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The line p is the polar of the point P , and each of the points on the line p has a corresponding polar that passes through P . As a point traverses the line p, its corresponding polar rotates around P .
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The symmetries between points and lines in planar projective geometry and between points and planes in solid projective geometry are striking features of these geometries that were extensively discussed during the nineteenth century under the labels “duality” or “reciprocity.” The aims of this article are, first, to provide a systematic analysis of...
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... Duality is a multi-faceted, fundamental phenomenon, which takes a central place in the philosophy of mathematics (Krömer and Corfield, 2014, Catren and Cukierman, 2022, Krömer and Haffner, 2024. The notion of duality possesses a "fine structure", namely, we recognize: i) Stone duality, which establishes that a category of "algebraic objects" (Boolean algebras) is the categorical dual of a category of "geometrical" objects (Stone spaces) (Awodey and Forssell, 2013;Gehrke, 2016); ii) variational duality (Mond and Hanson, 1967); iii) projective duality, which emerges from the symmetries between points and lines in planar projective geometry and between points and planes in solid projective (Eder, 2021, Tevelev, 2005; iv) Poincare duality, that links the dual notions of homology groups and cohomology groups for manifolds (Spivak, 1967); v) de Morgan logical duality (Gastaldi, 2024); vi) Fourier duality (Dutkay and Jorgensen, 2012); vii) linear programming duality (Bachem and Kern, 1992). ...
... Now we propose the procedure enabling conversion of mathematical schemes into the bicolored, complete graphs. We illustrate the idea with the set of theorems related to the projective geometry; however, the suggested scheme enables conversion of any (finite or infinite) set of propositions into the graph (Eder, 2021). In projective geometry, duality is a logical formalization of the symmetry of the roles fulfilled by points and lines in the definitions and theorems of projective planes. ...
... In projective geometry, duality is a logical formalization of the symmetry of the roles fulfilled by points and lines in the definitions and theorems of projective planes. In its simplest meaning the projective duality is interpreted as follows: in a projective plane a statement involving points, lines and incidence between them that is obtained from another such statement by interchanging the words "point" and "line" and making appropriate grammatical corrections, is considered the plane dual statement of the first statement (Eder, 2021). The plane dual statement of "Two points are on a unique line" is "Two lines meet at a unique point". ...
The meta-mathematic approach to the mathematical duality is introduced. Systems of mathematical theorems/propositions related each to other by the mutually exclusive relations of “duality” and “non-duality” are addressed. Theorems/propositions are considered as the vertices of the graph, and the relations of duality appear as the links connecting the vertices. Thus, the bi-colored, complete graph emerges. The coloring procedure is exemplified with the theorems of projective geometry. It is demonstrated, that that the graph built of the quartet of vertices/theorems of projective geometry, which contains no mono-colored triangle is possible. The monochromatic triangle will necessarily appear in the graph containing five vertices/theorems of projective geometry. The emerging graph is different from the traditional Ramsey graph. The graph representation is easily generalized for any system (finite or infinite) of theorems/propositions in which the relations of duality are established.
... Eder proposed that there is an obvious symmetry relationship between points and lines in projective geometry. To this end, he systematically analyzed the duality of point and line from the perspective of three-dimensional, and proposed new thinking about duality [9]. Santos et al. studied the laws of projective geometry in the context of quantum mechanics. ...
In recent years, education has been paid more and more attention by the society, and the concept of “education without discrimination” has gradually taken root in the hearts of the people. Special education is a form of education for special groups, which embody the fairness of education. Different from the conventional education model, special education often pays more attention to the physical and mental development of special populations, so the curriculum setting method of special education major is also different from the general method. Under this circumstance, how to carry out reasonable curriculum setting has become the core problem that needs to be solved in the reform of special education curriculum. Projective geometry is one of the methods of studying graph transformation, and its core is the principle of projective transformation invariance. Under the guidance of this theory, curriculum reform also presents many invariable characteristics. Based on this, this paper proposed a special education professional curriculum setting method integrating projective geometry, aiming to reform the special education professional curriculum setting strategy by using the invariant theory. In the evaluation of curriculum setting, the article analyzed the effect of new curriculum setting methods on special groups from different dimensions, and preliminarily formulated the initial special education curriculum. It can be concluded from the article in the evaluation grades that with the blessing of the new curriculum setting method, the student's health evaluation reached 2.7, a year-on-year increase of 42.1%. This fully shows that in the course of special education curriculum setting, projective geometry can provide novel ideas and directions for new curriculum setting methods.
In Principii di Geometria (1889b) and ‘Sui fondamenti della Geometria’ (1894) Peano offers axiomatic presentations of projective geometry. There seems to be a tension in Peano's construction of geometry in these two works: on the one hand, Peano insists that the basic components of geometry must be founded on intuition, and, on the other, he advocates the axiomatic method and an abstract understanding of the axioms. By studying Peano’s empiricist remarks and his conception of the notion of mathematical proof, and by discussing his critique of Segre’s foundation of hyperspace geometry, I will argue that the tension can be dissolved if these two seemingly contradictory positions are understood as compatible stages of a single process of construction rather than conflicting options.
To study Geometry and Representation from a theoretical point of view, it is certainly necessary to refer critically to the historical-anthropological climate of their genesis. This training approach allows to see and understand their evolution/impact for new approaches to the genesis of architectural/engineering artefacts, both from a configurative and structural point of view. On the other hand, if we consider the typical primitive/intuitive approaches of Representation, up to its rigorous elaborations based on a consistent knowledge of Optics and Geometry, it is possible to recognize the strong links between artistic experience, mathematical contribution and scientific elaboration. It is therefore possible to offer a broad overview of the “state of the art” relating to critical sector studies, conducted both in Italy and abroad, to underline: how awareness in the multiple fields of Geometry is expressed in the methods and process of realization of architecture and engineering, from conception to its realization;how Representation stands as a means between theory and construction. how awareness in the multiple fields of Geometry is expressed in the methods and process of realization of architecture and engineering, from conception to its realization; how Representation stands as a means between theory and construction. In this sense we focus on Topology and its genesis, as an area of Geometry that can be concretized in Structural Optimization, in order to verify the promising results that these procedures ensure in terms of reducing the use of material and design iterations, without neglecting the architectural/engineering quality, also in configurative terms. For this reason, one of the main factors in the growing popularity of this topic is the development, in computational terms, of modern computers, which allow to reliably solve complex analyzes based on FEM (Finite Element Analysis).
