کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
9662416 | 698668 | 2005 | 47 صفحه PDF | دانلود رایگان |
عنوان انگلیسی مقاله ISI
The topological structure of fractal tilings generated by quadratic number systems
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کلمات کلیدی
موضوعات مرتبط
مهندسی و علوم پایه
مهندسی کامپیوتر
علوم کامپیوتر (عمومی)
پیش نمایش صفحه اول مقاله
![عکس صفحه اول مقاله: The topological structure of fractal tilings generated by quadratic number systems The topological structure of fractal tilings generated by quadratic number systems](/preview/png/9662416.png)
چکیده انگلیسی
Let α be a root of an irreducible quadratic polynomial x2 + Ax + B with integer coefficients A, B and assume that α forms a canonical number system, i.e., each x â â¤[α] admits a representation of the shape x=a0+a1α+â¯+ahαh,with ai â {0, 1,â¦,|B| - 1}. It is possible to associate a tiling to such a number system in a natural way. If 2A < B + 3, then we show that the fractal boundary of the tiles of this tiling is a simple closed curve and its interior is connected. Furthermore, the exact set equation for the boundary of a tile is given. If 2A ⥠B + 3, then the topological structure of the tiles is quite involved. In this case, we prove that the interior of a tile is disconnected. Furthermore, we are able to construct finite labelled directed graphs which allow to determine the set of “neighbours” of a given tile T, i.e., the set of all tiles which have nonempty intersection with T. In a next step, we give the structure of the set of points, in which T coincides with L other tiles. In this paper, we use two different approaches: geometry of numbers and finite automata theory. Each of these approaches has its advantages and emphasizes different properities of the tiling. In particular, the conjecture in [1], that for A â 0 and 2A < B + 3 there exist exactly six points where T coincides with two other tiles, is solved in these two ways in Theorems 6.6 and 10.1.
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Computers & Mathematics with Applications - Volume 49, Issues 9â10, May 2005, Pages 1439-1485
Journal: Computers & Mathematics with Applications - Volume 49, Issues 9â10, May 2005, Pages 1439-1485
نویسندگان
Shigeki Akiyama, J.M. Thuswaldner,