Editing Reuleaux triangle (section)

== Applications ==

=== Reaching into corners ===
Several types of machinery take the shape of the Reuleaux triangle, based on its property of being able to rotate within a square.

The [[Watts Brothers Tool Works]] square [[drill bit]] has the shape of a Reuleaux triangle, modified with concavities to form cutting surfaces. When mounted in a special chuck which allows for the bit not having a fixed centre of rotation, it can drill a hole that is nearly square.<ref name="watts">{{citation
|publisher=[[Watts Brothers Tool Works]]
|title=How to drill square hexagon octagon pentagon holes
|location=Wilmerding, Pennsylvania
|year=1950–1951}} (27 page brochure).</ref> Although patented by Henry Watts in 1914, similar drills invented by others were used earlier.<ref name="gardner" /> Other Reuleaux polygons are used to drill pentagonal, hexagonal, and octagonal holes.<ref name="gardner" /><ref name="watts" />

[[Panasonic]]'s RULO [[robotic vacuum cleaner]] has its shape based on the Reuleaux triangle in order to ease cleaning up dust in the corners of rooms.<ref>{{citation|last1=Mochizuki|first1=Takashi|title=Panasonic Rolls Out Triangular Robot Vacuum|url=https://blogs.wsj.com/japanrealtime/2015/01/22/panasonic-rolls-out-triangular-robot-vacuum/|department=Japan Real Time|newspaper=[[Wall Street Journal]]|date=January 22, 2015}}.</ref><ref>{{citation|url=http://www.gizmag.com/panasonic-rulo/36378/|title=Panasonic enters the robo-vac game, with the triangular Rulo|first=Ben|last=Coxworth|date=March 3, 2015|magazine=Gizmag}}.</ref>

=== Rolling cylinders ===
[[File:Reuleaux triangle 54.JPG|thumb|Comparison of a cylindrical and Reuleaux triangle roller]]
Another class of applications of the Reuleaux triangle involves cylindrical objects with a Reuleaux triangle cross section.  Several pencils are manufactured in this shape, rather than the more traditional round or hexagonal barrels.<ref name="pencil">{{citation|url=http://www.pencilrevolution.com/2006/04/review-of-staedtler-noris-ergosoft-hb/|title=Review of Staedtler Noris Ergosoft HB|work=Pencil Revolution|date=April 26, 2006|access-date=2015-05-22|first=Johnny|last=Gamber|archive-date=2015-05-25|archive-url=https://web.archive.org/web/20150525033942/http://www.pencilrevolution.com/2006/04/review-of-staedtler-noris-ergosoft-hb/|url-status=dead}}.</ref>  They are usually promoted as being more comfortable or encouraging proper grip, as well as being less likely to roll off tables (since the center of gravity moves up and down more than a rolling hexagon).

A Reuleaux triangle (along with all other [[curve of constant width|curves of constant width]]) can [[rolling|roll]] but makes a poor wheel because it does not roll about a fixed center of rotation. An object on top of rollers that have Reuleaux triangle cross-sections would roll smoothly and flatly, but an axle attached to Reuleaux triangle wheels would bounce up and down three times per revolution.<ref name="gardner" /><ref>{{citation
 | last1 = Masferrer León | first1 = Claudia
 | last2 = von Wuthenau Mayer | first2 = Sebastián
 | date = December 2005
 | doi = 10.1007/bf02985852
 | issue = 4
 | journal = [[The Mathematical Intelligencer]]
 | pages = 7–13
 | title = Reinventing the wheel: Non-circular wheels
 | volume = 27}}.</ref> This concept was used in a science fiction short story by [[Poul Anderson]] titled "The Three-Cornered Wheel".<ref name="mj14" /><ref>{{citation|last1=Anderson|first1=Poul|title=The Three-Cornered Wheel|magazine=[[Analog Science Fiction and Fact|Analog]]|date=October 1963|pages=50–69}}</ref> A bicycle with floating axles and a frame supported by the rim of its Reuleaux triangle shaped wheel was built and demonstrated in 2009 by Chinese inventor Guan Baihua, who was inspired by pencils with the same shape.<ref>{{citation|first=Tyra|last=Dempster|url=https://www.reuters.com/article/us-china-bicycle-idUSTRE55G1GB20090617|publisher=Reuters|title=Chinese man reinvents the wheel|date=June 17, 2009}}</ref>

