Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 822.52004
Autor: Elekes, G.; Erdös, Paul
Title: Similar configurations and pseudo grids. (In English)
Source: Böröczky, K. (ed.) et al., Intuitive geometry. Proceedings of the 3rd international conference held in Szeged, Hungary, from 2 to 7 September, 1991. Amsterdam: North-Holland, Colloq. Math. Soc. János Bolyai. 63, 85-104 (1994).
Review: Given a finite set P of points in the plane, it is of interest to find another, n-point, set A which contains many similar or homothetic copies of P. When P is a square or equilateral triangle, it is easily seen that a suitable regular grid achieves c · n2 similar copies and c · n3/2 homothetic copies.
Such constructions might appear to depend on the fact that the square and equilateral triangle tile the plane in a lattice tiling. The authors show that this is not the case. If P is a triangle, or can be represented in the complex plane by a set of algebraic complex numbers {\xii}, then there exist ``pseudogrids'' containing c · n2 similar copies or c · n3/2 homothetic copies. The points of these pseudogrids are the values of polynomials in {\xii} with certain bounds on coefficients and degree. Even if P cannot be so represented, O(n2) similar copies and O(n3/2) homothetic copies may be achieved.
The results for homothetic copies are extended to higher dimensions. If Rd it is possible to construct pseudogrids which contain c · n(d+1)/d homothetic copies of a pattern P with algebraic coordinates, or O(n(d+1)/d) homothetic copies of an arbitrary pattern.
Reviewer: R.Dawson (Halifax)
Classif.: * 52A37 Other problems of combinatorial convexity
52C20 Tilings in 2 dimensions (discrete geometry)
Keywords: similar configurations; homothetic configurations; pseudogrids
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