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24 August 1998 Vol. 3, No. 34 THE MATH FORUM INTERNET NEWS Isohedral Tilings | maths online - Univ. of Vienna | MATHDI - ZDM CATALOG OF ISOHEDRAL TILINGS BY SYMMETRIC POLYGONAL TILES by Doris Schattschneider http://mathforum.org/dynamic/one-corona/ In tilings of the Euclidean plane by a single polygon, where three or more polygons meet at each vertex of each polygon, such a tiling is isohedral if and only if each polygon is "surrounded in the same way," or, more technically, the centered coronas of tiles are pairwise congruent. When the polygon tile is asymmetric, the proof of the result is a consequence of the local theorem for tilings, a general result that holds for tilings in d-dimensional space, so the result needs to be verified only for symmetric polygons. The proof is case-by-case, constructing all possible tilings with a single symmetric polygon tile that satisfies the condition on pairwise congruent coronas. All are shown to be isohedral. One consequence of the proof is the production of a complete catalog of isohedral tilings by symmetric polygon tiles. About one-quarter of the 42 different types of tilings are rigid or unique up to similarity. The remaining tilings are highly flexible, so that the shape of the tile can be deformed in a continuous manner to assume a variety of distinct shapes. This site allows you to view the whole catalog, and to explore the deformation of the flexible tilings interactively, using JavaSketchpad conversions of sketches constructed using The Geometer's Sketchpad. You can also download the Sketchpad files to your computer if you wish to explore them further. From the article "One Corona is Enough for the Euclidean Plane" by Doris Schattschneider and Nikolai Dolbilin, published in "Quasicrystals and Discrete Geometry," J. Patera, editor, The Fields Institute for Research in Mathematical Sciences Monograph Series, Vol. 10, AMS, Providence, RI, 1998, pp. 207-246. -|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|- MATHS ONLINE - University of Vienna, Austria http://www.univie.ac.at/future.media/moe/ A mathematics learning site on the Web, in English and German. Its gallery of multimedia learning units (Java applets), on math subjects for secondary school, high school, college, and university, offers guided exercises, solutions, and curricular applications of the exercises. Topics include: - The drawing plane and its coordinate system - Variables, terms, formulae, and identities - Equations (equivalence transformations, quadratics) - Analytic geometry (slope of a straight line, cycloids, logarithmic spirals) - Functions (function and graph, simple polynomial functions, functions containing negative numbers) - Trigonometric functions - Trigonometry (triangle and law of sines) - Differentiation (puzzles of matching functions with derivatives, nowhere differentiable functions) - Power series - Probability and statistics (Gauss distribution) - Integration (intuitively understanding the integral) - Fourier series - Model-building and simulation (Conway's game of Life, HighLife) Maths online also provides selected online tools, each in its own browser window so that it may be used with other pages. These include Java scientific calculators; tools for plotting 2D and 3D graphs, differentiating and integrating functions, and solving equations; and ways to perform computations using Mathematica. http://www.univie.ac.at/future.media/moe/onlinewerkzeuge.html The site also lists and describes math links that emphasize interactive learning material. http://www.univie.ac.at/future.media/moe/einzelthemen.html -|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|- MATHDI (MATHEMATICAL DIDACTICS) Online International Reviews on Mathematical Education http://www.emis.de/MATH/DI.html MATHDI is the Web version of International Reviews on Mathematical Education (the University of Karlsruhe's Zentralblatt fur Didaktik der Mathematik, ZDM). Its comprehensive bibliographic database of over 70,000 references for literature in math education and related fields includes approximately 500 volumes and other serials published worldwide from 1976 to the present. Subjects covered span research in math education, math teaching methodologies, math instruction from elementary school to university teaching and teacher training, elementary math and its applications, computer science education, and basic pedagogical and psychological issues for mathematics and science education. References are searchable by author, title, global index, source, ZDM classification, and publication year, and are available in full text. Citations, the majority with abstracts, are in English and German. Unregistered users may retrieve abstracts of up to three search results by e-mail, post, ariel, or fax, and can subscribe for full access to the online service or the print or CD-ROM versions of the database. -|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|- CHECK OUT OUR WEB SITE: The Math Forum http://mathforum.org/ Ask Dr. Math http://mathforum.org/dr.math/ Problems of the Week http://mathforum.org/pow/ Internet Resources http://mathforum.org/~steve/ Join the Math Forum http://mathforum.org/join.forum.html Send comments to the Math Forum Internet Newsletter editors _o \o_ __| \ / |__ o _ o/ \o/ __|- __/ \__/o \o | o/ o/__/ /\ /| | \ \ / \ / \ /o\ / \ / \ / | / \ / \ |

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