Reuleaux polygon

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A Reuleaux triangle replaces the sides of an equilateral triangle by circular arcs ReuleauxTriangle.svg
A Reuleaux triangle replaces the sides of an equilateral triangle by circular arcs
Reuleaux polygons.svg
Regular Reuleaux polygons
Reuleaux polygon construction.svg
Irregular Reuleaux heptagon
Gambian dalasi coin, a Reuleaux heptagon Gambia 1 dalasi.JPG
Gambian dalasi coin, a Reuleaux heptagon

In geometry, a Reuleaux polygon is a curve of constant width made up of circular arcs of constant radius. [1] These shapes are named after their prototypical example, the Reuleaux triangle, which in turn is named after 19th-century German engineer Franz Reuleaux. [2] The Reuleaux triangle can be constructed from an equilateral triangle by connecting each pair of adjacent vertices with a circular arc centered on the opposing vertex, and Reuleaux polygons can be formed by a similar construction from any regular polygon with an odd number of sides as well as certain irregular polygons. Every curve of constant width can be accurately approximated by Reuleaux polygons. They have been applied in coinage shapes.

Contents

Construction

If is a convex polygon with an odd number of sides, in which each vertex is equidistant to the two opposite vertices and closer to all other vertices, then replacing each side of by an arc centered at its opposite vertex produces a Reuleaux polygon. As a special case, this construction is possible for every regular polygon with an odd number of sides. [1]

Every Reuleaux polygon must have an odd number of circular-arc sides, and can be constructed in this way from a polygon, the convex hull of its arc endpoints. However, it is possible for other curves of constant width to be made of an even number of arcs with varying radii. [1]

Properties

The Reuleaux polygons based on regular polygons are the only curves of constant width whose boundaries are formed by finitely many circular arcs of equal length. [3]

Every curve of constant width can be approximated arbitrarily closely by a (possibly irregular) Reuleaux polygon of the same width. [1]

Four 15-sided Reinhardt polygons, formed from four different Reuleaux polygons with 9, 3, 5, and 15 sides Reinhardt 15-gons.svg
Four 15-sided Reinhardt polygons, formed from four different Reuleaux polygons with 9, 3, 5, and 15 sides

A regular Reuleaux polygon has sides of equal length. More generally, when a Reuleaux polygon has sides that can be split into arcs of equal length, the convex hull of the arc endpoints is a Reinhardt polygon. These polygons are optimal in multiple ways: they have the largest possible perimeter for their diameter, the largest possible width for their diameter, and the largest possible width for their perimeter. [4]

Applications

The constant width of these shapes allows their use as coins that can be used in coin-operated machines. For instance, the United Kingdom has made 20-pence and 50-pence coins in the shape of a regular Reuleaux heptagon. [5] The Canadian loonie dollar coin uses another regular Reuleaux polygon with 11 sides. [6] However, some coins with rounded-polygon sides, such as the 12-sided 2017 British pound coin, do not have constant width and are not Reuleaux polygons. [7]

Although Chinese inventor Guan Baihua has made a bicycle with Reuleaux polygon wheels, the invention has not caught on. [8]

Related Research Articles

A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.

In geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need to also be equiangular, but if it does then it is a regular polygon. If the number of sides is at least five, an equilateral polygon does not need to be a convex polygon: it could be concave or even self-intersecting.

<span class="mw-page-title-main">Star polygon</span> Regular non-convex polygon

In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations on regular simple or star polygons.

<span class="mw-page-title-main">Heptagon</span> Shape with seven sides

In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon.

<span class="mw-page-title-main">Midpoint</span> Point on a line segment which is equidistant from both endpoints

In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.

<span class="mw-page-title-main">Curve of constant width</span> Shape with width independent of orientation

In geometry, a curve of constant width is a simple closed curve in the plane whose width is the same in all directions. The shape bounded by a curve of constant width is a body of constant width or an orbiform, the name given to these shapes by Leonhard Euler. Standard examples are the circle and the Reuleaux triangle. These curves can also be constructed using circular arcs centered at crossings of an arrangement of lines, as the involutes of certain curves, or by intersecting circles centered on a partial curve.

<span class="mw-page-title-main">Reuleaux triangle</span> Curved triangle with constant width

A Reuleaux triangle is a curved triangle with constant width, the simplest and best known curve of constant width other than the circle. It is formed from the intersection of three circular disks, each having its center on the boundary of the other two. Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because its width is constant, the Reuleaux triangle is one answer to the question "Other than a circle, what shape can a manhole cover be made so that it cannot fall down through the hole?"

<span class="mw-page-title-main">Barbier's theorem</span>

In geometry, Barbier's theorem states that every curve of constant width has perimeter π times its width, regardless of its precise shape. This theorem was first published by Joseph-Émile Barbier in 1860.

<span class="mw-page-title-main">Semicircle</span> Geometric shape

In mathematics, a semicircle is a one-dimensional locus of points that forms half of a circle. It is a circular arc that measures 180°. It has only one line of symmetry.

<span class="mw-page-title-main">Convex polygon</span> Polygon that is the boundary of a convex set

In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon. Equivalently, a polygon is convex if every line that does not contain any edge intersects the polygon in at most two points.

<span class="mw-page-title-main">Simple polygon</span> Shape bounded by non-intersecting line segments

In geometry, a simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a piecewise-linear Jordan curve consisting of finitely many line segments. These polygons include as special cases the convex polygons, star-shaped polygons, and monotone polygons.

