Pseudosphere
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In geometry, a pseudosphere, or tractricoid in the traditional usage, is the result of revolving a tractrix about its asymptote. It has constant negative Gauss curvature, except at the cusp, and therefore is locally isometric to a hyperbolic plane (everywhere except for the cusp).
It also denotes the entire set of points of an infinite hyperbolic space; which is one of the three models of Riemannian geometry. This can be viewed as the extension of a saddle shape to infinity. The further outward from the starting point (x, y) = (0, 0); the more increasingly ruffled the manifold becomes. This makes it very hard to represent a pseudosphere in the Euclidean space of drawings. A trick mathematicians have come up with to represent it is called the Poincaré model of hyperbolic geometry. By increasingly shrinking the pseudosphere as it goes further out towards the edge, it will fit into a circle, called the Poincaré disk; with the "edge" representing infinity. This is usually tessellated with equilateral triangles, or other polygons; which become increasingly distorted towards the edges, such that some vertices are shared by more polygons than is normal under Euclidean geometry (In normal flat space only six triangles, for instance, can share a vertex, but on the Poincaré disk, some are shared by eight triangles, as the total of the angles is now less than 180°). Reverting the triangles back to their normal shape yields various sections of the pseudosphere. The central part yields the familiar saddle shape. A section that leads to the infinite edge, ends up becoming "wrapped" around and joined at its opposite sides, yielding the aforementioned "tractricoid" shape, which is also called a "Gabriel's Horn" (since it resembles a horn with the mouthpiece lying at infinity). Thus the tractricoid is only really a piece of the whole pseudosphere.
The name "pseudosphere" comes about because it is a two-dimensional surface of constant curvature. Just as the sphere has at every point, the positively curved geometry of a dome; the whole pseudosphere has at every point the negatively curved geometry of a saddle.
The M.C. Escher print Circle Limit IV is an artistic visualization of the Poincaré disk.
