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Reuleaux Triangle
by Daina Taimina & David W. Henderson

What is this triangle?

If an enormously heavy object has to be moved from one spot to another, it may not be practical to move it on wheels. Instead the object is placed on a flat platform that in turn rests on cylindrical rollers (Figure 1). As the platform is pushed forward, the rollers left behind are picked up and put down in front. An object moved this way over a flat horizontal surface does not bob up and down as it rolls along. The reason is that cylindrical rollers have a circular cross section, and a circle is closed curve "with constant width." What does that mean? If a closed convex curve is placed between two parallel lines and the lines are moved together until they touch the curve, the distance between the parallel lines is the curve's "width" in one direction. Because a circle has the same width in all directions, it can be rotated between two parallel lines without altering the distance between the lines.

Is a circle the only curve with constant width? Actually there are infinitely many such curves. The simplest noncircular such curve is named the Reuleaux triangle. Mathematicians knew it earlier (some references go back to Leonard Euler in the 18th century), but some curved triangles can be seen in Leonardo da Vinci's 15th-century Codex Madrid. In addition the 13th-century Notre Dame cathedral in Bruges, Belgium, has several windows in the clear shape of a Reuleaux triangle (Figure 2). But Franz Reuleaux was the first to demonstrate its constant-width properties and the first to use the triangle in mechanisms. See models B01, B02, B03, B04L01, L02, L03, L04, L05, and L06. A modern application of the Reuleaux triangle can be seen in the Wankel engine (Figure 3).

How to construct a Reuleaux triangle

To construct a Reuleaux triangle begin with an equilateral triangle of side s, and then replace each side by a circular arc with the other two original sides as radii (Figure 4).

The corners of a Reuleaux triangle are the sharpest possible on a curve with constant width. Extending each side of an equilateral triangle a uniform distance at each end can round these corners. The resulting curve has a width, in all directions, that is the sum of the same two radii (Figure 5).

Other symmetrical curves with constant width result if you start with a regular pentagon (or any regular polygon with an odd number of sides) and follow similar procedures. This construction is used in the design of some British coins (Figure 6).

This result can be generalized to the case when the polygon is not regular, but each of its vertices is the endpoint of two diagonals of the same length h (and the lengths of other diagonals are less than h). Of course, the polygon has to have an odd number of vertices.

Here is another really surprising method of constructing curves with constant width:

Draw as many straight lines as you like, but all mutually intersecting. Each arc of the curve will be drawn with the compass point at the intersection of the two lines that bound the arc. Start with any arc, then proceed around the curve, connecting each arc to the preceding one. If you do it carefully, the curve will close and will have a constant width. (You can try to prove it! It is not difficult at all.) The curves drawn in this way may have arcs of as many different circles as you wish. Here is one example (Figure 7), but you will really enjoy making your own! After you have made one, you can make more copies of it and check that your wheels really roll!

Another interesting result about curves of constant width is that the inscribed and circumscribed circles of an arbitrary figure of constant width h are concentric and the sum of their radii is equal to h.

Figure 1: Platform resting on cylindrical roller

Figure 2:  Window from Notre Dame Cathedral in Bruges, Belgium

Figure 3:  Wankel Engine (photo by Manfred Mornhinweg)

Figure 4: Replacing each side by circular arc with two original sides as radii

Figure 5: Extend side a uniform distance at each end to round corners

Figure 6: Practical application, British coins

Figure 7: Example, complex shape of constant width

Reuleaux Triangle Links

Problems about Reuleaux Triangles and other Curves with Constant Width

  • Find the area of a Reuleaux triangle. Does it exceed the area of the circle with the same width (diameter)?
  • Among curves with the same constant width, show that the circle bounds the region of greatest area and the Reuleaux triangle bounds the region of least area.
  • For the Reuleaux triangle, derive a formula linking the length of the perimeter to its width. This theorem is due to Joseph Emile Barbier (1839-1889): all shapes with constant width D have the same perimeter, L = π D. (For a proof of this theorem, see Lyusternik L.A. 1963. Convex figures and polyhedra. New York, Dover, pp. 31-35)
  • The angle between two intersecting curves is defined as the angle between their tangents at the point of intersection. Find internal angles of the Reuleaux triangle.
  • Assume that the side of an equilateral triangle is 50. Each side is then extended 10 units in each direction and the shape is rounded to construct a curve with constant width. What is the width of the resulting shape?
  • Show that for every point on the boundary of a figure with constant width there exists another boundary point whose distance from the original point is equal to the width of the shape.
  • Show that the distance between any two points inside a figure with constant width never exceeds its width.
  • Show that the length of the boundary of a figure with constant width depends only on the width of the figure.
  • Show that a curve with constant width has just one point in common with each of its supporting lines.
  • Show that the distance between any two points on a curve with constant width b is at most equal to b.
  • If a line joins the two points of contact of two parallel supporting lines of a curve with constant width, show that it is perpendicular to the supporting lines.
  • Show that there is at least one supporting line through every point on a curve with constant width.
  • Through every point P of a curve with constant width, show that a circle of radius b can be drawn that encloses the curve and that is tangent, at P, to the supporting lines of the curve, or to a predetermined supporting line if there are more than one.
  • If a circle has three (or more) points in common with a curve with constant width b, then show that the length of the radius of the circle is at most b.
  • Prove: The circle is the only curve with constant width with a center of symmetry. (Before proving this theorem it is useful first to prove the lemma: Any two diameters of a curve with constant width must intersect on the curve or in the interior of the curve. If they intersect on the curve, then the point of intersection is a corner point of the curve.)
  • A sequence of triangles and associated Reuleaux triangles can be drawn by using the midpoints of the sides of the original triangle as the vertices for the next triangle. The pattern can be repeated indefinitely, with each new triangle smaller than the preceding one. How does the total area of all smaller Reuleaux triangles compare with the area of the original Reuleaux triangle? The key observation that suggests a solution to the problem is realizing that, in addition to the sequence of Reuleaux triangles, there is a related sequence of equilateral triangles.

