Geometry

The article discusses the formula on the limitation imposed by a triangle ABC on an ellipse with foci P and Q. Topics include defining the Gregorian vertex of polygon, and discussions on the ellipse equation, hyperbola analysis which is similar to ellipse, and the parabola wherein four tangents outline a quadrilateral ABCD. Generalisations of the ellipse to triangle ABC are also presented.

Οι έννοιες της Ευκλείδειας Γεωμετρίας προκαλούν δυσκολίες στην κατανόησή τους από τους μαθητές. Στην παρούσα εργασία προτείνεται ένα εκπαιδευτικό σενάριο για τη διδασκαλία του τετραγώνου στην Ευκλείδεια Γεωμετρία. Μετά από μία σύντομη εισαγωγή, αναφέρονται ο τίτλος και η ταυτότητα του σεναρίου. Στη συνέχεια παρατίθεται το σκεπτικό του σεναρίου, όπου αναπτύσσονται οι καινοτομίες και η προστιθέμενη αξία του, καθώς και τα γνωστικά προβλήματα των μαθητών. Ακολουθεί το πλαίσιο εφαρμογής του σεναρίου και η ανάλυση των τεσσάρων δραστηριοτήτων, οι οποίες υλοποιούνται με χρήση των Τεχνολογιών Πληροφορικής και Επικοινωνιών και συγκεκριμένα του λογισμικού Χελωνόκοσμος. Τέλος, αναφέρονται προτάσεις για την επέκταση του παρόντος σεναρίου.

Oppositional geometry gives a mathematical model of oppositional phenomena through “oppositional structures” (logical squares, hexagons, cubes, …). Its so far known formal entities, the backbone of which are the “oppositional bi-simplexes (and poly-simplexes) of dimension m”, are distributed into three families (the alpha-, beta- and gamma-structures). However, some recent studies by different authors exhibit strange structures, notably strange variations of the notion of “oppositional hexagon” (or “logical hexagon”). In this paper we show that inside the oppositional tetrahexahedron, i.e. the beta3-structure (discovered in 1968 and rediscovered in 2008) – a 3D solid made of a logical cube and six logical “strong hexagons”, containing 14 vertices and 36 implication arrows – there are in fact C(6|14) = 30030 strange hexagons, which we call “hybrid hexagons”. In this paper, through a systematic study of those among them which have as invariant property a regular perimeter made of alternated arrows (henceforth “arrow-hexagons”), we show that they divide into a much smaller number of families, nine, each containing several isomorphic instances of the same oppositional pattern. An interesting result seems to be that when seen from the viewpoint of their mutual transformations (i.e. moving from one to another kind of arrow-hexagon, just by exchanging one of its 6 vertices with one among the remaining 14-6=8 vertices of the tetrahexahedron), these arrow-hexagonal patterns taken as points can be displayed into a new kind of 3D structure. The latter, by putting into order these points (each representing a family of arrow-hexagons), gives some kind of morphogenetic cartography of the arrow-hexagons of the beta3-structure. As we will argue, since several arrow-hexagons play the role of “attractors”, there are reasons for thinking that such a cartography could be very meaningful in the future for modelling “oppositional dynamics”, that is the systematic formal study of the situations where a given complex oppositional structure sees its shape change within time.

This paper will discuss results that arise in specific configurations pertaining to invariance under isoconjugation. The results lead to revolutionary theorems and crucial results in both plane and projective geometry. This paper also includes the first proof of an open problem proposed by Antreas Hitzapolakis to start with, and ends on a note demonstrating the power of the theorem developed in section 3.

Flaps can be detached from a thin film glued on a solid substrate by tearing and peeling. For flat substrates, it has been shown that these flaps spontaneously narrow and collapse in pointy triangular shapes. Here we show that various shapes, triangular, elliptic, acuminate, or spatulate, can be observed for the tears by adjusting the curvature of the substrate. From combined experiments and theoretical models, we show that the flap morphology is governed by simple geometric rules.

