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Double lattice
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<h2 id="double-lattice-packing">Double lattice packing</h2> <p>A packing that can be described as the orbit of a body under the action of a double lattice is called a double lattice packing. In many cases the highest known <a href="/facts/Packing_density/NY81NF80">packing density</a> for a body is achieved by a double lattice. Examples include the regular <a href="/facts/Pentagon/xoq5oJeQ">pentagon</a>, <a href="/facts/Heptagon/b0qIVNhY">heptagon</a>, and <a href="/facts/Nonagon/mdQnTG40">nonagon</a><a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> and the equilateral <a href="/facts/Triangular_bipyramid/fU4kIBTV">triangular bipyramid</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> <a href="/facts/W%25C5%2582odzimierz_Kuperberg/qUcpgaYW">Włodzimierz Kuperberg</a> and <a href="/facts/Greg_Kuperberg/q64KLJNM">Greg Kuperberg</a> showed that all convex planar bodies can pack at a density of at least √3/2 by using a double lattice.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p><p>In a preprint released in 2016, <a href="/facts/Thomas_Callister_Hales/c0cWXvC0">Thomas Hales</a> and Wöden Kusner announced a proof that the double lattice packing of the regular pentagon has the optimal density among all packings of regular pentagons in the plane.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> This packing has been used as a decorative pattern in China since at least 1900, and in this context has been called the "pentagonal ice-ray".<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> As of 2021, the proof of its optimality has not yet been refereed and published. </p><p>It has been conjectured that, among all convex shapes, the regular <a href="/facts/Heptagon/b0qIVNhY">heptagon</a> has the lowest packing density for its optimal double lattice packing, but this remains unproven.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>de Graaf, Joost; van Roij, René; Dijkstra, Marjolein (2011), "Dense regular packings of irregular nonconvex particles", Physical Review Letters, 107 (15): 155501, arXiv:1107.0603, Bibcode:2011PhRvL.107o5501D, doi:10.1103/PhysRevLett.107.155501, PMID 22107298 <a href="/wiki/Marjolein_Dijkstra" target="_blank">/wiki/Marjolein_Dijkstra</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Haji-Akbari, Amir; Engel, Michael; Glotzer, Sharon C. (2011), "Degenerate quasicrystal of hard triangular bipyramids", Phys. Rev. Lett., 107 (21): 215702, arXiv:1106.5561, Bibcode:2011PhRvL.107u5702H, doi:10.1103/PhysRevLett.107.215702, PMID 22181897 <a href="/wiki/Sharon_Glotzer" target="_blank">/wiki/Sharon_Glotzer</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Kuperberg, G.; Kuperberg, W. (1990), "Double-lattice packings of convex bodies in the plane", Discrete & Computational Geometry, 5 (4): 389–397, doi:10.1007/BF02187800, MR 1043721 <a href="/wiki/Greg_Kuperberg" target="_blank">/wiki/Greg_Kuperberg</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Hales, Thomas; Kusner, Wöden (September 2016), Packings of regular pentagons in the plane, arXiv:1602.07220 <a href="/wiki/Thomas_Callister_Hales" target="_blank">/wiki/Thomas_Callister_Hales</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Dye, Daniel Sheets (2012), Chinese Lattice Designs, Dover, pp. 307–309, ISBN 9780486146225 <a href="9780486146225" target="_blank">9780486146225</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Kallus, Yoav (2015), "Pessimal packing shapes", Geometry & Topology, 19 (1): 343–363, arXiv:1305.0289, doi:10.2140/gt.2015.19.343, MR 3318753 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> </ol>