Editing Reuleaux triangle (section)

=== Rotation within a square ===
[[File:Rotation of Reuleaux triangle.gif|thumb|Rotation of a Reuleaux triangle within a square, showing also the curve traced by the center of the triangle]]
Any curve of constant width can form a rotor within a [[square]], a shape that can perform a complete rotation while staying within the square and at all times touching all four sides of the square. However, the Reuleaux triangle is the rotor with the minimum possible area.<ref name="gardner" /> As it rotates, its axis does not stay fixed at a single point, but instead follows a curve formed by the pieces of four [[ellipse]]s.<ref>{{citation
 | last1 = Gleiftner | first1 = Winfried
 | last2 = Zeitler | first2 = Herbert
 | date = May 2000
 | doi = 10.1007/bf03322004
 | issue = 3–4
 | journal = [[Results in Mathematics]]
 | pages = 335–344
 | title = The Reuleaux triangle and its center of mass
 | volume = 37| s2cid = 119600507
 }}.</ref> Because of its 120° angles, the rotating Reuleaux triangle cannot reach some points near the sharper angles at the square's vertices, but rather covers a shape with slightly rounded corners, also formed by elliptical arcs.<ref name="gardner" />

{{multiple image|align=center|image1=Reuleaux triangle rotation center.svg|alt1=Reuleaux triangle in a square, with ellipse governing the path of motion of the triangle center|caption1=One of the four ellipses followed by the center of a rotating Reuleaux triangle in a square|image2=Reuleaux triangle rotation corners.svg|alt2=Reuleaux triangle in a square, with ellipse bounding the region swept by the triangle|caption2=Ellipse separating one of the corners (lower left) of a square from the region swept by a rotating Reuleaux triangle}}

At any point during this rotation, two of the corners of the Reuleaux triangle touch two adjacent sides of the square, while the third corner of the triangle traces out a curve near the opposite vertex of the square. The shape traced out by the rotating Reuleaux triangle covers approximately 98.8% of the area of the square.<ref>{{citation|author-link=Clifford A. Pickover|last=Pickover|first=Clifford A.|title=The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics|publisher=Sterling Publishing Company|year=2009|isbn= 978-1-4027-5796-9|pages=266|contribution=Reuleaux Triangle|url=https://books.google.com/books?id=JrslMKTgSZwC&pg=PA266}}.</ref>