<p>If <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13366_2024_783_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(Q_n^t\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>Q</mi> <mi>n</mi> <mi>t</mi> </msubsup> </math></EquationSource> </InlineEquation> is either a square of side length <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13366_2024_783_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">\(n^{-t}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mi>t</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> or an isosceles right triangle with legs of length <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13366_2024_783_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">\(n^{-t}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mi>t</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>, for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13366_2024_783_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="85" /> </InlineMediaObject> <EquationSource Format="TEX">\(n=1,2,\ldots \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> </mrow> </math></EquationSource> </InlineEquation>, then the sets <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13366_2024_783_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="100" /> </InlineMediaObject> <EquationSource Format="TEX">\(Q_1^t, Q_2^t, Q_3^t \ldots \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>Q</mi> <mn>1</mn> <mi>t</mi> </msubsup> <mo>,</mo> <msubsup> <mi>Q</mi> <mn>2</mn> <mi>t</mi> </msubsup> <mo>,</mo> <msubsup> <mi>Q</mi> <mn>3</mn> <mi>t</mi> </msubsup> <mo>…</mo> </mrow> </math></EquationSource> </InlineEquation> can be packed perfectly into a square as well as can be packed perfectly into an isosceles right triangle, provided that <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13366_2024_783_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="118" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/2&lt;t \le 37/72\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>&lt;</mo> <mi>t</mi> <mo>≤</mo> <mn>37</mn> <mo stretchy="false">/</mo> <mn>72</mn> </mrow> </math></EquationSource> </InlineEquation>. Based on this result, the perfect packing is considered using some convex and concave polygons, as well as polygons with holes.</p>

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Perfect packing by convex and concave polygons

  • Janusz Januszewski,
  • Łukasz Zielonka

摘要

If \(Q_n^t\) Q n t is either a square of side length \(n^{-t}\) n - t or an isosceles right triangle with legs of length \(n^{-t}\) n - t , for \(n=1,2,\ldots \) n = 1 , 2 , , then the sets \(Q_1^t, Q_2^t, Q_3^t \ldots \) Q 1 t , Q 2 t , Q 3 t can be packed perfectly into a square as well as can be packed perfectly into an isosceles right triangle, provided that \(1/2<t \le 37/72\) 1 / 2 < t 37 / 72 . Based on this result, the perfect packing is considered using some convex and concave polygons, as well as polygons with holes.