This necklace is one example of hyperbolic crochet. This necklace was made of cashmere yarn with a trailing thread of lurex. The hyperbolic balls are threaded on a crocheted chain and tied to form a necklace. Hyperbolic crochet blends crochet stitches with mathematical formulas to turn a 2D object (length by width)  into a 3d object (length by width by depth). In doing hyperbolic crochet, several chained stitches are formed into a circle, then each chain receives one or more single crochet stitches, continuing around and around following the same pattern of increases until a ball is formed due to mathematical principles of space. For the necklace, there were 3 increases in every stitch. It will also work with increases in 1 or two stitches and regular stitches in all others. The pattern must continue throughout the work. Depending on the yarn or thread, the stitch style (single, double, half double, etc.), the hook, and personal thread tension a vast array of balls, in many sizes, can be made. Various wavy shapes can also be made, simulating ocean coral.
<p>Absolutely love this!! Am just discovering hyperbolic crochet and this will be my first piece of jewelry to apply the technique. Thank you and please keep the ideas coming.</p>
Non-euclidean crochet. This is really something...
Katzsta: How long does it take you to make one of these necklaces?
It all depends on how big you want it and how frequently you do an increase. Considering that for the necklace, for one ball, the first round may have 6 stitches, the second round will have 18, the third will have 54, the fourth will have 162, etc. It really begins to add up. I would say each ball in the necklace was 1-2 hours.
Very lovely, and well explained.
<a href="http://theiff.org/oexhibits/05b.html" rel="nofollow">The Institute For Figuring</a> has a good resource for crocheted hyperbolic surfaces. Daina Taimina, a professional mathematician, originated the technique, and <a href="http://www.math.cornell.edu/~henderson/ExpGeom/" rel="nofollow">coauthored a text on non-Euclidean geometry</a> with her husband, also a mathematician, using some of her work as examples.
random: if I recall correctly, a rope or chain (say, one strung loosely between two posts) will hang according to a hyperbolic curve.
<a href="http://en.wikipedia.org/wiki/Catenary" rel="nofollow">Kinda.</a>

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