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• Author or Editor: Janusz Januszewski
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# Translative covering a convex body by its homothetic copies

Studia Scientiarum Mathematicarum Hungarica
Author: Janusz Januszewski

Any sequence of positive homothetic copies of a planar convex body C with total area not smaller than 6.5 times the area of C permits a translative covering of C.

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# Covering a Triangle with Sequences of its Homothetic Copies

Periodica Mathematica Hungarica
Author: Janusz Januszewski
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# Packing similar triangles into a triangle

Periodica Mathematica Hungarica
Author: Janusz Januszewski
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# A note on translative packing a triangle by sequences of its homothetic copies

Periodica Mathematica Hungarica
Author: Janusz Januszewski

## Summary

Every sequence of positive or negative homothetic copies of a triangle~\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} $T$ \end{document} 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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\frac{2}{9}$ \end{document} of the area 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} $T$ \end{document} can be translatively packed into \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} $T$ \end{document}. The bound 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} $\frac{2}{9}$ \end{document} cannot be improved upon here.

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# Optimal translative packing of homothetic triangles

Studia Scientiarum Mathematicarum Hungarica
Author: Janusz Januszewski

Any sequence of triangles homothetic to a fixed triangle T whose total area does not exceed one-half of the area of T can be packed translatively in T .

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# Efficient online packing of 4-dimensional cubes into the unit cube

Studia Scientiarum Mathematicarum Hungarica
Authors: Janusz Januszewski and Łukasz Zielonka

Any sequence of 4-dimensional cubes of total volume not greater than 1/8 can be online packed into the unit cube.

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