The three dimensional analog of a curve with constant width is the solid with constant width. A sphere is not the only such solid that will rotate within a cube, at all times touching all six sides of the cube; all solids of constant width share this property. Rotating a Reuleaux triangle around one of its axes of symmetry generates the simplest example of a nonspherical solid of this type. There are an infinite number of other such examples. The solids with constant width that have the smallest volumes are derived from the regular tetrahedron in somewhat the same way that the Reuleaux triangle is derived from the equilateral triangle: Spherical caps are first placed on each face of the tetrahedron, and then three of tghe edges must be slightly altered. These altered edges may either form a triangle or radiate from one corner. Since all curves with the same constant width have the same perimeter, it might be supposed that all solids width the same constant width have the same surface area. It was proved by Hermann Minkowski that all the shadows of solids with constant width are curves of the same constant width (when the projecting rays are parallel and the shadow falls on a plane perpendicular to the rays). All such shadows have equal perimeters. Michael Goldberg (1957, 1960, 1962) has introduced the term "rotor" for any convex figure that can be rotated inside a polygon or polyhedron while at all times touching every side or face. The Reuleaux triangle is the rotor of least area in a square. The least area rotor for the equilateral triangle is a biangle (a lens shaped figure) formed from two 60-degree arcs of the circle with radius equal to the triangle’s altitude. As the biangle rotates its corners trace the entire boundary of the triangle, with no rounding of corners. See B01.

Closely related to the theory of rotors is a famous problem named the Kakeya Needle Problem, which was first posed in 1917 by the Japanese mathematician Soichi Kakeya: What is the plane figure of least area, in which a line segment of length 1 can be rotated 360 degrees? The rotation obviously can be made inside a circle of unit diameter, but that is far from the smallest area. Ten years after the Kakeya problem was posed, the Russian mathematician Abram Besicovitch showed that there is no minimum area as an answer to Kakeya’s needle problem.

Reuleaux Tetrahedron Links

Kakeya Needle Problem Links

Where I can read more about Reuleaux triangle?

  • Blaschke Wilhelm, Kreis und Kugel. Leipzig, 1916; Berlin: W. de Gruyter, 1956.
  • Boltyanski Vladimir, Soifer Alexander, Geometric Etudes in Combinatorial Mathematics, Center for Excellence in Mathematical Education, Colorado Springs, 1991.
  • Cadwell J. H., Topics in Recreational Mathematics. Cambridge, England : Cambridge University Press, 1966, Ch.15.
  • Dossey John A., "What? A Roller With Corners? -Closed curves of constant width", Mathematics Teacher, No.65, 1972, pp. 720 - 724.
  • Gardner Martin, The Unexpected Hanging and Other Mathematical Diversions, Simon&Schuster, New York, 1969.
  • Goldberg Michael, "N-gon Rotors Making n+1 Contacts with Fixed Simple Curves", American Mathematical Monthly, Vol.69, June-July 1962, pp. 486-91.
  • Goldberg Michael, "Rotors in Polygons and Polyhedra," Mathematical Tables and Other Aids to Computation, Vol. 14, July 1960, pp.229-39.
  • Goldberg Michael, "Trammel Rotors in Regular Polygons," American Mathematical Monthly, Vol. 64, February 1957, pp. 71-78.
  • Rademacher Hans, Toeplitz Otto, The Enjoyment of Mathematics, Princeton, N. J.: Princeton University Press, 1957, pp 163-77, 203.
  • Reuleaux Franz, The Kinematics of Machinery. New York: Macmillan, 1876; Dover Publications, 1964, pp.129-46.
  • Smart James R., " Problem Solving in Geometry--a Sequence of Reuleaux Triangles: Investigation of area relations for a sequence of Reuleaux triangles associated with an equilateral triangle and a sequence of medial triangles", Mathematics Teacher, No.79, 1986, pp.11 - 14.
  • Smith Stanley A., "Rolling Curves - Activities involving curves of constant width", Mathematics Teacher, No.67, 1974, pp. 239 - 242.
  • Smith, Scott G., "Drilling square holes: Using a Reuleaux triangle", Mathematics Teacher, No.86, 1993, pp. 579 - 583.
  • Yaglom, I., M., Boltyanskii V. G., Convex Figures, New York: Holt, Rinehart& Winston, 1961,Ch. 7 and 8.

Where can I learn more about Kakeya's needle problem?

  • Besicovitch, A. S., "The Kakeya Problem", American Mathematical Monthly, Vol. 70, August-September 1963, pp. 697-706.
  • Blank A. A., "A Remark on the Kakeya Problem," American Mathematical Monthly, Vol. 70, August- September 1963, pp. 706-11.
  • Cadwell J. H., Topics in Recreational Mathematics, Cambridge, England: Cambridge University Press, 1966. Pp. 96-99.
  • Yaglom I. M., Boltyanskii, V. G., Convex Figures, New York: Holt, Rinehart& Winston, 1961. Pp. 61-62, 226-27.

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