Math Forum Internet News

Volume 3, Number 34

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24 August 1998                                     Vol. 3, No. 34


Isohedral Tilings | maths online - Univ. of Vienna | MATHDI - ZDM

                  by Doris Schattschneider

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 

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, 

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.

The site also lists and describes math links that emphasize 
interactive learning material.


   Online International Reviews on Mathematical Education 


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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