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In this snapshot, we will consider the problem of finding the number of solutions to a given system of polynomial equations. This question leads to the theory of Newton polytopes and Newton-Okounkov bodies of which we will give a basic notion.
Submitted by Bianca Violet on
Was haben das Mineral Pyrit, der fünfte platonische K̈örper (genannt Dodekaeder) und ein Polynom vom Grad 16 gemeinsam? Dieser Artikel untersucht diesen Zusammenhang mit Hilfe der kostenlosen Software SURFER der Plattform IMAGINARY - open mathematics. Daraus entstehen faszinierende Bilder, die zeigen, wie ein Würfel sich nacheinander in einen Dodekaeder, einen Rhombendodekaeder und einen Oktaeder verwandelt, alles dank einer einzigen Formel. Es wird ein Überblick über die Ideen und die Mathematik hinter diesen Visualisierungen gegeben.
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This snapshot is about zero-dimensional symmetry. Thanks to recent discoveries we now understand such symmetry better than previously imagined possible. While still far from complete, a picture of zero-dimensional symmetry is beginning to emerge.
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Platonic solids, Felix Klein, H. S. M. Coxeter and a flap of a swallowtail: The five Platonic solids tetrahedron, cube, octahedron, icosahedron and dodecahedron have always attracted much curiosity from mathematicians, not only for their sheer beauty but also because of their many symmetry properties. In this snapshot we will start from these symmetries, move on to groups, singularities, and finally find the connection between a tetrahedron and a “swallowtail”.
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We discuss certain open problems in the context of arrangements of lines in the plane.
Submitted by IMAGINARY on
This Snapshot is about the generalization of “>” from ordinary numbers to so-called fields. At the end, I will touch on some ideas in recent research.
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Friezes have occured as architectural ornaments for many centuries. In this snapshot, we consider the mathematical analogue of friezes as introduced in the 1970s by Conway and Coxeter. Recently, infinite versions of such friezes have appeared in current research. We are going to describe them and explain how they can be classified using some nice geometric pictures.
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Leonhard Euler (1707–1783) – one of the greatest mathematicians of the eighteenth century and of all times – often corresponded with a friend of his, Christian Goldbach (1690–1764), an amateur and polymath who lived and worked in Russia, just like Euler himself. In a letter written in June 1742, Goldbach made a conjecture – that is, an educated guess – on prime numbers:
“Es scheinet wenigstens, dass eine jede Zahl, die größer ist als 2, ein aggregatum trium numerorum primorum sey.”
Submitted by IMAGINARY on
The algebraic geometer Duco van Straten, Johannes Gutenberg-University Mainz, Germany gives a beautiful short introduction into the field of algebraic geometry - from lines, curves to cusps. It shows many pictures and also links its content to historical events and applications.
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SURFER is a program designed to make everybody feel like a mathematician. The program is a bridge between art and math. Everybody can participate in the dialogue between algebraic equations and pictures of algebraic surfaces in an interactive and aesthetic way. In this paper we will introduce the program and its potential in math art, education and communication. The program was originally developed for the IMAGINARY exhibition, a project by the Mathematisches Forschungsinstitut Oberwolfach.