=== Mechanism design ===
[[File:Luch2 greifer.gif|thumb|upright|left|Film advance mechanism in the Soviet Luch-2 8mm film projector based on a Reuleaux triangle]]
Another class of applications of the Reuleaux triangle involves using it as a part of a [[Linkage (mechanical)|mechanical linkage]] that can convert [[rotation around a fixed axis]]
into [[reciprocating motion]].<ref name="onup">{{citation|title=Old and New Unsolved Problems in Plane Geometry and Number Theory|volume=11|series=Dolciani mathematical expositions|first1=Victor|last1=Klee|author1-link=Victor Klee|first2=S.|last2=Wagon|author2-link=Stan Wagon|publisher=Cambridge University Press|year=1991|isbn=978-0-88385-315-3|page=21|url=https://books.google.com/books?id=tRdoIhHh3moC&pg=PA21}}.</ref> These mechanisms were studied by Franz Reuleaux. With the assistance of the Gustav Voigt company, Reuleaux built approximately 800 models of mechanisms, several of which involved the Reuleaux triangle.<ref name="cornell" /> Reuleaux used these models in his pioneering scientific investigations of their motion.<ref>{{citation|title=Mathematics and the Aesthetic: New Approaches to an Ancient Affinity|series=CMS Books in Mathematics|editor1-first=Nathalie|editor1-last=Sinclair|editor1-link=Nathalie Sinclair|editor2-first=David|editor2-last=Pimm|editor3-first=William|editor3-last=Higginson|publisher=Springer|year=2007|isbn=978-0-387-38145-9|contribution=Experiencing meanings in geometry|first1=David W.|last1=Henderson|first2=Daina|last2=Taimina|author2-link=Daina Taimina|pages=58–83|doi=10.1007/978-0-387-38145-9_4|hdl=1813/2714|hdl-access=free}}. See in particular [https://books.google.com/books?id=GJBKLnkYyi0C&pg=PA81 p. 81].</ref> Although most of the Reuleaux–Voigt models have been lost, 219 of them have been collected at [[Cornell University]], including nine based on the Reuleaux triangle.<ref name="cornell">{{citation|title=The Reuleaux Collection of Kinematic Mechanisms at Cornell University|author1-link=Francis C. Moon|first=Francis C.|last=Moon|date=July 1999|publisher=Cornell University Library|url=https://www.asme.org/wwwasmeorg/media/resourcefiles/aboutasme/who%20we%20are/engineering%20history/landmarks/232-reuleaux-collection-of-kinematic-mechanisms-at-cornell-university.pdf|archive-url=https://web.archive.org/web/20200614022856/https://www.asme.org/wwwasmeorg/media/resourcefiles/aboutasme/who%20we%20are/engineering%20history/landmarks/232-reuleaux-collection-of-kinematic-mechanisms-at-cornell-university.pdf|archive-date=June 14, 2020}}.</ref><ref name="moon241" /> However, the use of Reuleaux triangles in mechanism design predates the work of Reuleaux; for instance, some [[steam engine]]s from as early as 1830 had a [[Cam (mechanism)|cam]] in the shape of a Reuleaux triangle.<ref>{{harvtxt|Moon|2007|page=240}}</ref><ref name="mathtrek">{{citation|first=Ivars|last=Peterson|author-link=Ivars Peterson|title=Rolling with Reuleaux|date=October 19, 1996|work=MathTrek|url=https://www.sciencenews.org/article/rolling-reuleaux|publisher=[[ScienceNews]]}}. Reprinted in {{citation|title=Mathematical Treks: From Surreal Numbers to Magic Circles|series=MAA spectrum|first=Ivars|last=Peterson|author-link=Ivars Peterson|publisher=[[Mathematical Association of America]]|year=2002|isbn=978-0-88385-537-9|pages=141–144|url=https://books.google.com/books?id=4gWSAraVhtAC&pg=PA141}}.</ref>

