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Acta Mathematica Hungarica
Volume/Issue:
Volume 10: Issue 1-2
DOI:
https://doi.org/10.1007/bf02063295
Author:
A. Heppes
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Acta Mathematica Hungarica
Volume/Issue:
Volume 12: Issue 1-2
DOI:
https://doi.org/10.1007/bf02066683
Author:
A. Heppes
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Periodica Mathematica Hungarica
Volume/Issue:
Volume 53: Issue 1-2
DOI:
https://doi.org/10.1007/s10998-006-0029-9
Author:
A. Heppes

Summary  

The problem of covering a circle, a square or a regular triangle with \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $n$ \end{document} congruent circles of minimum diameter (the {\it circle covering} problem) has been investigated by a number of authors and the smallest diameter has been found for several values of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $n$ \end{document} . This paper is devoted to the study of an analogous problem, the {\it diameter covering} problem, in which the shape and congruence of the covering pieces is relaxed and -- invariably -- the maximal diameter of the pieces is minimized. All cases are considered when the solution of the first problem is known and in all but one case the diameter covering problem is solved.

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Periodica Mathematica Hungarica
Volume/Issue:
Volume 36: Issue 2-3
DOI:
https://doi.org/10.1023/a:1004681709417
Author:
A. Heppes
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Periodica Mathematica Hungarica
Volume/Issue:
Volume 36: Issue 2-3
DOI:
https://doi.org/10.1023/a:1004633826255
Author:
A. Heppes
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Periodica Mathematica Hungarica
Volume/Issue:
Volume 45: Issue 1-2
DOI:
https://doi.org/10.1023/a:1022345929586
Author:
A. Heppes

Abstract  

In the present paper lattice packings of open unit discs are considered in the Euclidean plane. Usually, efficiency of a packing is measured by its density, which in case of lattice packings is the quotient of the area of the discs and the area of the fundamental domain of the packing. In this paper another measure, the expandability radius is introduced and its relation to the density is studied. The expandability radius is the radius of the largest disc which can be used to substitute a disc of the packing without overlapping the rest of the packing. Lower and upper bounds are given for the density of a lattice packing of given expandability radius for any feasible value. The bounds are sharp and the extremal configurations are also presented. This packing problem is related to a covering problem studied by Bezdek and Kuperberg [BK97].

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Acta Mathematica Hungarica
Volume/Issue:
Volume 7: Issue 3-4
DOI:
https://doi.org/10.1007/bf02020535
Author:
A. Heppes
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Acta Mathematica Hungarica
Volume/Issue:
Volume 7: Issue 3-4
DOI:
https://doi.org/10.1007/bf02020540
Author:
A. Heppes
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Acta Mathematica Hungarica
Volume/Issue:
Volume 91: Issue 3
DOI:
https://doi.org/10.1023/a:1010623307014
Author:
A. Heppes
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Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 36: Issue 3-4
DOI:
https://doi.org/10.1556/sscmath.36.2000.3-4.15
Author:
A. Heppes

The main goal of this paper is to establish the long-sta di g conjecture that in the Euclidea plane no arrangeme t of discs of radius 1 and .2 -1 ca have larger packi g de sitythan that of the set of incircles of the semiregular (Archimedean)tessellation (8, 8, 4)

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