Editing Pseudosphere (section)

== Tractroid ==
[[Image:Pseudosphere.png|right|frame|Tractroid]]
The same surface can be also described as the result of [[surface of revolution|revolving]] a [[tractrix]] about its [[asymptote]].
For this reason the pseudosphere is also called a  '''tractroid'''. As an example, the (half) pseudosphere (with radius 1) is the surface of revolution of the tractrix parametrized by<ref>{{cite book |title=Low-dimensional geometry: from Euclidean surfaces to hyperbolic knots
|first1=Francis
|last1=Bonahon
|publisher=AMS Bookstore
|year=2009
|isbn=978-0-8218-4816-6
|page=108
|url=https://books.google.com/books?id=YZ1L8S4osKsC}}, [https://books.google.com/books?id=YZ1L8S4osKsC&pg=PA108 Chapter 5, page 108]
</ref>
: <math>t \mapsto \left( t - \tanh t, \operatorname{sech}\,t \right), \quad \quad 0 \le t < \infty.</math>

It is a [[Mathematical singularity|singular space]] (the equator is a singularity), but away from the singularities, it has constant negative [[Gaussian curvature]] and therefore is locally [[isometry|isometric]] to a [[Hyperbolic space|hyperbolic plane]].

The name "pseudosphere" comes about because it has a [[dimension|two-dimensional]] [[Surface (topology)|surface]] of constant negative Gaussian curvature, just as a sphere has a surface with constant positive Gaussian curvature.
Just as the [[sphere]] has at every point a [[positive number|positively]] curved geometry of a [[dome]] the whole pseudosphere has at every point the [[negative number|negatively]] curved geometry of a [[saddle surface|saddle]].

As early as 1693 [[Christiaan Huygens]] found that the volume and the surface area of the pseudosphere are finite,<ref>{{cite book |title=Mathematics and Its History |edition=revised, 3rd |first1=John |last1=Stillwell |publisher=Springer Science & Business Media |year=2010 |isbn=978-1-4419-6052-8 |page=345 |url=https://books.google.com/books?id=V7mxZqjs5yUC}}, [https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA345 extract of page 345]</ref> despite the infinite extent of the shape along the axis of rotation. For a given edge [[radius]] {{mvar|R}}, the [[area]] is {{math|4π''R''<sup>2</sup>}} just as it is for the sphere, while the [[volume]] is {{math|{{sfrac|2|3}}π''R''<sup>3</sup>}} and therefore half that of a sphere of that radius.<ref>{{cite book
|title=Great Currents of Mathematical Thought, Vol. II: Mathematics in the Arts and Sciences
|edition=2
|first1=F.
|last1=Le Lionnais
|publisher=Courier Dover Publications
|year=2004
|isbn=0-486-49579-5
|page=154
|url=https://books.google.com/books?id=pCYDhbhu1O0C}}, [https://books.google.com/books?id=pCYDhbhu1O0C&pg=PA154 Chapter 40, page 154]
</ref><ref>{{MathWorld|title=Pseudosphere|urlname=Pseudosphere}}</ref>

The pseudosphere is an important geometric precursor to mathematical [[fabric arts]] and [[pedagogy]].<ref>{{cite news | url=https://www.nytimes.com/2024/01/15/science/mathematics-crochet-coral.html | title=The Crochet Coral Reef Keeps Spawning, Hyperbolically | work=The New York Times | date=15 January 2024 | last1=Roberts | first1=Siobhan }}</ref>