One application of this principle arises in a [[film projector]]. In this application, it is necessary to advance the film in a jerky, stepwise motion, in which each frame of film stops for a fraction of a second in front of the projector lens, and then much more quickly the film is moved to the next frame. This can be done using a mechanism in which the rotation of a Reuleaux triangle within a square is used to create a motion pattern for an actuator that pulls the film quickly to each new frame and then pauses the film's motion while the frame is projected.<ref>{{harvtxt|Lay|2007}}, [https://books.google.com/books?id=U9eOPjmaH90C&pg=PA83 p.&nbsp;83].</ref>

The rotor of the [[Wankel engine]] is shaped as a curvilinear triangle that is often cited as an example of a Reuleaux triangle.<ref name="icons"/><ref name="howround"/><ref name= "gardner" /><ref name="mathtrek" /> However, its curved sides are somewhat flatter than those of a Reuleaux triangle and so it does not have constant width.<ref>{{harvtxt|Gruber|1983|page=80}}</ref><ref>{{citation | last = Nash | first = David H.
 | date = March 1977
 | doi = 10.1080/0025570x.1977.11976621
 | issue = 2
 | journal = Mathematics Magazine
 | pages = 87–89
 | title = Rotary engine geometry
 | volume = 50}}</ref><ref> {{citation | last1 = Badr | first1 = O.
 | last2 = Naik | first2 = S.
 | last3 = O'Callaghan | first3 = P. W.
 | last4 = Probert | first4 = S. D.
 | doi = 10.1016/0306-2619(91)90063-4
 | issue = 1
 | journal = Applied Energy
 | pages = 59–76
 | title = Rotary Wankel engines as expansion devices in steam Rankine-cycle engines
 | volume = 39
 | year = 1991| bibcode = 1991ApEn...39...59B
 }}.</ref>

=== Architecture ===
[[File:Reuleaux triangle shaped window of Onze-Lieve-Vrouwekerk, Bruges.jpg|thumb|Reuleaux triangle shaped window of the [[Church of Our Lady, Bruges]] in Belgium]]
In [[Gothic architecture]], beginning in the late 13th century or early 14th century,<ref name="mcwte">{{citation|url=https://books.google.com/books?id=SY7aHx8KwK0C&pg=PA63|pages=63–64|title=Medieval Church Window Tracery in England|first=Stephen|last=Hart|publisher=Boydell & Brewer Ltd|year=2010|isbn=978-1-84383-533-2}}.</ref> the Reuleaux triangle became one of several curvilinear forms frequently used for windows, window [[tracery]], and other architectural decorations.<ref name="icons" /> For instance, in [[English Gothic architecture]], this shape was associated with the decorated period, both in its geometric style of 1250–1290 and continuing into its curvilinear style of 1290–1350.<ref name="mcwte" /> It also appears in some of the windows of the [[Milan Cathedral]].<ref>{{citation|last1=Marchetti|first1=Elena|last2=Costa|first2=Luisa Rossi|editor1-last=Williams|editor1-first=Kim|editor2-link=Kim Williams (architect)|editor2-last=Ostwald|editor2-first=Michael J.|contribution=What geometries in Milan Cathedral?|doi=10.1007/978-3-319-00137-1_35|pages=509–534|publisher=Birkhäuser|title=Architecture and Mathematics from Antiquity to the Future, Volume I: Antiquity to the 1500s|year=2014|isbn=978-3-319-00136-4 }}</ref> In this context, the shape is sometimes called a ''spherical triangle'',<ref name="mcwte" /><ref>{{citation|title= A glossary of terms used in Grecian, Roman, Italian, and Gothic architecture|volume=1|first=John Henry|last=Parker|edition=5th|year=1850|page=202|url=https://books.google.com/books?id=uXtZAAAAYAAJ&pg=PA202|location=London|publisher=David Rogue}}.</ref><ref>{{citation|title=Practical plane geometry|first=E. S.|last=Burchett|year=1876|at=Caption to Plate LV, Fig. 6|url=https://books.google.com/books?id=oDcDAAAAQAAJ&pg=RA1-PA94|location=London and Glasgow|publisher=William Collins, Sons, and Co.}}.</ref> which should not be confused with [[spherical triangle]] meaning a triangle on the surface of a [[sphere]]. In its use in Gothic church architecture, the three-cornered shape of the Reuleaux triangle may be seen both as a symbol of the [[Trinity]],<ref>{{citation|title=The Symbolism of Churches and Church Ornaments: A Translation of the First Book of the Rationale Divinorum Officiorum|first=Guillaume|last=Durand|edition=3rd|publisher=Gibbings|year=1906|url=https://books.google.com/books?id=vRknpENyAlQC&pg=PR88|page=lxxxviii}}.</ref> and as "an act of opposition to the form of the circle".<ref>{{citation|title=Gothic Architecture|volume=19|series=Pelican history of art|first1=Paul|last1=Frankl|first2=Paul|last2=Crossley|publisher=Yale University Press|year=2000|isbn=978-0-300-08799-4|page=146|url=https://books.google.com/books?id=LBZ6781vvOwC&pg=PA146}}.</ref>

