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((short description|Mathematical thought experiment posed by Henri Poincaré))
((Refimprove|date=September 2014))

The idea of a '''sphere-world''' was constructed by French mathematician [[Henri Poincaré]] who, while pursuing his argument for [[conventionalism]] (see [[philosophy of space and time]]),  offered a [[thought experiment]] about a [[sphere]] with strange properties.<ref>((cite book|last1=Poincaré|first1=Henri|url=http://www.gutenberg.org/ebooks/37157|via=[[Project Gutenberg]]|title=Science and hypothesis|date=1952|publisher=Dover Publications|location=New York, N.Y.|isbn=0486602214|page=51|accessdate=17 September 2014))</ref>

==The concept==
Poincaré asks us to imagine a sphere of radius ''R''.  The temperature of the sphere decreases from its maximum at the center to absolute zero at its extremity such that a body’s temperature at a distance ''r'' from the center is proportional to <math>R^2-r^2</math>.

In addition, all bodies have the same [[coefficient of thermal expansion|coefficient of dilatation]] so every body shrinks and expands in similar proportion as they move about the sphere.  To finish the story, Poincaré states that the [[refractive index|index of refraction]] will also vary with the distance ''r'', in [[inverse proportion]] to <math>R^2-r^2</math>.

How will this world look to inhabitants of this sphere? 

In many ways it will look ''normal''. Bodies will remain intact upon transfer from place to place, as well as seeming to remain the same size (the Spherians would shrink along with them).  The geometry, on the other hand, would seem quite different.  Supposing the inhabitants were to view rods believed to be rigid, or measure [[distance]] with [[light beam|light rays]]. They would find that a [[geodesic]] is not a straight line, and that the ratio of a circle’s circumference to its radius is greater than <math>2\pi</math>.

These inhabitants would in fact determine that their universe is not ruled by [[Euclidean geometry]], but instead by [[hyperbolic geometry]].

==Commentary==

This [[thought experiment]] is discussed in [[Roberto Torretti]]'s book ''Philosophy of Geometry from Riemann to Poincaré''<ref>((cite book|last1=Torretti|first1=Roberto|title=Philosophy of geometry from Riemann to Poincaré|date=1984|publisher=D. Reidel Pub. Co.|location=Dordrecht, Holland|isbn=9027709203|page=[https://archive.org/details/philosophyofgeom0000torr/page/320 320]|edition=[New ed.]|url=https://archive.org/details/philosophyofgeom0000torr/page/320))</ref> and in [[Jeremy Gray]]'s article "Epistemology of Geometry" in the [[Stanford Encyclopedia of Philosophy]].<ref>((cite web|title=Epistemology of Geometry |author1=Jeremy Gray|url=http://plato.stanford.edu/entries/epistemology-geometry/|publisher=[[Stanford Encyclopedia of Philosophy]]|accessdate=19 September 2014))</ref> This sphere-world is also described in [[Ian Stewart (mathematician)|Ian Stewart]]'s book ''[[Flatterland]]'' (chapter 10, Platterland).

==References==
((reflist))

==See also==
* [[Flatland]]

((DEFAULTSORT:Sphere-World))
[[Category:Thought experiments]]
[[Category:Hyperbolic geometry]]
[[Category:Philosophy of physics]]
[[Category:Henri Poincaré]]


((hyperbolic-geometry-stub))

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