Gyroid

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{{Cite journal|last=Karcher|first=Hermann|title=The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions|journal=Manuscripta Mathematica|language=en|volume=64|issue=3|pages=291–357|doi=10.1007/BF01165824|issn=0025-2611|year=1989|s2cid=119894224}} gave a different, more contemporary treatment of the surface in 1989 using conjugate surface construction.

In 1996 Große-Brauckmann and Wohlgemuth{{Cite journal|last1=Große-Brauckmann|first1=Karsten|last2=Meinhard|first2=Wohlgemuth|title=The gyroid is embedded and has constant mean curvature companions|journal=Calculus of Variations and Partial Differential Equations|language=en|volume=4|issue=6|pages=499–523|doi=10.1007/BF01261761|issn=0944-2669|year=1996|hdl=10068/184059|s2cid=120479308|hdl-access=free}} 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).{{Cite journal|last1=Lambert|first1=Charla A.|last2=Radzilowski|first2=Leonard H.|last3=Thomas|first3=Edwin L.|title=Triply periodic level surfaces for cubic tricontinuous block copolymer morphologies|journal=Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences|language=en|volume=354|issue=1715|pages=2009–2023|doi=10.1098/rsta.1996.0089|issn=1471-2962|year=1996|s2cid=121697168}} 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.

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

:$\backslash sin\; x\backslash cos\; y+\backslash sin\; y\backslash cos\; z+\backslash sin\; z\backslash cos\; x=0$

The gyroid structure is closely related to the K₄ crystal (Laves' graph of girth ten).{{cite journal|first=T. |last=Sunada|author-link=Toshikazu Sunada|title= Crystals that nature might miss creating|journal=Notices of the American Mathematical Society|volume=55|year=2008|pages=208–215|url=https://www.ams.org/notices/200802/tx080200208p.pdf}}

File:TiO2 alternating gyroid and Ta2O5 double gyroid.png

In nature, self-assembled gyroid structures are found in certain surfactant or lipid mesophases{{cite journal | last1=Longley | first1=William | last2=McIntosh | first2=Thomas J. | title=A bicontinuous tetrahedral structure in a liquid-crystalline lipid | journal=Nature | publisher=Springer Science and Business Media LLC | volume=303 | issue=5918 | year=1983 | issn=0028-0836 | doi=10.1038/303612a0 | pages=612–614| bibcode=1983Natur.303..612L | s2cid=4315071 }} 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.{{cite journal |last1=Bates |first1=Frank |title=Block Copolymers—Designer Soft Materials |journal=Physics Today |date=February 1999 |volume=52 |issue=2 |page=32 |doi=10.1063/1.882522|bibcode=1999PhT....52b..32B |doi-access=free }} Such self-assembled polymer structures have found applications in experimental supercapacitors,{{cite journal | last1=Wei | first1=Di | last2=Scherer | first2=Maik R. J. | last3=Bower | first3=Chris | last4=Andrew | first4=Piers | last5=Ryhänen | first5=Tapani | last6=Steiner | first6=Ullrich | title=A Nanostructured Electrochromic Supercapacitor | journal=Nano Letters | publisher=American Chemical Society (ACS) | volume=12 | issue=4 | date=2012-03-15 | issn=1530-6984 | doi=10.1021/nl2042112 | pages=1857–1862| pmid=22390702 | bibcode=2012NanoL..12.1857W }} solar cells{{cite journal | last1=Crossland | first1=Edward J. W. | last2=Kamperman | first2=Marleen | last3=Nedelcu | first3=Mihaela | last4=Ducati | first4=Caterina |author-link4=Caterina Ducati| last5=Wiesner | first5=Ulrich | last6=Smilgies | first6=Detlef -M. | last7=Toombes | first7=Gilman E. S. | last8=Hillmyer | first8=Marc A. | last9=Ludwigs | first9=Sabine | last10=Steiner | first10=Ullrich | last11=Snaith | first11=Henry J. |display-authors=5| title=A Bicontinuous Double Gyroid Hybrid Solar Cell | journal=Nano Letters | publisher=American Chemical Society (ACS) | volume=9 | issue=8 | date=2009-08-12 | issn=1530-6984 | doi=10.1021/nl803174p | pages=2807–2812| pmid=19007289 | bibcode=2009NanoL...9.2807C | hdl=1813/17055 | hdl-access=free }} photocatalysts,{{cite journal |last1=Dörr |first1=Tobias Sebastian |last2=Deilmann |first2=Leonie |last3=Haselmann |first3=Greta |last4=Cherevan |first4=Alexey |last5=Zhang |first5=Peng |last6=Blaha |first6=Peter |last7=de Oliveira |first7=Peter William |last8=Kraus |first8=Tobias |last9=Eder |first9=Dominik |title=Ordered Mesoporous TiO 2 Gyroids: Effects of Pore Architecture and Nb-Doping on Photocatalytic Hydrogen Evolution under UV and Visible Irradiation |journal=Advanced Energy Materials |date=December 2018 |volume=8 |issue=36 |doi=10.1002/aenm.201802566|s2cid=106140475 |doi-access=free }} and nanoporous membranes.{{cite journal | last1=Li | first1=Li | last2=Schulte | first2=Lars | last3=Clausen | first3=Lydia D. | last4=Hansen | first4=Kristian M. | last5=Jonsson | first5=Gunnar E. | last6=Ndoni | first6=Sokol | title=Gyroid Nanoporous Membranes with Tunable Permeability | journal=ACS Nano | publisher=American Chemical Society (ACS) | volume=5 | issue=10 | date=2011-09-14 | issn=1936-0851 | doi=10.1021/nn200610r | pmid=21866958 | pages=7754–7766| s2cid=5467282 | url=https://backend.orbit.dtu.dk/ws/files/7968876/nn200610r.pdf }}