The Reuleaux triangle has also been used in other styles of architecture. For instance, [[Leonardo da Vinci]] sketched this shape as the plan for a fortification.<ref name="moon241">{{harvtxt|Moon|2007|page=241}}.</ref> Modern buildings that have been claimed to use a Reuleaux triangle shaped floorplan include the [[Kresge Auditorium|MIT Kresge Auditorium]], the [[Kölntriangle]], the [[Donauturm]], the [[Torre de Collserola]], and the [[Mercedes-Benz Museum]].<ref name=conti/> However in many cases these are merely rounded triangles, with different geometry than the Reuleaux triangle.

=== Mapmaking ===
{{main | octant projection }}
Another early application of the Reuleaux triangle, [[da Vinci's world map]] from circa 1514, was a [[world map]] in which the spherical surface of the earth was divided into eight octants, each flattened into the shape of a Reuleaux triangle.<ref name="snyder">{{citation|title=Flattening the Earth: Two Thousand Years of Map Projections|first=John P.|last=Snyder|publisher=University of Chicago Press|year=1997|isbn=978-0-226-76747-5|page=40|url=https://books.google.com/books?id=0UzjTJ4w9yEC&pg=PA40}}.</ref><ref>{{citation
 | last = Keuning | first = Johannes
 | date = January 1955
 | doi = 10.1080/03085695508592085
 | issue = 1
 | journal = [[Imago Mundi]]
 | jstor = 1150090
 | pages = 1–24
 | title = The history of geographical map projections until 1600
 | volume = 12}}.</ref><ref name="dee">{{citation
 | last = Bower | first = David I.
 | date = February 2012
 | doi = 10.1179/1743277411y.0000000015
 | issue = 1
 | journal = [[The Cartographic Journal]]
 | pages = 55–61
 | title = The unusual projection for one of John Dee's maps of 1580
 | url = http://dibower.co.uk/data/documents/Dee-projn-paper.pdf
 | volume = 49| bibcode = 2012CartJ..49...55B
 | s2cid = 129873912
 }}.</ref>
[[File:Leonardo da Vinci’s Mappamundi.jpg|thumb|center|upright=2|Leonardo [[da Vinci's world map]] in eight Reuleaux-triangle quadrants]]

Similar maps also based on the Reuleaux triangle were published by [[Oronce Finé]] in 1551 and by [[John Dee]] in 1580.<ref name="dee" />

=== Other objects ===
[[File:V-Pick Psychos.JPG|thumb|Reuleaux triangle shaped [[guitar pick]]s]]
Many [[guitar pick]]s employ the Reuleaux triangle, as its shape combines a sharp point to provide strong articulation, with a wide tip to produce a warm timbre. Because all three points of the shape are usable, it is easier to orient and wears less quickly compared to a pick with a single tip.<ref name="picks">{{citation|last=Hoover|first=Will|date=November 1995|title= Picks!: The Colorful Saga of Vintage Celluloid Guitar Plectrums|pages=32–33|publisher=Backbeat Books|isbn=978-0-87930-377-8}}.</ref>

