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Compass and Straightedge in the Poincaré Disk
, 2001
"... The spirit of this article belongs to another age. Today, “geometry” is most often analytic; this is especially suited to the making of nice pictures by computer. But here we give a synthetic approach to the development of hyperbolic geometry; our constructions use only a Euclidean compass and Eucli ..."
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The spirit of this article belongs to another age. Today, “geometry” is most often analytic; this is especially suited to the making of nice pictures by computer. But here we give a synthetic approach to the development of hyperbolic geometry; our constructions use only a Euclidean compass and Euclidean straightedge, and can be carried out by hand. Indeed, M.C. Escher used something like the methods we give here to produce his well known Circle Limit I, II, III, and IV prints [9]. In [6], H.S.M. Coxeter describes a remarkable correspondence with Escher. Having met at the 1954 International Congress of Mathematics in Amsterdam, Coxeter apparently sent Escher a paper in which a drawing of part of a tiling of the Poincaré disk appeared. Coxeter must have been quite pleased and surprised to find a print of Circle Limit I in his mail in December 1958. It is quite remarkable that the drawing in the paper Coxeter had sent Escher was not even as detailed as our Figure 1, and did not show the “scaffolding ” Coxeter had used in its construction. Nonetheless Escher deduced and generalized the technique of its construction, producing incredibly fine tesselations of the Poincarédisk.
Advanced MathematicalThinking at Any Age: Its Nature and Its Development
"... This article argues that advanced mathematical thinking, usually conceived as thinking in advanced mathematics, might profitably be viewed as advanced thinking in mathematics (advanced mathematicalthinking). Hence, advanced mathematicalthinking can properly be viewed as potentially starting in ele ..."
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This article argues that advanced mathematical thinking, usually conceived as thinking in advanced mathematics, might profitably be viewed as advanced thinking in mathematics (advanced mathematicalthinking). Hence, advanced mathematicalthinking can properly be viewed as potentially starting in elementary school. The definition of mathematical thinking entails considering the epistemological and didactical obstacles to a particular way of thinking. The interplay between ways of thinking and ways of understanding gives a contrast between the two, to make clearer the broader view of mathematical thinking and to suggest implications for instructional practices. The latter are summarized with a description of the DNR system (Duality, Necessity, and Repeated Reasoning). Certain common assumptions about instruction are criticized (in an effort to be provocative) by suggesting that they can interfere with growth in mathematical thinking. The reader may have noticed the unusual location of the hyphen in the title of this article. We relocated the hyphen in “advancedmathematical thinking ” (i.e., thinking in advanced mathematics) so that the phrase reads, “advanced mathematicalthinking” (i.e., mathematical thinking of an advanced nature). This change in emphasis is to argue that a student’s growth in mathematical thinking is an evolving process, and that the nature of mathematical thinking should be studied so as to
On HamiltonJacobi theory: its geometry and relation to pilotwave theory
, 2002
"... This paper is an exposition, mostly following Rund, of some aspects of classical HamiltonJacobi theory, especially in relation to the calculus of variations. The last part of the paper briefly describes the application to geometric optics and the opticomechanicsal analogy; and reports recent work ..."
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This paper is an exposition, mostly following Rund, of some aspects of classical HamiltonJacobi theory, especially in relation to the calculus of variations. The last part of the paper briefly describes the application to geometric optics and the opticomechanicsal analogy; and reports recent work of Holland providing a Hamiltonian formulation of the pilotwave theory. The treatment is technically elementary, and provides an introduction to the subject for newcomers. Submitted to Contemporary Physics ‘Dont worry, young man: in mathematics, none of us really understands any idea—we just get used to them.’ John von Neumann, after explaining (no doubt very quickly!) the method of characteristics (i.e. HamiltonJacobi theory) to a young physicist, as a way to solve his problem; to which the physicist had replied ‘Thank you very much; but I’m afraid I still don’t understand this method.’ 1
The calculus as algebraic analysis: Some observations on mathematical analysis in the 18th century. Archive for History of Exact Sciences
, 1989
"... (a) Equality of mixed partial differentials.................. 319 ..."
