The hyperbolic plane is sometimes described as a surface in which the space expands. Like a Euclidean plane it is open and infinite, but it has a more complex and counterintuitive geometry. A hyperbolic plane is a surface in which the space curves away from itself at every point. On a sphere, the surface curves in on itself and is closed. One way of understanding it is that it’s the geometric opposite of the sphere. In formal geometric terms, it is “a simply connected Riemannian manifold with negative Gaussian curvature.“ In higher-level mathematics courses it is often defined as the geometry that is described by the upper half-plane model. Margaret Wertheim, founder of the Institute for Figuring in Los Angeles, spoke to them about hyperbolic geometry and crocheting.Ĭabinet: What exactly is a hyperbolic plane?ĭavid Henderson: There are many ways of describing the hyperbolic plane. Taimina and her husband, David Henderson, a geometer at Cornell, are the co-authors of Experiencing Geometry, a widely used textbook on both Euclidean and non-Euclidean spaces. Eventually, in 1997, Daina Taimina, a mathematician at Cornell University, made the first useable physical model of the hyperbolic plane-a feat many mathematicians had believed was impossible-using, of all things, crochet. Starting in the 1950s, they began to suggest possibilities for constructing such surfaces. For more than a century, mathematicians searched in vain for a physical surface with hyperbolic geometry. The discovery of hyperbolic space in the 1820s and 1830s by the Hungarian mathematician János Bolyai and the Russian mathematician Nicholay Lobatchevsky marked a turning point in mathematics and initiated the formal field of non-Euclidean geometry. It was therefore a deep shock to their community to find that there existed in principle a completely other spatial structure whose existence was discerned only by overturning a two-thousand-year-old prejudice about “parallel” lines. Until the nineteenth century, mathematicians knew about only two kinds of geometry: the Euclidean plane and the sphere. Wolfgang Bolyai (1775–1856) to his son János Bolyai regarding the study of hyperbolic geometry Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life.
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