hyperbolic plane


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Related to hyperbolic plane: hyperbolic geometry

hyperbolic plane

[¦hī·pər‚bäl·ik ′plān]
(mathematics)
A two-dimensional vector space E on which there is a nondegenerate, symmetric or alternating form ƒ(x,y) such that there exists a nonzero element w in E for which ƒ(w,w) = 0.
References in periodicals archive ?
Geometry teachers usually try to explain the hyperbolic plane via flat models that wildly distort its geometry--making lines look like semicircles, for instance.
As a result, the hyperbolic plane is somewhat like a carpet that, too big for its room, buckles and flares out more and more as it grows.
Taimina realized that she could crochet a durable model of the hyperbolic plane using a simple rule: Increase the number of stitches in each row by a fixed factor, by adding a new stitch after, for instance, every two (or three or four or n) stitches.
In 1901, mathematician David Hilbert proved that because of this buckling, it's impossible to build a smooth model of the hyperbolic plane.
In the 1970s, William Thurstou, now also at Cornell, described a way to build an approximate physical model of the hyperbolic plane by taping together paper arcs into rings whose circumferences grow exponentially.
I have met so many people now who don't have a math background, but who want to understand what these hyperbolic planes mean," Taimina says.
Like Taimina's hyperbolic planes, Osinga's Lorenz manifold has taken to the road frequently since its construction, making appearances at mathematical conferences, at art shows, and even on television news.
In addition to Taimina's hyperbolic planes and a Lorenz surface crocheted by Yackel, the exhibit featured Mobius strips, which are twisted rings that have only one side, and Klein bottles, which are closed surfaces that have no inside.
Taimina's hyperbolic planes have also attracted interest from art lovers.
Remarkably, this may be the first detailed, explicit synthetic construction of triangle tilings of the hyperbolic plane to appear," Goodman-Strauss notes.
Neither did Henri Poincare and other 19th-century mathematicians who drew various pictures of the hyperbolic plane.
He offers techniques and instructions for drawing by hand some tilings of the Poincare model of the hyperbolic plane.