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# Optimal translative packing of homothetic triangles

Studia Scientiarum Mathematicarum Hungarica
Author: Janusz Januszewski

Januszewski, J. , A simple method of translative packing triangles in a triangle, Geombinatorics , 12(2) (2002), 61–68. MR 2003h :52020 Januszewski J. A simple method of

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# Packing a triangle with positive homothetical copies

Studia Scientiarum Mathematicarum Hungarica
Authors: Marek Lassak and Joanna Ściesińska
We consider packing a triangle with a number of equal positive homothetical copies. In particular, we show that every triangle can be packed with 7 copies of ratio
\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} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{2}{7}$$ \end{document}
, with 8 copies of ratio
\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} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{3}{{11}}$$ \end{document}
, and with 9 and 10 copies of ratio
\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} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{1}{4}$$ \end{document}
. All these ratios cannot be enlarged. We also present hypothetically best packings by greater number of copies.
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# Packing similar triangles into a triangle

Periodica Mathematica Hungarica
Author: Janusz Januszewski
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# Parallel packing and covering of an equilateral triangle with sequences of squares

Acta Mathematica Hungarica
Author: J. Januszewski

## Abstract

An equilateral triangle T e of sides 1 can be parallel covered with any sequence of squares whose total area is not smaller than 1:5. Moreover, any sequence of squares whose total area does not exceed
\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} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac{3} {4}(2 - \sqrt 3 )$$ \end{document}
(2 − √3) can be parallel packed into T e.
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# Nanometer wide ribbons and triangles by STM lithography of graphene

Nanopages
Authors: Gergely Dobrik, Levente Tapasztó, and László Biró

In this article, we present a method, which is suitable for the realization of graphene-based nanostructures by scanning tunneling microscopy lithography. Graphene nanoribbons (GNRs) and other more complicated nanoarchitectures like: GNR networks, triangular quantum-billiards, etc. can be created with controlled shape and crystallographic orientation. The cutting process operates with nanometer accuracy. After the lithography process the same STM tip is suitable for acquiring atomic resolution images on the nanoarchitectures created by STM lithography. The experimental observation of long range electronic superstructures indicates the long phase coherence length in graphene nanostructures even at room temperature.

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# Möbius and triangle semigroups

Periodica Mathematica Hungarica
Authors: Keunbae Choi and Yongdo Lim

## Abstract

The purpose of this paper is to describe the structures of the Möbius semigroup induced by the Möbius transformation group (ℝ, SL(2,ℝ)). In particular, we study stabilizer subsemigoups of Möbius semigroup via the triangle semigroup. In this work, we obtained a geometric interpretation of the least contraction coefficient function of the Möbius semigroup via the triangle semigroup and investigated an extension of stabilizer subsemigoups of the Möbius semigroup. Finally, we obtained a factorization of our stabilizer subsemigoups of the Möbius semigroup.

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# On Sequences of Nested Triangles

Periodica Mathematica Hungarica
Authors: Dan Ismailescu and Jacqueline Jacobs

## Summary

For a given triangle, we consider several sequences of nested triangles obtained via iterative procedures. We are interested in the limiting behavior of these sequences. We briefly mention the relevant known results and prove that the triangle determined by the feet of the angle bisectors converges in shape towards an equilateral one. This solves a problem raised by Trimble~[5].

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# A gallbladder with the “hidden cystic duct”: A brief overview of various surgical techniques of the Calot’s triangle dissection

Interventional Medicine and Applied Science
Authors: Jakub Kaczynski and Joanna Hilton

74 961 966 Sekimoto M, Tomita N, Tamura S, Ohsato H, Monden M: New retraction technique to allow better visualization of Calot’s triangle during

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# Reverse triangle inequality. Antinorms and semi-antinorms

Studia Scientiarum Mathematicarum Hungarica
Authors: Maria Moszyńska and Wolf-Dieter Richter

Ansari, A. H. and Moslehian, M. S. , Refinements of reverse triangle inequalities in inner product spaces, J. Inequalities in Pure and Applied Math. , 6(3) , Article 64 (2005). MR 2164305

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# Covering a planar domain with sets of small diameter

Periodica Mathematica Hungarica
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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