Dini's surface
Encyclopedia
In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, Dini's surface is a surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

 with constant negative curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

 that can be created by twisting a pseudosphere
Pseudosphere
In geometry, the term pseudosphere is used to describe various surfaces with constant negative gaussian curvature. Depending on context, it can refer to either a theoretical surface of constant negative curvature, to a tractricoid, or to a hyperboloid....

. It is named after Ulisse Dini and described by the following parametric equation
Parametric equation
In mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....

s:


Another description is a helicoid
Helicoid
The helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for every point on the helicoid there is a helix contained in the helicoid which passes through...

 constructed from the tractrix
Tractrix
Tractrix is the curve along which a small object moves, under the influence of friction, when pulled on a horizontal plane by a piece of thread and a puller that moves at a right angle to the initial line between the object and the puller at an infinitesimal speed. It is therefore a curve of...

.

Uses

Renditions of Dini's surface have appeared on the covers of Western Kentucky University
Western Kentucky University
Western Kentucky University is a public university in Bowling Green, Kentucky, USA. It was formally founded by the Commonwealth of Kentucky in 1906, though its roots reach back a quarter-century earlier....

's Graduate Study in Mathematics, Gray's Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition, and volume 2, number 3 of La Gaceta de la Real Sociedad Matemática Española.
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