Volume 3, Number 34
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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.
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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
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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.
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