terminology of Assur graphs as graphs with special properties...

In this article we review how constraint theory can be applied to proteins to give useful information about the rigid and flexible regions. This approach includes all the constraints in a biomolecule that are important at room temperature. A rigid region decomposition determines the rigid regions (both stressed and unstressed) and the flexible regi...

These pages serve two purposes. First, they are notes to accompany the talk "Hyperbolic and projective geometry in constraint programming for CAD" by Walter Whiteley at the "Janos Bolyai Conference on Hyperbolic Geometry", 8--12 July 2002, in Budapest, Hungary. Second, they sketch results that will be included in a forthcoming paper that will prese...

In this paper, we consider using angle of arrival information (bearing) for sensor network localization. The essential property we require in this paper is that a node can infer heading information from its neighbors. We address the uniqueness of network localization solutions by the theory of globally rigid graphs. We show that while the parallel...

In this paper, we provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring the distances to their neighbors. We construct grounded graphs to model network localization and apply graph rigidity theory to test the conditions for unique loca...

In this paper, we provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring the distances to their neighbors. We construct grounded graphs to model network localization and apply graph rigidity theory to test the conditions for unique loca...

There is a lot of anecdotal and observational evidence for the extensive role of visuals and diagrams in the practice of mathematics. The classical book of Hadamard records the recollections of a number of well-known mathematicians on how they made significant discoveries. The book of Brown surveys this over a longer period from a more philosophica...

We outline the mathematical models, and the related counting algorithms, that are the basis for fast computations to predict biomolecular flexibility and rigidity. Within these mathematical models, we describe the snap-shot flexibility (instantaneous motions) of biomolecules, extracted from a single snap-shot of the molecule and the connection to l...

This paper is concerned with rigid formations of mobile autonomous agents that have leader-follower architecture. In a previous paper, Baillieul and Suri gave a proposition as a necessary condition for stable rigidity. They also gave a separate theorem as a sufficient condition for stable rigidity. This paper suggests an approach to analyze rigid f...