Gyroid

Infinitely connected triply periodic minimal surface

For the polyhedron, see Gyroid (crystallography).

For the video game character, see Animal Crossing (video game).

A gyroid minimal surface, coloured to show the Gaussian curvature at each point 3D model of a gyroid unit cell

A gyroid is an infinitely connected triply periodic minimal surface discovered by Alan Schoen in 1970.[1][2] It arises naturally in polymer science and biology, as an interface with high surface area.

## History and properties

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The gyroid is the unique non-trivial embedded member of the associate family of the Schwarz P and D surfaces. Its angle of association with respect to the D surface is approximately 38.01°. The gyroid is similar to the lidinoid.

The gyroid was discovered in 1970 by NASA scientist Alan Schoen. He calculated the angle of association and gave a convincing demonstration of pictures of intricate plastic models, but did not provide a proof of embeddedness. Schoen noted that the gyroid contains neither straight lines nor planar symmetries. Karcher[3] gave a different, more contemporary treatment of the surface in 1989 using conjugate surface construction. In 1996 Große-Brauckmann and Wohlgemuth[4] proved that it is embedded, and in 1997 Große-Brauckmann provided CMC (constant mean curvature) variants of the gyroid and made further numerical investigations about the volume fractions of the minimal and CMC gyroids.

The gyroid separates space into two oppositely congruent labyrinths of passages. The gyroid has space group I4132 (no. 214).[5] Channels run through the gyroid labyrinths in the (100) and (111) directions; passages emerge at 70.5 degree angles to any given channel as it is traversed, the direction at which they do so gyrating down the channel, giving rise to the name "gyroid". One way to visualize the surface is to picture the "square catenoids" of the P surface (formed by two squares in parallel planes, with a nearly circular waist); rotation about the edges of the square generate the P surface. In the associate family, these square catenoids "open up" (similar to the way the catenoid "opens up" to a helicoid) to form gyrating ribbons, then finally become the Schwarz D surface. For one value of the associate family parameter the gyrating ribbons lie in precisely the locations required to have an embedded surface.

The gyroid refers to the member that is in the associate family of the Schwarz P surface, but in fact the gyroid exists in several families that preserve various symmetries of the surface; a more complete discussion of families of these minimal surfaces appears in triply periodic minimal surfaces.

Curiously, like some other triply periodic minimal surfaces, the gyroid surface can be trigonometrically approximated by a short equation:

sin ⁡ x cos ⁡ y + sin ⁡ y cos ⁡ z + sin ⁡ z cos ⁡ x = 0 {\displaystyle \sin x\cos y+\sin y\cos z+\sin z\cos x=0}

The gyroid structure is closely related to the K4 crystal (Laves' graph of girth ten).[6]

## Applications

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SEM micrograph of TiO2 alternating gyroid nanostructure (top) and Ta2O5 double gyroid nanostructure (bottom).

In nature, self-assembled gyroid structures are found in certain surfactant or lipid mesophases[7] and block copolymers. In a typical A-B diblock copolymer phase diagram, the gyroid phase can be formed at intermediate volume fractions between the lamellar and cylindrical phases. In A-B-C block copolymers, the double and alternating-gyroid phases can be formed.[8] Such self-assembled polymer structures have found applications in experimental supercapacitors,[9] solar cells[10] photocatalysts,[11] and nanoporous membranes.[12] Gyroid membrane structures are occasionally found inside cells.[13] Gyroid structures have photonic band gaps that make them potential photonic crystals.[14] Single gyroid photonic crystals have been observed in biological structural coloration such as butterfly wing scales and bird feathers, inspiring work on biomimetic materials.[15][16][17] The gyroid mitochondrial membranes found in the retinal cone cells of certain tree shrew species present a unique structure which may have an optical function.[18]

In 2017, MIT researchers studied the possibility of using the gyroid shape to turn bi-dimensional materials, such as graphene, into a three-dimensional structural material with low density, yet high tensile strength.[19]

Researchers from Cambridge University have shown the controlled chemical vapor deposition of sub–60 nm graphene gyroids. These interwoven structures are one of the smallest free-standing graphene 3D structures. They are conductive, mechanically stable, and easily transferable, and are of interest for a wide range of applications.[20]

The gyroid pattern has also found use in 3D printing for lightweight internal structures, due to its high strength, combined with speed and ease of printing using an FDM 3D printer.[21][22]