{{multiple image|total_width=420|image1=Philadelphia fire hydrant.jpg|image2=Tux Hydrant.jpg|footer=Illicit use of a fire hydrant, Philadelphia, 1996, and a newer Philadelphia hydrant with a Reuleaux triangle shaped nut to prevent such use.}}
The Reuleaux triangle has been used as the shape for the cross section of a [[fire hydrant]] valve nut. The constant width of this shape makes it difficult to open the fire hydrant using standard parallel-jawed wrenches; instead, a wrench with a special shape is needed. This property allows the fire hydrants to be opened only by firefighters (who have the special wrench) and not by other people trying to use the hydrant as a source of water for other activities.<ref>{{citation
 | last1 = Martini | first1 = Horst
 | last2 = Montejano | first2 = Luis
 | last3 = Oliveros | first3 = Déborah | author3-link = Déborah Oliveros
 | doi = 10.1007/978-3-030-03868-7
 | isbn = 978-3-030-03866-3
 | mr = 3930585
 | page = 3
 | publisher = Birkhäuser
 | title = Bodies of Constant Width: An Introduction to Convex Geometry with Applications
 | year = 2019| s2cid = 127264210
 }}</ref>

[[File:Smithsonian Submillimeter Array.jpg|thumb|The [[Submillimeter Array]], with seven of its eight antennae arranged on an approximate Reuleaux triangle]]
Following a suggestion of {{harvtxt|Keto|1997}},<ref name="keto">{{citation
 | last = Keto | first = Eric
 | bibcode = 1997ApJ...475..843K
 | doi = 10.1086/303545
 | issue = 2
 | journal = [[The Astrophysical Journal]]
 | pages = 843–852
 | title = The shapes of cross-correlation interferometers
 | volume = 475
 | year = 1997| doi-access = free
 }}.</ref> the antennae of the [[Submillimeter Array]], a radio-wave astronomical observatory on [[Mauna Kea]] in [[Hawaii]], are arranged on four nested Reuleaux triangles.<ref name="blundell">{{citation
 | last = Blundell | first = Raymond
 | contribution = The submillimeter array
 | contribution-url = https://www.cfa.harvard.edu/sma/general/IEEE.pdf
 | doi = 10.1109/mwsym.2007.380132
 | title = Proc. 2007 IEEE/MTT-S International Microwave Symposium
 | pages = 1857–1860
 | year = 2007| isbn = 978-1-4244-0687-6
 | s2cid = 41312640
 }}.</ref><ref name="hml04">{{citation
 | last1 = Ho | first1 = Paul T. P.
 | last2 = Moran | first2 = James M.
 | last3 = Lo | first3 = Kwok Yung
 | bibcode = 2004ApJ...616L...1H
 | doi = 10.1086/423245
 | issue = 1
 | journal = [[The Astrophysical Journal]]
 | pages = L1–L6
 | title = The submillimeter array
 | volume = 616
 | year = 2004| arxiv = astro-ph/0406352| s2cid = 115133614
 }}.</ref> Placing the antennae on a curve of constant width causes the observatory to have the same spatial resolution in all directions, and provides a circular observation beam. As the most asymmetric curve of constant width, the Reuleaux triangle leads to the most uniform coverage of the plane for the [[Fourier transform]] of the signal from the array.<ref name="keto"/><ref name="hml04"/> The antennae may be moved from one Reuleaux triangle to another for different observations, according to the desired angular resolution of each observation.<ref name="blundell"/><ref name="hml04"/> The precise placement of the antennae on these Reuleaux triangles was optimized using a [[neural network]]. In some places the constructed observatory departs from the preferred Reuleaux triangle shape because that shape was not possible within the given site.<ref name="hml04"/>

=== Signs and logos ===
The shield shapes used for many signs and corporate logos feature rounded triangles. However, only some of these are Reuleaux triangles.

