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Curves which, when rotated in a square, make contact with all four sides. 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!
See also Delta Curve, Kakeya Needle Problem, Reuleaux Triangle
References
Bogomolny, A. ``Shapes of Constant Width.''
http://www.cut-the-knot.com/do_you_know/cwidth.html.
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.
Fischer, G. (Ed.). Plates 98-102 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 89 and 96, 1986.
Gardner, M. Ch. 18 in The Unexpected Hanging and Other Mathematical Diversions. Chicago, IL: Chicago University
Press, 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.
Yaglom, I. M. and Boltyanski, V. G. Convex Figures. New York: Holt, Rinehart, and Winston, 1961.