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  • 1990-1994  (5)
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  • 1
    Electronic Resource
    Electronic Resource
    Tiling a polygon with parallelograms (1993)
    Springer
    Algorithmica 9 (1993), S. 382-397 
    Details
    ISSN: 1432-0541
    Keywords: Tiling ; NP-complete
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Given a simple polygon in the plane we devise a quadratic algorithm for determining the existence of, and constructing, a tiling of the polygon with parallelograms. We also show that any two parallelogram tilings can be obtained from one another by a sequence of “rotations,” and give a condition for the uniqueness of such a tiling. Three generalizations of this problem, that of tiling by a fixed set of triangles, a fixed set of trapezoids, or parallelogram tiling for polygonal regions with holes, are shown to be NP-complete.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01228510
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    How to take short cuts (1992)
    Springer
    Discrete & computational geometry 8 (1992), S. 251-264 
    Details
    ISSN: 1432-0444
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We present an elementary construction for adding short cuts on a rectifiable curve of the plane, so that any two points on the curve are linked by a path along the curve and short cuts, of total length at most a constant times their Euclidean distance. The total length of the short cuts is a constant times the perimeter of the curve.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02293048
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Intersecting random translates of invariant Cantor sets (1991)
    Springer
    Inventiones mathematicae 104 (1991), S. 601-629 
    Details
    ISSN: 1432-1297
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Given two Cantor setsX andY in [0, 1), invariant under the mapx→b x mod 1, the Hausdorff dimension of (X+t)∩Y is constant almost everywhere. WhenX,Y are defined by admissible digits in baseb, and more generally by sofic systems, we compute this dimension in terms of the largest Lyapunov exponent of a random product of matrices. The results are extended to higher dimensions and multiple intersections.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01245092
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Rigidity of planar tilings (1992)
    Springer
    Inventiones mathematicae 107 (1992), S. 637-651 
    Details
    ISSN: 1432-1297
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Aperturbation of a tiling of a region inR n is a set of isometries, one applied to each tile, so that the images of the tiles tile the same region. We show that a locally finite tiling of an open region inR 2 with tiles which are closures of their interiors isrigid in the following sense: any sufficiently small perturbation of the tiling must have only “earthquake”-type discontinuities, that is, the discontinuity set consists of straight lines and arcs of circles, and the perturbation near such a curve shifts points along the direction of that curve. We give an example to show that this type of rigidity does not hold inR n , forn〉2. Using rigidity in the plane we show that any tiling problem with a finite number of tile shapes (which are topological disks) is equivalent to a polygonal tiling problem, i.e. there is a set of polygonal shapes with equivalent tiling combinatorics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01231905
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    Rigidity of planar tilings (1993)
    Springer
    Inventiones mathematicae 112 (1993), S. 223-223 
    Details
    ISSN: 1432-1297
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01232432
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    Paper (German National Licenses)
    Fulltext
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