Triply periodic minimal surface

Nearly all studied TPMS are free of self-intersections (i.e. embedded in $\backslash mathbb\{R\}^3$): from a mathematical standpoint they are the most interesting (since self-intersecting surfaces are trivially abundant).{{cite journal |first1=Hermann|last1=Karcher|first2=Konrad|last2=Polthier| title=Construction of triply periodic minimal surfaces | journal=Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences | publisher=The Royal Society | volume=354 | issue=1715 | date=1996-09-16 | issn=1364-503X | doi=10.1098/rsta.1996.0093 | pages=2077–2104|arxiv=1002.4805|bibcode=1996RSPTA.354.2077K|s2cid=15540887|url=http://www.polthier.info/articles/triply/triply_withoutApp.pdf}}

All connected TPMS have genus ≥ 3, and in every lattice there exist orientable embedded TPMS of every genus ≥3.{{cite journal | last=Traizet | first=M. | title=On the genus of triply periodic minimal surfaces | journal=Journal of Differential Geometry | publisher=International Press of Boston | volume=79 | issue=2 | year=2008 | issn=0022-040X | doi=10.4310/jdg/1211512641 | pages=243–275|doi-access=free|url=http://www.lmpt.univ-tours.fr/~traizet/triply.pdf}}

Embedded TPMS are orientable and divide space into two disjoint sub-volumes (labyrinths). If they are congruent the surface is said to be a balance surface.{{Cite web|url=http://staff-www.uni-marburg.de/~fischerw/nonself/nonsi.htm|archive-url = https://web.archive.org/web/20070222230616/http://staff-www.uni-marburg.de/~fischerw/nonself/nonsi.htm|archive-date = 2007-02-22|title = Without self-intersections}}

File:Schwarz P Surface.png

The first examples of TPMS were the surfaces described by Schwarz in 1865, followed by a surface described by his student E. R. Neovius in 1883.H. A. Schwarz, Gesammelte Mathematische Abhandlungen, Springer, Berlin, 1933.E. R. Neovius, "Bestimmung zweier spezieller periodischer Minimal Flachen", Akad. Abhandlungen, Helsingfors, 1883.

In 1970 Alan Schoen came up with 12 new TPMS based on skeleton graphs spanning crystallographic cells.Alan H. Schoen, Infinite periodic minimal surfaces without self-intersections, NASA Technical Note TN D-5541 (1970){{cite web |url=https://schoengeometry.com/e-tpms-media/19700020472_1970020472[1].pdf |title=Infinite periodic minimal surfaces without self-intersections by Alan H. Schoen|accessdate=2019-04-12 |url-status=live|archiveurl=https://web.archive.org/web/20180413003718/http://schoengeometry.com/e-tpms-media/19700020472_1970020472[1].pdf |archivedate=2018-04-13 }}

{{cite web |url= https://schoengeometry.com/e-tpms.html |title=Triply-periodic minimal surfaces by Alan H. Schoen |accessdate=2019-04-12 |url-status=live|archiveurl=https://web.archive.org/web/20181022143254/http://schoengeometry.com/e-tpms.html|archivedate=2018-10-22 }} While Schoen's surfaces became popular in natural science the construction did not lend itself to a mathematical existence proof and remained largely unknown in mathematics, until H. Karcher proved their existence in 1989.{{cite journal |last1= Karcher |first1=Hermann |date=1989-03-05 |title=The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions |journal=Manuscripta Mathematica |volume=64 |issue=3 |pages=291–357 |doi=10.1007/BF01165824 |s2cid=119894224 }}

Using conjugate surfaces many more surfaces were found. While Weierstrass representations are known for the simpler examples, they are not known for many surfaces. Instead methods from Discrete differential geometry are often used.

The classification of TPMS is an open problem.

TPMS often come in families that can be continuously deformed into each other. Meeks found an explicit 5-parameter family for genus 3 TPMS that contained all then known examples of genus 3 surfaces except the gyroid.William H. Meeks, III. The Geometry and the Conformal Structure of Triply Periodic Minimal Surfaces in R3. PhD thesis, University of California, Berkeley, 1975. Members of this family can be continuously deformed into each other, remaining embedded in the process (although the lattice may change). The gyroid and lidinoid are each inside a separate 1-parameter family.Adam G. Weyhaupt. New families of embedded triply periodic minimal surfaces of genus three in euclidean space. PhD thesis, Indiana University, 2006

Another approach to classifying TPMS is to examine their space groups. For surfaces containing lines the possible boundary polygons can be enumerated, providing a classification.{{cite journal |first1=W. |last1=Fischer |first2=E. |last2=Koch| title=Spanning minimal surfaces | journal=Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences | publisher=The Royal Society | volume=354 | issue=1715 | date=1996-09-16 | issn=1364-503X | doi=10.1098/rsta.1996.0094 | pages=2105–2142|bibcode=1996RSPTA.354.2105F |s2cid=118170498 }}

Periodic minimal surfaces can be constructed in S³{{cite journal | last1=Karcher | first1=H. | last2=Pinkall | first2=U. | last3=Sterling | first3=I. | author2-link=Ulrich Pinkall | title=New minimal surfaces in S³ | journal=Journal of Differential Geometry | publisher=International Press of Boston | volume=28 | issue=2 | year=1988 | issn=0022-040X | doi=10.4310/jdg/1214442276 | pages=169–185|doi-access=free}} and H³.K. Polthier. New periodic minimal surfaces in h3. In G. Dziuk, G. Huisken, and J. Hutchinson, editors, Theoretical and Numerical Aspects of Geometric Variational Problems, volume 26, pages 201–210. CMA Canberra, 1991.

It is possible to generalise the division of space into labyrinths to find triply periodic (but possibly branched) minimal surfaces that divide space into more than two sub-volumes.{{cite journal | last1=Góźdź | first1=Wojciech T. | last2=Hołyst | first2=Robert | title=Triply periodic surfaces and multiply continuous structures from the Landau model of microemulsions | journal=Physical Review E | publisher=American Physical Society (APS) | volume=54 | issue=5 | date=1996-11-01 | issn=1063-651X | doi=10.1103/physreve.54.5012 | pages=5012–5027| pmid=9965680 | bibcode=1996PhRvE..54.5012G }}

Quasiperiodic minimal surfaces have been constructed in $\backslash mathbb\{R\}^2\; \backslash times\; \backslash textbf\{S\}^1$.Laurent Mazet, Martin Traizet, A quasi-periodic minimal surface, Commentarii Mathematici Helvetici, pp. 573–601, 2008 [https://arxiv.org/abs/math/0609489] It has been suggested but not been proven that minimal surfaces with a quasicrystalline order in $\backslash mathbb\{R\}^3$ exist.{{cite journal | last1=Sheng | first1=Qing | last2=Elser | first2=Veit | title=Quasicrystalline minimal surfaces | journal=Physical Review B | publisher=American Physical Society (APS) | volume=49 | issue=14 | date=1994-04-01 | issn=0163-1829 | doi=10.1103/physrevb.49.9977 | pages=9977–9980| pmid=10009804 | bibcode=1994PhRvB..49.9977S }}

• TPMS at the Minimal Surface Archive [https://minimal.sitehost.iu.edu/archive/Triply/index.html]
• Periodic minimal surfaces gallery [http://www-klinowski.ch.cam.ac.uk/pmsgal1.html]

{{reflist}}

{{Minimal surfaces}}

Category:Differential geometry

Category:Minimal surfaces