 TOPICS # Curve of Constant Width

Curves which, when rotated in a square, make contact with all four sides. Such curves are sometimes also known as rollers.

The "width" of a closed convex curve is defined as the distance between parallel lines bounding it ("supporting lines"). Every curve of constant width is convex. Curves of constant width have the same "width" regardless of their orientation between the parallel lines. In fact, they also share the same perimeter (Barbier's theorem). Examples include the circle (with largest area), and Reuleaux triangle (with smallest area) but there are an infinite number. A curve of constant width can be used in a special drill chuck to cut square "holes."

A generalization gives solids of constant width. These do not have the same surface area for a given width, but their shadows are curves of constant width with the same width!

Delta Curve, Kakeya Needle Problem, Reuleaux Triangle, Solid of Constant Width

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## References

Blaschke, W. "Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts." Math. Ann. 76, 504-513, 1915.Bogomolny, A. "Shapes of Constant Width." http://www.cut-the-knot.org/do_you_know/cwidth.shtml.Bogomolny, A. "Star Construction of Shapes of Constant Width." http://www.cut-the-knot.org/Curriculum/Geometry/CWStar.shtml.Böhm, J. "Convex Bodies of Constant Width." Ch. 4 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 96-100, 1986.Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 7, 1991.Fischer, G. (Ed.). Plates 98-102 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, pp. 89 and 96, 1986.Gardner, M. "Mathematical Games: Curves of Constant Width, One of which Makes it Possible to Drill Square Holes." Sci. Amer. 208, 148-156, Feb. 1963.Gardner, M. "Curves of Constant Width." Ch. 18 in The Unexpected Hanging and Other Mathematical Diversions. Chicago, IL: Chicago University Press, pp. 212-221, 1991.Goldberg, M. "Circular-Arc Rotors in Regular Polygons." Amer. Math. Monthly 55, 393-402, 1948.Kelly, P. Convex Figures. New York: Harcourt Brace, 1995.Rademacher, H. and Toeplitz, O. The Enjoyment of Mathematics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, 1957.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 150-151, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 219-220, 1991.Yaglom, I. M. and Boltyanski, V. G. Convex Figures. New York: Holt, Rinehart, and Winston, 1961.

## Referenced on Wolfram|Alpha

Curve of Constant Width

## Cite this as:

Weisstein, Eric W. "Curve of Constant Width." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CurveofConstantWidth.html