We start by recalling how the main answers to the traditional metaphysical question “why is there something rather than nothing?” tend to subscribe to a “desert metaphysics”: nothingness is mysteriously felt as metaphysically more natural than existence. This prejudice can be traced back to some fundamental proto-structuralist ideas of Gestalt-theory. So we introduce “oppositional geometry”, a recent candidate to the status of new branch of mathematics that generalizes the traditional theory of opposition and seemingly re-launches the structuralist program. We show that, convergent with it, there is currently the emergence of a promising “neo-Gestaltian approach” to psychology. Using this toolkit we propose to derive the notion of “structural metaphysics”. As an example, an interesting structuralist alternative to the aforementioned desert metaphysics can be introduced by the notion of “jungle metaphysics”, inspired by Meinong’s and Sylvan’s famous “jungle ontology”. Such a structuralist view suggests that the (meta)physical mystery of the apparent “ek-sistence” of something out of a presupposed nothingness is possibly due, rather than to a “creation”, to a systematic perceptual illusion, combinatorially rooted in some hidden, holist structural hyper-symmetry of the (meta)physical. In recent times at least three authors have powerfully developed such a jungle view, by some kind of paradoxical looking (but cogent) “extreme ontologies”: the quantum physicist H. Everett (with his “many-worlds interpretation of quantum mechanics”), the logician and analytical philosopher D. Lewis (with his logic for counterfactuals and the “modal realism” it implies) and the continental philosopher E. Severino (with his hetero-Parmenidian theory of the “eternity of all beings”). After putting them in perspective we end discussing possible future generalizations and deepenings of these ideas.

In this note, we analyze the Liar Paradox. We re-write the problem in order to
properly include it in the body of Science and prove that the problem is an
allurement to show the complexity of the human mind in the same sense that the
Sorites Paradox is an allurement to show the complexity of the human verbal
expression.

Geometry is an a priori science. However, its apriority is saddled with problems.
The aim of this paper will be to show 1) how Kant understands that the contents
of geometry are synthetic a priori judgments in the Critique of Pure Reason, and 2) if it’s
still relevant to study Kant’s theory of geometry after the challenges posed by non-
Euclidian theories of space. With respect to point 1: Kant understands geometry as the
discipline that objectifies the pure intuition of space. Every geometric concept is built
upon the pure intuition of space through a synthetic ostensive process. Furthermore,
the pure intuition of space is the form of external experiences. Thus, geometry and
external phenomena share a common ground – pure space. This common ground is
what provides an answer to the question of the possibility of mathematics as a universal
and a priori science. With respect to point 2: the relevance of studying Kant’s
theory of geometry lies not only in the fact that geometry can serve as an example
to philosophy based on the fact that it establishes its propositions a priori, but also
because the object-study of geometry – the pure intuition of space– forces the reader
to review Kant’s thoughts about sensibility and its relation to space. The analysis
of Kant’s theory of geometry then amounts to studying Kant’s theory of sensibility.

We present basic notions of Pier's system of geometry in an intuitive way and from a heuristic point of view, trying to focus on the "essence" of Italian mathematician's constructions. However, it is done without the detriment for formal precision of the presentation, we hope. We also point to some important metamathematical dependencies between Pieri's and Hilbert's approach to Euclidean geometry.

In this paper we give probably an exhaustive analysis of the geometry of solids which was sketched by Tarski in his short paper "Foundations of the geometry of solids" (1929).

We show that in order to prove theorems stated in the paper one must enrich Tarski's theory with a new postulate asserting that the universe of discourse of the geometry of solids coincides with arbitrary mereological sums of balls, i.e. with solids. We show that once having adopted such a solution one of Tarski's postulates can be omitted. We also prove that the equivalence of some important postulates is not provable in any theory whose domain contains objects other than solids. Moreover, we show that the concentricity relation as defined by Tarski must be transitive in the largest class of structures satisfying Tarski's axioms.

We build a model (in three-dimensional Euclidean space) of the theory of so called T$^\star$-structures and present the proof of the fact that this is the only (up to isomorphism) model of the theory.

This article is devoted to the problem of ontological foundations of three-dimensional Euclidean geometry. Starting from Bertrand Russell’s intuitions concerning the sensual world we try to show that it is possible to build a foundation for pure geometry by means of the so called regions of space.

It is not our intention to present mathematically developed theory, but rather demonstrate basic assumptions, tools and techniques that are used in construction of systems of point-free geometry and topology by means of mereology (resp. Boolean algebras) and Whitehead-like connection structures. We list and briefly analyze axioms for mereological structures, as well as those for connection structures. We argue that mereology is a good tool to model so called spatial relations. We also try to justify our choice of axioms for connection relation.

Finally, we briefly discuss two theories: Grzegorczyk’s point-free topology and Tarski’s geometry of solids.