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Register a new userA Topological Approach to Creating any Pulli Kolam, an Artform from Southern India
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Pulli kolam is a ubiquitous art form in south India. It involves drawing a line looped around a collection of dots (pullis) place on a plane such that three mandatory rules are followed: all line orbits should be closed, all dots are encircled and no two lines can overlap over a finite length. The mathematical foundation for this art form has attracted attention over the years. In this work, we propose a simple 5-step topological method by which one can systematically draw all possible kolams for any number of dots N arranged in any spatial configuration on a surface.
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Abstract axiomatic formulation of mathematical structures are extensively used to describe our physical world. We take here the reverse way. By making basic assumptions as starting point, we reconstruct some features of both geometry and topology in a fully operational manner. Curiously enough, primitive concepts such as points, spaces, straight lines, planes are all defined within our formalism. Our construction breaks down with the usual literature, as our axioms have deep connection with nature. Besides that, we hope this operational approach could also be of pedagogical interest.
The first part of this article intends to present the role played by Thom in diffusing Smales ideas about immersion theory, at a time (1957) where some famous mathematicians were doubtful about them: it is clearly impossible to make the sphere inside out! Around a decade later, M. Gromov transformed Smales idea in what is now known as the h-principle. Here, the h stands for homotopy.Shortly after the astonishing discovery by Smale, Thom gave a conference in Lille (1959) announcing a theorem which would deserve to be named a homological h-principle. The aim of our second part is to comment about this theorem which was completely ignored by the topologists in Paris, but not in Leningrad. We explain Thoms statement and answer the question whether it is true. The first idea is combinatorial. A beautiful subdivision of the standard simplex emerges from Thoms article. We connect it with the jiggling technique introduced by W. Thurston in his seminal work on foliations.
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This article is an updated version of an entry of the Encyclopedia of Mathematics Education (2018). In the same time, it is the seed of the HAL collection DAD-MULTILINGUAL, constituted by the translation of this entry in various languages.The documentational approach to didactics is a theory in mathematics education. Its first aim is to understand teachers professional development by studying their interactions with the resources they use and design in/for their teaching. In this text we briefly describe the emergence of the approach, its theoretical sources, its main concepts and the associated methodology. We illustrate these aspects with examples from different research projects. This synthetic presentation is written for researchers, but also for non-specialists (e.g. master students) interested in a first discovery of the documentational approach.
We prove that any cyclic quadrilateral can be inscribed in any closed convex $C^1$-curve. The smoothness condition is not required if the quadrilateral is a rectangle.
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