Gyroid membrane structures are occasionally found inside cells.{{cite book |last1=Hyde |first1=S. |last2=Blum |first2=Z. |last3=Landh |first3=T. |last4=Lidin |first4=S. |last5=Ninham |first5=B. W. |last6=Andersson |first6=S. |last7=Larsson |first7=K. |title=The Language of Shape: The Role of Curvature in Condensed Matter: Physics, Chemistry and Biology |date=1996 |publisher=Elsevier |isbn=978-0-08-054254-6 |url=https://books.google.com/books?id=1LZlSZ7ORrQC&q=gyroid |language=en}}

Gyroid structures have photonic band gaps that make them potential photonic crystals.{{cite journal | last1=Martín-Moreno | first1=L. | last2=García-Vidal | first2=F. J. | last3=Somoza | first3=A. M. | title=Self-Assembled Triply Periodic Minimal Surfaces as Molds for Photonic Band Gap Materials | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=83 | issue=1 | date=1999-07-05 | issn=0031-9007 | doi=10.1103/physrevlett.83.73 | pages=73–75|arxiv=cond-mat/9810299| bibcode=1999PhRvL..83...73M | s2cid=54803711 }} Single gyroid photonic crystals have been observed in biological structural coloration such as butterfly wing scales and bird feathers, inspiring work on biomimetic materials.{{cite journal | last1=Saranathan | first1=V. | last2=Narayanan | first2=S. | last3=Sandy | first3=A. | last4=Dufresne | first4=E. R. | last5=Prum | first5=R. O. | title=Evolution of single gyroid photonic crystals in bird feathers | journal=Proceedings of the National Academy of Sciences | volume=118 | issue=23 | date=2021-06-01 | issn=1091-6490 | doi=10.1073/pnas.2101357118 | pages=e2101357118| pmid=34074782 | pmc=8201850 | bibcode=2021PNAS..11801357S |doi-access=free}}{{cite journal | last1=Saranathan | first1=V. | last2=Osuji | first2=C. O. | last3=Mochrie | first3=S. G. J. | last4=Noh | first4=H. | last5=Narayanan | first5=S. | last6=Sandy | first6=A. | last7=Dufresne | first7=E. R. | last8=Prum | first8=R. O. | title=Structure, function, and self-assembly of single network gyroid ($I4\_\{1\}32$) photonic crystals in butterfly wing scales | journal=Proceedings of the National Academy of Sciences | volume=107 | issue=26 | date=2010-06-14 | issn=0027-8424 | doi=10.1073/pnas.0909616107 | pages=11676–11681|doi-access=free | pmid=20547870 | bibcode=2010PNAS..10711676S | pmc=2900708}}{{cite journal | last1=Michielsen | first1=K | last2=Stavenga | first2=D.G | title=Gyroid cuticular structures in butterfly wing scales: biological photonic crystals | journal=Journal of the Royal Society Interface | publisher=The Royal Society | volume=5 | issue=18 | date=2007-06-13 | issn=1742-5689 | doi=10.1098/rsif.2007.1065 | pmid=17567555 | pages=85–94| doi-access=free | pmc=2709202 }} 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.{{cite journal | last1=Almsherqi | first1=Zakaria | last2=Margadant | first2=Felix | last3=Deng | first3=Yuru | title=A look through 'lens' cubic mitochondria | journal=Interface Focus | publisher=The Royal Society | volume=2 | issue=5 | date=2012-03-07 | issn=2042-8898 | doi=10.1098/rsfs.2011.0120 | pages=539–545| pmid=24098837 | pmc=3438578 | doi-access=free }}