In an in silico study, researchers from the university hospital Charité in Berlin investigated the potential of gyroid architecture when used as a scaffold in a large bone defect in a rat femur. When comparing the regenerated bone within a gyroid scaffold compared to a traditional strut-like scaffold, they found that gyroid scaffolds led to less bone formation and attributed this reduced bone formation to the gyroid architecture hindering cell penetration.[23]

## References

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1. ^ Schoen, Alan H. (May 1970). Infinite periodic minimal surfaces without self-intersections (PDF) (Technical report). NASA Technical Note. NASA. NASA TN D-5541.
2. ^ Hoffman, David (June 25 – July 27, 2001). "Computing Minimal Surfaces". Global Theory of Minimal Surfaces. Proceedings of the Clay Mathematics Institute. Berkeley, California: Mathematical Sciences Research Institute. ISBN 9780821835876. OCLC 57134637.
3. ^ Karcher, Hermann (1989). "The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions". Manuscripta Mathematica. 64 (3): 291–357. doi:10.1007/BF01165824. ISSN 0025-2611. S2CID 119894224.
4. ^ Große-Brauckmann, Karsten; Meinhard, Wohlgemuth (1996). "The gyroid is embedded and has constant mean curvature companions". Calculus of Variations and Partial Differential Equations. 4 (6): 499–523. doi:10.1007/BF01261761. hdl:10068/184059. ISSN 0944-2669. S2CID 120479308.
5. ^ Lambert, Charla A.; Radzilowski, Leonard H.; Thomas, Edwin L. (1996). "Triply periodic level surfaces for cubic tricontinuous block copolymer morphologies". Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences. 354 (1715): 2009–2023. doi:10.1098/rsta.1996.0089. ISSN 1471-2962. S2CID 121697168.
6. ^ Sunada, T. (2008). "Crystals that nature might miss creating" (PDF). Notices of the American Mathematical Society. 55: 208–215.
7. ^ Longley, William; McIntosh, Thomas J. (1983). "A bicontinuous tetrahedral structure in a liquid-crystalline lipid". Nature. 303 (5918). Springer Science and Business Media LLC: 612–614. Bibcode:1983Natur.303..612L. doi:10.1038/303612a0. ISSN 0028-0836. S2CID 4315071.
8. ^ Bates, Frank (February 1999). "Block Copolymers—Designer Soft Materials". Physics Today. 52 (2): 32. Bibcode:1999PhT....52b..32B. doi:10.1063/1.882522.
9. ^ Wei, Di; Scherer, Maik R. J.; Bower, Chris; Andrew, Piers; Ryhänen, Tapani; Steiner, Ullrich (2012-03-15). "A Nanostructured Electrochromic Supercapacitor". Nano Letters. 12 (4). American Chemical Society (ACS): 1857–1862. Bibcode:2012NanoL..12.1857W. doi:10.1021/nl2042112. ISSN 1530-6984. PMID 22390702.
10. ^ Crossland, Edward J. W.; Kamperman, Marleen; Nedelcu, Mihaela; Ducati, Caterina; Wiesner, Ulrich; et al. (2009-08-12). "A Bicontinuous Double Gyroid Hybrid Solar Cell". Nano Letters. 9 (8). American Chemical Society (ACS): 2807–2812. Bibcode:2009NanoL...9.2807C. doi:10.1021/nl803174p. hdl:1813/17055. ISSN 1530-6984. PMID 19007289.
11. ^ Dörr, Tobias Sebastian; Deilmann, Leonie; Haselmann, Greta; Cherevan, Alexey; Zhang, Peng; Blaha, Peter; de Oliveira, Peter William; Kraus, Tobias; Eder, Dominik (December 2018). "Ordered Mesoporous TiO 2 Gyroids: Effects of Pore Architecture and Nb-Doping on Photocatalytic Hydrogen Evolution under UV and Visible Irradiation". Advanced Energy Materials. 8 (36). doi:10.1002/aenm.201802566. S2CID 106140475.
12. ^ Li, Li; Schulte, Lars; Clausen, Lydia D.; Hansen, Kristian M.; Jonsson, Gunnar E.; Ndoni, Sokol (2011-09-14). "Gyroid Nanoporous Membranes with Tunable Permeability" (PDF). ACS Nano. 5 (10). American Chemical Society (ACS): 7754–7766. doi:10.1021/nn200610r. ISSN 1936-0851. PMID 21866958. S2CID 5467282.