<span class="mw-page-title-main">Inscribed figure</span> Geometric figure which is "snugly enclosed" by another figure

In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about figure F". A circle or ellipse inscribed in a convex polygon is tangent to every side or face of the outer figure. A polygon inscribed in a circle, ellipse, or polygon has each vertex on the outer figure; if the outer figure is a polygon or polyhedron, there must be a vertex of the inscribed polygon or polyhedron on each side of the outer figure. An inscribed figure is not necessarily unique in orientation; this can easily be seen, for example, when the given outer figure is a circle, in which case a rotation of an inscribed figure gives another inscribed figure that is congruent to the original one.

<span class="mw-page-title-main">Four-vertex theorem</span> Closed curves have ≥4 extremes of curvature

The four-vertex theorem of geometry states that the curvature along a simple, closed, smooth plane curve has at least four local extrema. The name of the theorem derives from the convention of calling an extreme point of the curvature function a vertex. This theorem has many generalizations, including a version for space curves where a vertex is defined as a point of vanishing torsion.

<span class="mw-page-title-main">Reuleaux tetrahedron</span> Shape formed by intersecting four balls

The Reuleaux tetrahedron is the intersection of four balls of radius s centered at the vertices of a regular tetrahedron with side length s. The spherical surface of the ball centered on each vertex passes through the other three vertices, which also form vertices of the Reuleaux tetrahedron. Thus the center of each ball is on the surfaces of the other three balls. The Reuleaux tetrahedron has the same face structure as a regular tetrahedron, but with curved faces: four vertices, and four curved faces, connected by six circular-arc edges.

<span class="mw-page-title-main">Line segment</span> Part of a line that is bounded by two distinct end points; line with two endpoints

In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. It is a special case of an arc, with zero curvature. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using an overline (vinculum) above the symbols for the two endpoints, such as in AB.

Reuleaux may refer to:

<span class="mw-page-title-main">Blaschke–Lebesgue theorem</span> Plane geometry theorem on least area of all curves of given constant width

In plane geometry the Blaschke–Lebesgue theorem states that the Reuleaux triangle has the least area of all curves of given constant width. In the form that every curve of a given width has area at least as large as the Reuleaux triangle, it is also known as the Blaschke–Lebesgue inequality. It is named after Wilhelm Blaschke and Henri Lebesgue, who published it separately in the early 20th century.

In geometry, a circular triangle is a triangle with circular arc edges.

In geometry, a polycon is a kind of a developable roller. It is made of identical pieces of a cone whose apex angle equals the angle of an even sided regular polygon. In principle, there are infinitely many polycons, as many as there are even sided regular polygons. Most members of the family have elongated spindle like shapes. The polycon family generalizes the sphericon. It was discovered by the Israeli inventor David Hirsch in 2017

<span class="mw-page-title-main">Reinhardt polygon</span> Polygon with many longest diagonals

In geometry, a Reinhardt polygon is an equilateral polygon inscribed in a Reuleaux polygon. As in the regular polygons, each vertex of a Reinhardt polygon participates in at least one defining pair of the diameter of the polygon. Reinhardt polygons with sides exist, often with multiple forms, whenever is not a power of two. Among all polygons with sides, the Reinhardt polygons have the largest possible perimeter for their diameter, the largest possible width for their diameter, and the largest possible width for their perimeter. They are named after Karl Reinhardt, who studied them in 1922.

References

  1. 1 2 3 4 Martini, Horst; Montejano, Luis; Oliveros, Déborah (2019), "Section 8.1: Reuleaux Polygons", Bodies of Constant Width: An Introduction to Convex Geometry with Applications, Birkhäuser, pp. 167–169, doi:10.1007/978-3-030-03868-7, ISBN   978-3-030-03866-3, MR   3930585, S2CID   127264210
  2. Alsina, Claudi; Nelsen, Roger B. (2011), Icons of Mathematics: An Exploration of Twenty Key Images, Dolciani Mathematical Expositions, vol. 45, Mathematical Association of America, p. 155, ISBN   978-0-88385-352-8
  3. Firey, W. J. (1960), "Isoperimetric ratios of Reuleaux polygons", Pacific Journal of Mathematics , 10 (3): 823–829, doi: 10.2140/pjm.1960.10.823 , MR   0113176
  4. Hare, Kevin G.; Mossinghoff, Michael J. (2019), "Most Reinhardt polygons are sporadic", Geometriae Dedicata , 198: 1–18, arXiv: 1405.5233 , doi:10.1007/s10711-018-0326-5, MR   3933447, S2CID   119629098
  5. Gardner, Martin (1991), "Chapter 18: Curves of Constant Width", The Unexpected Hanging and Other Mathematical Diversions, University of Chicago Press, pp. 212–221, ISBN   0-226-28256-2
  6. Chamberland, Marc (2015), Single Digits: In Praise of Small Numbers, Princeton University Press, pp. 104–105, ISBN   9781400865697
  7. Freiberger, Marianne (December 13, 2016), "New £1 coin gets even", Plus Magazine
  8. du Sautoy, Marcus (May 27, 2009), "A new bicycle reinvents the wheel, with a pentagon and triangle", The Times. See also Newitz, Annalee (September 30, 2014), "Inventor creates seriously cool wheels", Gizmodo