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.{{cite web|url=https://news.mit.edu/2017/3-d-graphene-strongest-lightest-materials-0106 |title=Researchers design one of the strongest, lightest materials known|website=MIT news|author=David L. Chandler|date=2017-01-06|access-date=2020-01-09}}

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.{{cite journal|doi=10.1063/1.4997774|title=Chemical vapour deposition of freestanding sub-60 nm graphene gyroids|journal=Appl. Phys. Lett.|volume=111|issue=25|pages=253103|year=2017|last1=Cebo|first1=T.|last2=Aria|first2=A. I.|last3=Dolan|first3=J.A.|last4=Weatherup|first4=R. S.|last5=Nakanishi|first5=K.|last6=Kidambi|first6=P. R. |last7=Divitini|first7=G.|last8=Ducati|first8=C.|last9=Steiner|first9=U.|last10=Hofmann|first10=S.|hdl=1826/13396|bibcode=2017ApPhL.111y3103C|url=https://zenodo.org/record/3540712|hdl-access=free}}

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.{{Cite web|url=https://mattshub.com/blogs/blog/gyroid-infill/|title=Introducing Gyroid Infill|last=Harrison|first=Matthew|date=2018-03-15|website=Matt's Hub|language=en|access-date=2019-01-05}}{{Cite web |last=By |date=2022-10-31 |title=3D Printed Heat Exchanger Uses Gyroid Infill For Cooling |url=https://hackaday.com/2022/10/31/3d-printed-heat-exchanger-uses-gyroid-infill-for-cooling/ |access-date=2022-11-05 |website=Hackaday |language=en-US}}

In a study in silico, 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.{{cite journal | last1=Jaber| first1=Mahdi| last2=S. P. Poh| first2=Patrina | last3=N Duda| first3=Georg| last4=Checa| first4=Sara | title=PCL strut-like scaffolds appear superior to gyroid in terms of bone regeneration within a long bone large defect: An in silico study | journal=Front. Bioeng. Biotechnol.| date=2022-09-23| volume=10| page=995266| doi=10.3389/fbioe.2022.995266 | pmid=36213070| pmc=9540363| doi-access=free}}

In the shoulder of the wings of blue-winged leafbirds (Chloropsis moluccensis), gyroid crystals cause a very special-looking mesh of blue hues due to interference patterns. It is the only known species that has this adaptation.https://www.pnas.org/doi/10.1073/pnas.2101357118 Vinodkumar, Narayanan, Sandy, Prum

{{reflist|30em}}

• [http://schoengeometry.com/e-tpms.html Triply Periodic Minimal Surfaces at schoengeometry.com]
• [http://mathworld.wolfram.com/Gyroid.html Gyroid] at MathWorld
• [http://www.bathsheba.com/math/gyroid Rotatable picture of a gyroid's period]
• [https://minimal.sitehost.iu.edu/archive/Triply/genus3/Gyroid%20small/web/index.html The gyroid at Bloomington's Virtual Minimal Surface Museum]
• [https://books.google.com/books?id=eLBR7ONRzUMC&pg=PA92 Electrochemical Nanofabrication: Principles and Applications]

{{Minimal surfaces}}

Category:Minimal surfaces