13. ^ Hyde, S.; Blum, Z.; Landh, T.; Lidin, S.; Ninham, B. W.; Andersson, S.; Larsson, K. (1996). The Language of Shape: The Role of Curvature in Condensed Matter: Physics, Chemistry and Biology. Elsevier. ISBN 978-0-08-054254-6.
14. ^ Martín-Moreno, L.; García-Vidal, F. J.; Somoza, A. M. (1999-07-05). "Self-Assembled Triply Periodic Minimal Surfaces as Molds for Photonic Band Gap Materials". Physical Review Letters. 83 (1). American Physical Society (APS): 73–75. arXiv:cond-mat/9810299. Bibcode:1999PhRvL..83...73M. doi:10.1103/physrevlett.83.73. ISSN 0031-9007. S2CID 54803711.
15. ^ Saranathan, V.; Narayanan, S.; Sandy, A.; Dufresne, E. R.; Prum, R. O. (2021-06-01). "Evolution of single gyroid photonic crystals in bird feathers". Proceedings of the National Academy of Sciences. 118 (23): e2101357118. Bibcode:2021PNAS..11801357S. doi:10.1073/pnas.2101357118. ISSN 1091-6490. PMC 8201850. PMID 34074782.
16. ^ Saranathan, V.; Osuji, C. O.; Mochrie, S. G. J.; Noh, H.; Narayanan, S.; Sandy, A.; Dufresne, E. R.; Prum, R. O. (2010-06-14). "Structure, function, and self-assembly of single network gyroid ( I 4 1 32 {\displaystyle I4_{1}32} ) photonic crystals in butterfly wing scales". Proceedings of the National Academy of Sciences. 107 (26): 11676–11681. Bibcode:2010PNAS..10711676S. doi:10.1073/pnas.0909616107. ISSN 0027-8424. PMC 2900708. PMID 20547870.
17. ^ Michielsen, K; Stavenga, D.G (2007-06-13). "Gyroid cuticular structures in butterfly wing scales: biological photonic crystals". Journal of the Royal Society Interface. 5 (18). The Royal Society: 85–94. doi:10.1098/rsif.2007.1065. ISSN 1742-5689. PMC 2709202. PMID 17567555.
18. ^ Almsherqi, Zakaria; Margadant, Felix; Deng, Yuru (2012-03-07). "A look through 'lens' cubic mitochondria". Interface Focus. 2 (5). The Royal Society: 539–545. doi:10.1098/rsfs.2011.0120. ISSN 2042-8898. PMC 3438578. PMID 24098837.
19. ^ David L. Chandler (2017-01-06). "Researchers design one of the strongest, lightest materials known". MIT news. Retrieved 2020-01-09.
20. ^ Cebo, T.; Aria, A. I.; Dolan, J.A.; Weatherup, R. S.; Nakanishi, K.; Kidambi, P. R.; Divitini, G.; Ducati, C.; Steiner, U.; Hofmann, S. (2017). "Chemical vapour deposition of freestanding sub-60 nm graphene gyroids". Appl. Phys. Lett. 111 (25): 253103. Bibcode:2017ApPhL.111y3103C. doi:10.1063/1.4997774. hdl:1826/13396.
21. ^ Harrison, Matthew (2018-03-15). "Introducing Gyroid Infill". Matt's Hub. Retrieved 2019-01-05.
22. ^ By (2022-10-31). "3D Printed Heat Exchanger Uses Gyroid Infill For Cooling". Hackaday. Retrieved 2022-11-05.
23. ^ Jaber, Mahdi; S. P. Poh, Patrina; N Duda, Georg; Checa, Sara (2022-09-23). "PCL strut-like scaffolds appear superior to gyroid in terms of bone regeneration within a long bone large defect: An in silico study". Front. Bioeng. Biotechnol. 10: 995266. doi:10.3389/fbioe.2022.995266. PMC 9540363. PMID 36213070.

## External links

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* Triply Periodic Minimal Surfaces at schoengeometry.com
* Gyroid at MathWorld
* Rotatable picture of a gyroid's period
* The gyroid at Bloomington's Virtual Minimal Surface Museum
* Electrochemical Nanofabrication: Principles and Applications

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Minimal surfaces

* Associate family
* Bour's
* Catalan's
* Catenoid
* Chen–Gackstatter
* Costa's
* Enneper
* Gyroid
* Helicoid
* Henneberg
* k-noid
* Lidinoid
* Neovius
* Richmond
* Riemann's
* Saddle tower
* Scherk
* Schwarz
* Triply periodic

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