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Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands and are globally symmetric. This paper considers the symmetric tangling of multiple 2-periodic honeycomb networks. We do this using a constructive methodology borrowing elements of graph theory, low-dimensional topology and geometry. The result is a wide-ranging enumeration of symmetric tangled honeycomb networks, providing a foundation for their exploration in both the chemistry lab and the geometers toolbox.
Conventional embeddings of the edge-graphs of Platonic polyhedra, {f,z}, where f,z denote the number of edges in each face and the edge-valence at each vertex, respectively, are untangled in that they can be placed on a sphere (S-2) such that distinct edges do not intersect, analogous to unknotted loops, which allow crossing-free drawings of S-1 on the sphere. The most symmetric (flag-transitive) realizations of those polyhedral graphs are those of the classical Platonic polyhedra, whose symmetries are *2fz, according to Conway's two-dimensional (2D) orbifold notation (equivalent to Schonflies symbols I-h, O-h, and T-d). Tangled Platonic {f,z} polyhedra-which cannot lie on the sphere without edge-crossings-are constructed as windings of helices with three, five, seven,... strands on multigenus surfaces formed by tubifying the edges of conventional Platonic polyhedra, have (chiral) symmetries 2fz (I, O, and T), whose vertices, edges, and faces are symmetrically identical, realized with two flags. The analysis extends to the "theta(z)" polyhedra, {2,z}. The vertices of these symmetric tangled polyhedra overlap with those of the Platonic polyhedra; however, their helicity requires curvilinear (or kinked) edges in all but one case. We show that these 2fz polyhedral tangles are maximally symmetric; more symmetric embeddings are necessarily untangled. On one hand, their topologies are very constrained: They are either self-entangled graphs (analogous to knots) or mutually catenated entangled compound polyhedra (analogous to links). On the other hand, an endless variety of entanglements can be realized for each topology. Simpler examples resemble patterns observed in synthetic organometallic materials and clathrin coats in vivo.
Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands and are globally symmetric. This paper considers the symmetric tangling of multiple 2-periodic honeycomb networks. We do this using a constructive methodology borrowing elements of graph theory, low-dimensional topology and geometry. The result is a wide-ranging enumeration of symmetric tangled honeycomb networks, providing a foundation for their exploration in both the chemistry lab and the geometers toolbox.
In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.
In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.
We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This generalizes the enumeration using Delaney--Dress combinatorial tiling theory of combinatorial classes of tilings to isotopy classes of tilings. To accomplish this, we derive an action of the mapping class group of the orbifold associated to the symmetry group of a tiling on the set of tilings. We explicitly give descriptions and presentations of semipure mapping class groups and of tilings as decorations on orbifolds. We apply this enumerative result to generate an array of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations that are commensurate with the threedimensional symmetries of the primitive, diamond, and gyroid triply periodic minimal surfaces, which have relevance to a variety of physical systems.
Have you already swiped or liked this morning? Have you taken part in a video conference at work, used or programmed a database? Have you paid with your smartphone on the way home, listened to a podcast, or extended the lending of books you borrowed from the library? And in the evening, have you filled out your tax return application on ELSTER.de on your tablet, shopped online, or paid invoices before you were tempted to watch a series on a streaming platform?
Our lives are entirely digitalized.
These changes make many things faster, easier, and more efficient. But keeping pace with these changes demands a lot from us, and not everyone succeeds. There are people who prefer to go to the bank to make a transfer, leave the programming to the experts, send their tax return by mail, and only use their smartphone to make phone calls. They don’t want to keep pace, or maybe they can’t. They haven’t learned these things. Others, younger people, grow up as “digital natives” surrounded by digital devices, tools, and processes. But does that mean they really know how to use them? Or do they also need digital education?
But what does successful digital education actually look like? Does it teach us how to use a tablet, how to google properly, and how to write Excel spreadsheets? Perhaps it’s about more than that. It’s about understanding the comprehensive change that has been taking hold of our world since it was broken down into digital ones and zeros and rebuilt virtually. But how do we learn to live in a world of digitality – with all that it entails, and to our benefit?
For the new issue of “Portal Wissen”, we looked around at the university and interviewed researchers about the role that the connection between digitalization and learning plays in the research of various disciplines. We spoke to Katharina Scheiter, Professor of Digital Education, about the future of German schools and had several experts show us examples of how digital tools can improve learning in schools. We also talked to computer science and agricultural researchers about how even experienced farmers can still learn a lot about their land and their work thanks to digital tools. We spoke to educational researchers who are using big data to analyze how boys and girls learn and what the possible causes for differences are. Education and political scientist Nina Kolleck, on the other hand, looks at education against the backdrop of globalization and relies on the analysis of large amounts of social media data.
Of course, we don’t lose sight of the diversity of research at the University of Potsdam. We learn, for example, what alternatives to antibiotics could soon be available. This magazine also looks at stress and how it makes us ill as well as the research into sustainable ore extraction.
A new feature of our magazine is a whole series of shorter articles that invite you to browse and read: from research news and photographic insights into laboratories to simple explanations of complex phenomena and outlooks into the wider world of research to a small scientific utopia and a personal thanks to research. All this in the name of education, of course. Enjoy your read!