Curves With Infinite Length
As mentioned above, some curves are non-rectifiable, that is, there is no upper bound on the lengths of polygonal approximations; the length can be made arbitrarily large. Informally, such curves are said to have infinite length. There are continuous curves on which every arc (other than a single-point arc) has infinite length. An example of such a curve is the Koch curve. Another example of a curve with infinite length is the graph of the function defined by f(x) = x sin(1/x) for any open set with 0 as one of its delimiters and f(0) = 0. Sometimes the Hausdorff dimension and Hausdorff measure are used to "measure" the size of such curves.
Read more about this topic: Arc Length
Famous quotes containing the words curves, infinite and/or length:
“At the end of every diet, the path curves back toward the trough.”
—Mason Cooley (b. 1927)
“In nature’s infinite book of secrecy
A little I can read.”
—William Shakespeare (1564–1616)
“What though the traveler tell us of the ruins of Egypt, are we so sick or idle that we must sacrifice our America and today to some man’s ill-remembered and indolent story? Carnac and Luxor are but names, or if their skeletons remain, still more desert sand and at length a wave of the Mediterranean Sea are needed to wash away the filth that attaches to their grandeur. Carnac! Carnac! here is Carnac for me. I behold the columns of a larger
and purer temple.”
—Henry David Thoreau (1817–1862)