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)
“Whatever we have got has been by infinite labour, and search, and ranging through every corner of nature; the difference is that instead of dirt and poison, we have rather chosen to fill our hives with honey and wax, thus furnishing mankind with the two noblest of things, which are sweetness and light.”
—Jonathan Swift (1667–1745)
“At length he would call to let us know where he was waiting for us with his canoe, when, on account of the windings of the stream, we did not know where the shore was, but he did not call often enough, forgetting that we were not Indians.... This was not because he was unaccommodating, but a proof of superior manners. Indians like to get along with the least possible communication and ado. He was really paying us a great compliment all the while, thinking that we preferred a hint to a kick.”
—Henry David Thoreau (1817–1862)