The corporate logo of [[Petrofina]] (Fina), a Belgian oil company with major operations in Europe, North America and Africa, used a Reuleaux triangle with the Fina name from 1950 until Petrofina's merger with ''Total S.A.'' (today [[TotalEnergies]]) in 2000.<ref>{{citation|title=Interesting Stuff: Curves of Constant Width|first=Sam|last=Gwillian|url=http://newportcityradio.org/culture/interesting-stuff-curves-of-constant-width/all-pages|archive-url=https://archive.today/20160616012426/http://newportcityradio.org/culture/interesting-stuff-curves-of-constant-width/all-pages|url-status=dead|archive-date=June 16, 2016|publisher=Newport City Radio|date=May 16, 2015}}</ref><ref>{{citation|title=Fina Logo History: from Petrofina to Fina|url=http://www.total.com/en/about-total/group-presentation/group-history/fina-logo-history-922651.html|work=Total: Group Presentation|publisher=Total S.A.|access-date=31 October 2015|archive-url=https://web.archive.org/web/20121226182719/http://www.total.com/en/about-total/group-presentation/group-history/fina-logo-history-922651.html|archive-date=December 26, 2012}}.</ref>
Another corporate logo framed in the Reuleaux triangle, the south-pointing [[compass]] of [[Bavaria Brewery (Netherlands)|Bavaria Brewery]], was part of a makeover by design company Total Identity that won the SAN 2010 Advertiser of the Year award.<ref>{{citation|url=https://totalidentity.com/global-bavaria|title=Global: Bavaria, Fundamental Rebranding Operation at Bavaria|website=Total Identity|access-date=2015-06-27|url-status=unfit|archive-url=https://web.archive.org/web/20150630071933/https://totalidentity.com/global-Bavaria|archive-date=2015-06-30}}</ref> The Reuleaux triangle is also used in the logo of [[Colorado School of Mines]].<ref>{{citation|title=M-blems: Explaining the logo|url=http://magazine.mines.edu/BackIssues/PDF_Archives/vol_92_num_2.pdf|page=29|first=Roland B.|last=Fisher|volume=92|number=2|date=Spring 2002|magazine=Mines: The Magazine of Colorado School of Mines|archive-url=https://web.archive.org/web/20100710165916/http://magazine.mines.edu/BackIssues/PDF_Archives/vol_92_num_2.pdf|archive-date=2010-07-10|url-status=usurped}}</ref>

In the United States, the [[National Trails System]] and [[United States Bicycle Route System]] both mark routes with Reuleaux triangles on signage.<ref>{{citation|title=Information: MUTCD — Interim Approval for the Optional Use of an Alternative Design for the U.S. Bicycle Route (M1-9) Sign (IA-15)|work=Manual on Uniform Traffic Control Devices for Streets and Highways: Resources|first=Jeffrey A.|last=Lindley|date=June 1, 2012|access-date=August 20, 2018|url=https://mutcd.fhwa.dot.gov/resources/interim_approval/ia15/|publisher=US Department of Transportation, Federal Highway Administration}}</ref>

=== In nature ===
[[File:Reuleaux foam.svg|thumb|upright|The Reuleaux triangle as the central bubble in a mathematical model of a four-bubble planar soap bubble cluster]]
According to [[Plateau's laws]], the circular arcs in two-dimensional [[soap bubble]] clusters meet at 120° angles, the same angle found at the corners of a Reuleaux triangle. Based on this fact, it is possible to construct clusters in which some of the bubbles take the form of a Reuleaux triangle.<ref name="foam">{{citation
 | last1 = Modes | first1 = Carl D.
 | last2 = Kamien | first2 = Randall D.
 | arxiv = 0810.5724
 | doi = 10.1039/c3sm51585k
 | issue = 46
 | journal = [[Soft Matter (journal)|Soft Matter]]
 | pages = 11078–11084
 | title = Spherical foams in flat space
 | volume = 9
 | year = 2013| bibcode = 2013SMat....911078M
 | s2cid = 96591302
 }}.</ref>

The shape was first isolated in crystal form in 2014 as Reuleaux triangle disks.<ref>{{citation |first1=C. H. B. |last1= Ng |first2=W. Y. |last2=Fan |title=Reuleaux triangle disks: New shape on the block |journal=[[Journal of the American Chemical Society]] |volume=136 |issue=37 |year=2014 |pages=12840–12843 |doi=10.1021/ja506625y|pmid= 25072943 |bibcode= 2014JAChS.13612840N }}.</ref> Basic [[bismuth nitrate]] disks with the Reuleaux triangle shape were formed from the [[hydrolysis]] and [[precipitation]] of bismuth nitrate in an ethanol–water system in the presence of 2,3-bis(2-pyridyl)pyrazine.