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:
“For a hundred and fifty years, in the pasture of dead horses,
roots of pine trees pushed through the pale curves of your ribs,
yellow blossoms flourished above you in autumn, and in winter
frost heaved your bones in the ground—old toilers, soil makers:
O Roger, Mackerel, Riley, Ned, Nellie, Chester, Lady Ghost.”
—Donald Hall (b. 1928)
“His hair has the long jesuschrist look. He is wearing the costume clothes. But most of all, he now has a very tolerant and therefore withering attitude toward all those who are still struggling in the old activist political ways ... while he, with the help of psychedelic chemicals, is exploring the infinite regions of human consciousness.”
—Tom Wolfe (b. 1931)
“Whoever aims publicly at great things and at length perceives secretly that he is too weak to achieve them, has usually also insufficient strength to renounce his aims publicly, and then inevitably becomes a hypocrite.”
—Friedrich Nietzsche (1844–1900)