I looked over at a lamp I've been using for years. The lampshade is conical and covered with loose-woven (in the sense that there are gaps for light between the threads) thin twine cloth glued onto a paper backing. And I said, "No, it can't be..." Yup. At the vertical seam where the cloth wraps around and meets itself, the threads on the two sides continue in the same directions-- +45 and -45 degrees (not perfectly matched, but it's clear they could have been). on the far side of the shade, the threads are going vertical and horizontal. Following a bottom thread at the seam it arcs up, comes back down and crosses itself at a right angle at the seam. Everybody knows you can make a cone out of paper. That means, despite the curving pattern, the threads of the weave are geodesics... at least locally. And that means that putting any thread under tension wouldn't make it want to slip up or down the cone. Questions: Imagine you took apart the lampshade, cut straight through the vertical joint (so there was no overlap), and laid the cone material out flat.  Maybe fill in the center part where the top hole was cut out. The material would form some pie-chart fraction. Measuring from the center/tip, an arbitrary unfolded cone could cover <180, 180 or >180 degrees of the pie. What is that angle for *my* lampshade? Considering whole cones again, how many times does a geodesic of a cone cross itself? Can that number be infinite? Can it be finite? If finite, what does the number of crossings depend on and how? What are the relative distances of the crossings from the tip of the cone? Are the even and odd crossings spaced differently? What about the non- self-crossing point where the curve is closest to the tip? Maybe a nice diagram? Does the 3D curve have a name? If you draw verticals from the curve down to a horizontal plane, what is that curve? If you project horizontally to a vertical plane what kinds of curves do you get? Can the threads be arranged so that they consistently follow the over-under weave pattern all the way around? If so, show how. Hint: imagine making the cone out of graph paper. Can you imagine a machine to weave it, or even a jig to help weave it by hand? If the consistent weave is impossible, prove it. --Steve