Schwarz minimal surface

<h2 id="schwarz-p-primitive">Schwarz P ("Primitive")</h2>

<p>Schoen named this surface 'primitive' because it has two intertwined congruent labyrinths, each with the shape of an inflated tubular version of the simple cubic lattice. While the standard P surface has cubic symmetry the unit cell can be any rectangular box, producing a family of minimal surfaces with the same topology.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>
</p><p>It can be approximated by the implicit surface
</p>

cos
        ⁡
        (
        x
        )
        +
        cos
        ⁡
        (
        y
        )
        +
        cos
        ⁡
        (
        z
        )
        =
        0
         
      
    
    {\displaystyle \cos(x)+\cos(y)+\cos(z)=0\ }
  
.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>
<p>The P surface has been considered for prototyping <a href="/facts/Tissue_engineering/Tx77KG1S">tissue scaffolds</a> with a high surface-to-volume ratio and porosity.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>
</p>

<h2 id="schwarz-d-diamond">Schwarz D ("Diamond")</h2>

<p>Schoen named this surface 'diamond' because it has two intertwined congruent labyrinths, each having the shape of an inflated tubular version of the <a href="/facts/Diamond_lattice/6dK7DpGB">diamond bond structure</a>. It is sometimes called the F surface in the literature.
</p><p>It can be approximated by the implicit surface
</p>

cos
        ⁡
        (
        x
        )
        cos
        ⁡
        (
        y
        )
        cos
        ⁡
        (
        z
        )
        −
        sin
        ⁡
        (
        x
        )
        sin
        ⁡
        (
        y
        )
        sin
        ⁡
        (
        z
        )
        =
        0
         
      
    
    {\displaystyle \cos(x)\cos(y)\cos(z)-\sin(x)\sin(y)\sin(z)=0\ }
  
.
<p>An exact expression exists in terms of <a href="/facts/Elliptic_integrals/q8EGYzQ3">elliptic integrals</a>, based on the <a href="/facts/Weierstrass_representation/2eJxe3wY">Weierstrass representation</a>.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a>
</p>

<h2 id="schwarz-h-hexagonal">Schwarz H ("Hexagonal")</h2>

<p>The H surface is similar to a <a href="/facts/Catenoid/f094Ud1m">catenoid</a> with a triangular boundary, allowing it to tile space.
</p>

<h2 id="schwarz-clp-crossed-layers-of-parallels">Schwarz CLP ("Crossed layers of parallels")</h2>

<h2 id="illustrations">Illustrations</h2>
<ul><li><a href="https://web.archive.org/web/20010627215357/http://www.susqu.edu/brakke/evolver/examples/periodic/periodic.html">Susquehanna University - Triply Periodic Minimal Surfaces <i>(Archived)</i></a></li>
<li><a href="https://web.archive.org/web/20090614123646/http://www.indiana.edu/~minimal/archive/Triply/genus3.html">Indiana University - Triply Periodic Surfaces of Genus 3 <i>(Archived)</i></a></li>
<li><a href="http://www.thphys.uni-heidelberg.de/~biophys/index.php?lang=e&n1=research_tpms">Ruprecht-Karls-Universität Heidelberg - Bicontinuous cubic phases based on triply periodic minimal surfaces</a></li>
<li><a href="https://web.archive.org/web/20160225062057/http://homepages.ulb.ac.be/~morahman/gallery/schwartz.html">Université libre de Bruxelles - Schwartz's Surface <i>(Archived)</i></a></li>
<li><a href="http://virtualmathmuseum.org/Surface/gallery_m.html">Virtual Math Museum - 3DXM Minimal Surface Gallery</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>H. A. Schwarz, Gesammelte Mathematische Abhandlungen, Springer, Berlin, 1933. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>E. R. Neovius, "Bestimmung zweier spezieller periodischer Minimalflächen", Akad. Abhandlungen, Helsingfors, 1883. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Alan H. Schoen, Infinite periodic minimal surfaces without self-intersections, NASA Technical Note TN D-5541 (1970)[1] <a href="https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19700020472.pdf" target="_blank">https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19700020472.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Hermann Karcher, Konrad Polthier, "Construction of Triply Periodic Minimal Surfaces", Phil. Trans. R. Soc. Lond. A 16 September 1996 vol. 354 no. 1715 2077–2104 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>"Alan Schoen geometry". <a href="http://schoengeometry.com/e-tpms.html" target="_blank">http://schoengeometry.com/e-tpms.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Mackay, Alan L. (April 1985). "Periodic minimal surfaces". Nature. 314 (6012): 604–606. Bibcode:1985Natur.314..604M. doi:10.1038/314604a0. S2CID 4267918. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Terrones, H.; Mackay, A. L. (December 1994). "Negatively curved graphite and triply periodic minimal surfaces". Journal of Mathematical Chemistry. 15 (1): 183–195. doi:10.1007/BF01277558. S2CID 123561096. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>W. H. Meeks. The theory of triply-periodic minimal surfaces. Indiana University Math. Journal, 39 (3):877-936, 1990. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>"Triply Periodic Level Surfaces". Archived from the original on 2019-02-12. Retrieved 2019-02-10. <a href="http://www.msri.org/publications/sgp/jim/geom/level/library/triper/index.html" target="_blank">http://www.msri.org/publications/sgp/jim/geom/level/library/triper/index.html</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Jaemin Shin, Sungki Kim, Darae Jeong, Hyun Geun Lee, Dongsun Lee, Joong Yeon Lim, and Junseok Kim, Finite Element Analysis of Schwarz P Surface Pore Geometries for Tissue-Engineered Scaffolds, Mathematical Problems in Engineering, Volume 2012, Article ID 694194, doi:10.1155/2012/694194 <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Paul J.F. Gandy, Djurdje Cvijović, Alan L. Mackay, Jacek Klinowski, Exact computation of the triply periodic D (`diamond') minimal surface, Chemical Physics Letters, Volume 314, Issues 5–6, 10 December 1999, Pages 543–551 <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
</ol>

Schwarz minimal surface open-in-new

Schwarz minimal surface