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(a) Equality of mixed partial differentials.................. 319
Technology as a Transformative Force in Education: What Else Is Needed to Make It Work? 12
"... Looking Beyond the RearView Mirror Computational technology is a profoundly transformative force across our society. In importance, it is analogous to technological inventions such as the printing press or the internal combustion engine, and such representational inventions such as alphabetic writi ..."
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Looking Beyond the RearView Mirror Computational technology is a profoundly transformative force across our society. In importance, it is analogous to technological inventions such as the printing press or the internal combustion engine, and such representational inventions such as alphabetic writing. Impacts of such enormous
Gröbner Bases and Invariant Theory
"... This paper was Hilbert's quick answer to those of his fellow mathematicians who harshly criticized the nonconstructiveness of his first proof. The second paper contains the Nullstellensatz, the HilbertMumford criterion ..."
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This paper was Hilbert's quick answer to those of his fellow mathematicians who harshly criticized the nonconstructiveness of his first proof. The second paper contains the Nullstellensatz, the HilbertMumford criterion
The Changing Relationships Between Science and Mathematics: From Being Queen of Sciences to Servant of Sciences
, 2003
"... this paper, we will give a brief historical overview of the relationship between science and mathematics then we will move on to the impact of computers in this relationship and finally we will talk about the future of this relationship. The origin of the sciences is rooted in tool making and agricu ..."
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this paper, we will give a brief historical overview of the relationship between science and mathematics then we will move on to the impact of computers in this relationship and finally we will talk about the future of this relationship. The origin of the sciences is rooted in tool making and agriculture. It is fair to say that making and using tools and the cultural transmission of scientific knowledge became essential to the existence of the human species and was practiced in all human societies. The history of science and mathematics starts with the Neolithic era. In the Neolithic Revolution, although mathematical knowledge was limited to counting and arithmetical operations, scientific knowledge was more advanced than mathematical knowledge; for example, potters possessed practical knowledge of the behavior of clay and fire, and, although they may not have had explanations for the phenomena of their crafts, they toiled without any 2 systematic science of materials or selfconscious
The Bridge Between The Continuous And The Discrete Via Original Sources
"... this paper we summarize the story told through original sources from our chapter on the relationship between the continuous and the discrete, hinging historically on two interlocking themes: the search for formulas for sums of numerical powers, and Euler's development of his summation formula i ..."
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this paper we summarize the story told through original sources from our chapter on the relationship between the continuous and the discrete, hinging historically on two interlocking themes: the search for formulas for sums of numerical powers, and Euler's development of his summation formula in relation to sums of infinite series
On Two Complementary Types of Total Time Derivative in Classical Field Theories and Maxwell’s Equations
, 2005
"... Close insight into mathematical and conceptual structure of classical field theories shows serious inconsistencies in their common basis. In other words, we claim in this work to have come across two severe mathematical blunders in the very foundations of theoretical hydrodynamics. One of the defect ..."
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Close insight into mathematical and conceptual structure of classical field theories shows serious inconsistencies in their common basis. In other words, we claim in this work to have come across two severe mathematical blunders in the very foundations of theoretical hydrodynamics. One of the defects concerns the traditional treatment of time derivatives in Eulerian hydrodynamic description. The other one resides in the conventional demonstration of the socalled Convection Theorem. Both approaches are thought to be necessary for crossverification of the standard differential form of continuity equation. Any revision of these fundamental results might have important implications for all classical field theories. Rigorous reconsideration of time derivatives in Eulerian description shows that it evokes Minkowski metric for any flow field domain without any previous postulation. Mathematical approach is developed within the framework of congruences for general 4dimensional differentiable manifold and the final result is formulated in form of a theorem. A modified version of the Convection Theorem provides a necessary crossverification for a reconsidered differential form of continuity equation. Although the approach is developed for onecomponent (scalar) flow field, it can be easily generalized to any tensor field. Some possible implications for classical electrodynamics are also explored.