In mathematics, and in particular the study of Hilbert spaces, a crinkled arc is a type of continuous curve. The concept is usually credited to Paul Halmos.
Specifically, consider where is a Hilbert space with inner product We say that is a crinkled arc if it is continuous and possesses the crinkly property: if then that is, the chords and are orthogonal whenever the intervals and are non-overlapping.
Halmos points out that if two nonoverlapping chords are orthogonal, then "the curve makes a right-angle turn during the passage between the chords' farthest end-points" and observes that such a curve would "seem to be making a sudden right angle turn at each point" which would justify the choice of terminology. Halmos deduces that such a curve could not have a tangent at any point, and uses the concept to justify his statement that an infinite-dimensional Hilbert space is "even roomier than it looks".
Writing in 1975, Richard Vitale considers Halmos's empirical observation that every attempt to construct a crinkled arc results in essentially the same solution and proves that is a crinkled arc if and only if, after appropriate normalizations,