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Chapters 6, 7, and 8 cover specific geometric objects with negatively-curved surfaces, including the pseudosphere, helicoid, and catenoid, investigate mathematical toys, and use these crocheted models "to explore otherwise hard to visualize objects". A final chapter covers the applications of hyperbolic geometry and its ongoing research ...
The pseudosphere and its relation to three other models of hyperbolic geometry. The half pseudosphere of curvature −1 is covered by the interior of a horocycle.In the Poincaré half-plane model one convenient choice is the portion of the half-plane with y ≥ 1. [7]
Crocheted hyperbolic pseudosphere from the IFF Collection The IFF's collection of crocheted hyperbolic planes, in imitation of a coral reef IFF hyperbolic crochet cacti and kelp at the LA County Fair, 2006
Of these, the tractoid (or pseudosphere) is the best known; using the tractoid as a model of the hyperbolic plane is analogous to using a cone or cylinder as a model of the Euclidean plane. However, the entire hyperbolic plane cannot be embedded into Euclidean space in this way, and various other models are more convenient for abstractly ...
Hyperbolic crochet [ edit ] While attending a geometry workshop at Cornell University about teaching geometry for university professors in 1997, Taimiņa was presented with a fragile paper model of a hyperbolic plane, made by the professor in charge of the workshop, David Henderson (designed by geometer William Thurston . [ 4 ] )
The metric of the model on the half-plane, { , >}, is: = + ()where s measures the length along a (possibly curved) line. The straight lines in the hyperbolic plane (geodesics for this metric tensor, i.e., curves which minimize the distance) are represented in this model by circular arcs perpendicular to the x-axis (half-circles whose centers are on the x-axis) and straight vertical rays ...
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However, not the entire hyperbolic plane can be placed onto the pseudosphere as a model, only a portion of the hyperbolic plane. [2] Poincare disc with hyperbolic parallel lines. The entire hyperbolic plane can also be placed on a Poincaré disk and maintain its angles. However, the lines will turn into circular arcs, which warps them.
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