During my high-school studies, I did a Math Project, titled "Crystallographic Restriction Theorem in the Euclidean Plane", under the supervision of a Technion Math Ph.D. student. Originally the project has been written in Hebrew using the LaTeX2ε document preparation system. Later the project has been translated to English by me with the help of the supervisor, and shortened. The math project was presented at the "Israeli Young Scientist Contest 2002" and at the Intel International Science and Engineering Fair 2002, and in both contests the project has received awards, including a honorable mention from the American Mathematics Society.
The Crystallographic Restriction Theorem in the Euclidean plane lists all possible types of tessellation groups in the Euclidean plane. There are exactly 17 isomorphism equivalence-classes of tessellation groups. Using the broadest definition of a tessellation group, we present a completely self-contained exposition of the subject, the novelty being our usage of modern techniques in the analysis of this classical result from a synthetic point of view.
A tessellation group is subgroup of Isom (E2) possessing a connected compact, "tile" P, having the following properties: the G-translates of the tile cover the plane and any two such translates have disjoint interiors.
This definition is not easily used. Therefore we use topological geometric considerations to find an equivalent definition, which is more readily applied. First, the tile is replaced with a "standard" one associated with the tessellation group. Second, the tessellation group is defined as a subgroup of a symmetry-group of a lattice, which is shown to have the structure of a semidirect-product. As a result, our proof is easily applied to any given tiling.
The techniques and language used in this project are easily extended to other cases of interest (spherical geometry, higher-dimensional Euclidean etc.).
All downloads are RAR archives.
*Note: IsomDemon is a graphical DOS program (compiled with Borland's BGI)
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