TPMS are of relevance in natural science. TPMS have been observed as biological membranes,[2] as block copolymers,[3] equipotential surfaces in crystals[4] etc. They have also been of interest in architecture, design and art.
Properties
Nearly all studied TPMS are free of self-intersections (i.e. embedded in ): from a mathematical standpoint they are the most interesting (since self-intersecting surfaces are trivially abundant).[5]
All connected TPMS have genus ≥ 3,[6] and in every lattice there exist orientable embedded TPMS of every genus ≥3.[7]
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.[8]
In 1970 Alan Schoen came up with 12 new TPMS based on skeleton graphs spanning crystallographic cells.[11][12] 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.[13]
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.[6] 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.[14]
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.[8][15]
Generalisations
Periodic minimal surfaces can be constructed in S3[16] and H3.[17]
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.[18]
Quasiperiodic minimal surfaces have been constructed in .[19] It has been suggested but not been proven that minimal surfaces with a quasicrystalline order in exist.[20]
External galleries of images
TPMS at the Minimal Surface Archive [2]
Periodic minimal surfaces gallery [3]
References
^"Triply Periodic Minimal surfaces". Mathematics of the EPINET Project. Archived from the original on 2023-02-28.
^Deng, Yuru; Mieczkowski, Mark (1998). "Three-dimensional periodic cubic membrane structure in the mitochondria of amoebae Chaos carolinensis". Protoplasma. 203 (1–2). Springer Science and Business Media LLC: 16–25. doi:10.1007/bf01280583. ISSN 0033-183X. S2CID 25569139.
^Jiang, Shimei; Göpfert, Astrid; Abetz, Volker (2003). "Novel Morphologies of Block Copolymer Blends via Hydrogen Bonding". Macromolecules. 36 (16). American Chemical Society (ACS): 6171–6177. Bibcode:2003MaMol..36.6171J. doi:10.1021/ma0342933. ISSN 0024-9297.
^Mackay, Alan L. (1985). "Periodic minimal surfaces". Physica B+C. 131 (1–3). Elsevier BV: 300–305. Bibcode:1985PhyBC.131..300M. doi:10.1016/0378-4363(85)90163-9. ISSN 0378-4363. S2CID 4267918.
^ a bKarcher, Hermann; Polthier, Konrad (1996-09-16). "Construction of triply periodic minimal surfaces" (PDF). Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences. 354 (1715). The Royal Society: 2077–2104. arXiv:1002.4805. Bibcode:1996RSPTA.354.2077K. doi:10.1098/rsta.1996.0093. ISSN 1364-503X. S2CID 15540887.
^ a bWilliam H. Meeks, III. The Geometry and the Conformal Structure of Triply Periodic Minimal Surfaces in R3. PhD thesis, University of California, Berkeley, 1975.
^Traizet, M. (2008). "On the genus of triply periodic minimal surfaces" (PDF). Journal of Differential Geometry. 79 (2). International Press of Boston: 243–275. doi:10.4310/jdg/1211512641. ISSN 0022-040X.
^ a b"Without self-intersections". Archived from the original on 2007-02-22.
^H. A. Schwarz, Gesammelte Mathematische Abhandlungen, Springer, Berlin, 1933.
^Alan H. Schoen, Infinite periodic minimal surfaces without self-intersections, NASA Technical Note TN D-5541 (1970)"Infinite periodic minimal surfaces without self-intersections by Alan H. Schoen" (PDF). Archived (PDF) from the original on 2018-04-13. Retrieved 2019-04-12.
^"Triply-periodic minimal surfaces by Alan H. Schoen". Archived from the original on 2018-10-22. Retrieved 2019-04-12.
^Karcher, Hermann (1989-03-05). "The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions". Manuscripta Mathematica. 64 (3): 291–357. doi:10.1007/BF01165824. S2CID 119894224.
^Adam G. Weyhaupt. New families of embedded triply periodic minimal surfaces of genus three in euclidean space. PhD thesis, Indiana University, 2006
^Fischer, W.; Koch, E. (1996-09-16). "Spanning minimal surfaces". Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences. 354 (1715). The Royal Society: 2105–2142. Bibcode:1996RSPTA.354.2105F. doi:10.1098/rsta.1996.0094. ISSN 1364-503X. S2CID 118170498.
^Karcher, H.; Pinkall, U.; Sterling, I. (1988). "New minimal surfaces in S3". Journal of Differential Geometry. 28 (2). International Press of Boston: 169–185. doi:10.4310/jdg/1214442276. ISSN 0022-040X.
^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.
^Góźdź, Wojciech T.; Hołyst, Robert (1996-11-01). "Triply periodic surfaces and multiply continuous structures from the Landau model of microemulsions". Physical Review E. 54 (5). American Physical Society (APS): 5012–5027. Bibcode:1996PhRvE..54.5012G. doi:10.1103/physreve.54.5012. ISSN 1063-651X. PMID 9965680.
^Laurent Mazet, Martin Traizet, A quasi-periodic minimal surface, Commentarii Mathematici Helvetici, pp. 573–601, 2008 [1]
^Sheng, Qing; Elser, Veit (1994-04-01). "Quasicrystalline minimal surfaces". Physical Review B. 49 (14). American Physical Society (APS): 9977–9980. Bibcode:1994PhRvB..49.9977S. doi:10.1103/physrevb.49.9977. ISSN 0163-1829. PMID 10009804.