In this paper, we offer an overview of a number of results on the static rigidity and infinitesim... more 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 projective invariance of infinitesimal rigidity, coning between dimensions, transfer to the spherical metric, slide joints and pure conditions for singular configurations. Part II extends the results, tools and concepts from Part I to additional types of rigid structures including body-bar, body–hinge and rod-bar frameworks, all drawing on projective representations, transformations and insights. Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance. These are another fundamental example of abstract rigidity matroids with deep analogies to rigi...
A three-dimensional model and geometry software can help develop students' spatial reasoning ... more A three-dimensional model and geometry software can help develop students' spatial reasoning and visualization skills.
We consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks ... more We consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks and point-hyperplane frameworks in $${\mathbb {R}}^d$$ R d . In particular, for a graph $$G=(V,E)$$ 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 $$X\subset V$$ X ⊂ V realised on the equator and point-hyperplane frameworks with the vertices in X representing hyperplanes are equivalent. We then show, again under forced or incidental symmetry, that infinitesimal rigidity properties under certain symmetry groups can be paired, or clustered, under inversion on the sphere so that infinitesimal rigidity with a given group is equivalent to infinitesimal rigidity under a paired group. The fundamental basic example is that mirror symmetric rigidity is equivalent to half-turn symmetric rigidity on the 2-sphere. With these results in hand we also deduce some combinatorial conseque...
For a tensegrity framework with bars, cables (unilateral tension members) and struts (unilateral ... more For a tensegrity framework with bars, cables (unilateral tension members) and struts (unilateral compression members), this paper presents a sequence of four mathematical concepts for detecting its rigidity. In order from the strongest to the weakest, these concepts are called infinitesimal (or static) rigidity, prestress stability, second-order rigidity and rigidity. Emphasis is placed on pre-stress stability, which lies between infinitesimal rigidity and second-order rigidity.
Publisher Summary There is a lot of anecdotal and observational evidence for the extensive role o... more Publisher Summary 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 philosophical point of view. The interviews of current mathematicians by the Mathematics educator Burton have found a lot of conversation about intuition. This is the kind of word that people use when there are no other words—for example, when the process is visual but not sharp in form. These differences and gaps between the visual experience of the teacher and the visual experience of the student are important in understanding why and how visuals in the classroom can fail. Many individuals and groups have recognized that visualization has the potential to improve student learning or student performance in mathematics education, and then been disappointed at the actual observations and difficulties in the classroom. Dynamic geometry programs, such as Cabri Geometrie, Geometers SketchPad, and Cinderella are now used for the teaching of school geometry.
In this paper, we offer an overview of a number of results on the static rigidity and infinitesim... more 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 projective invariance of infinitesimal rigidity, coning between dimensions, transfer to the spherical metric, slide joints and pure conditions for singular configurations. Part II extends the results, tools and concepts from Part I to additional types of rigid structures including body-bar, body–hinge and rod-bar frameworks, all drawing on projective representations, transformations and insights. Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance. These are another fundamental example of abstract rigidity matroids with deep analogies to rigi...
A three-dimensional model and geometry software can help develop students' spatial reasoning ... more A three-dimensional model and geometry software can help develop students' spatial reasoning and visualization skills.
We consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks ... more We consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks and point-hyperplane frameworks in $${\mathbb {R}}^d$$ R d . In particular, for a graph $$G=(V,E)$$ 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 $$X\subset V$$ X ⊂ V realised on the equator and point-hyperplane frameworks with the vertices in X representing hyperplanes are equivalent. We then show, again under forced or incidental symmetry, that infinitesimal rigidity properties under certain symmetry groups can be paired, or clustered, under inversion on the sphere so that infinitesimal rigidity with a given group is equivalent to infinitesimal rigidity under a paired group. The fundamental basic example is that mirror symmetric rigidity is equivalent to half-turn symmetric rigidity on the 2-sphere. With these results in hand we also deduce some combinatorial conseque...
For a tensegrity framework with bars, cables (unilateral tension members) and struts (unilateral ... more For a tensegrity framework with bars, cables (unilateral tension members) and struts (unilateral compression members), this paper presents a sequence of four mathematical concepts for detecting its rigidity. In order from the strongest to the weakest, these concepts are called infinitesimal (or static) rigidity, prestress stability, second-order rigidity and rigidity. Emphasis is placed on pre-stress stability, which lies between infinitesimal rigidity and second-order rigidity.
Publisher Summary There is a lot of anecdotal and observational evidence for the extensive role o... more Publisher Summary 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 philosophical point of view. The interviews of current mathematicians by the Mathematics educator Burton have found a lot of conversation about intuition. This is the kind of word that people use when there are no other words—for example, when the process is visual but not sharp in form. These differences and gaps between the visual experience of the teacher and the visual experience of the student are important in understanding why and how visuals in the classroom can fail. Many individuals and groups have recognized that visualization has the potential to improve student learning or student performance in mathematics education, and then been disappointed at the actual observations and difficulties in the classroom. Dynamic geometry programs, such as Cabri Geometrie, Geometers SketchPad, and Cinderella are now used for the teaching of school geometry.
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Papers by Walter Whiteley