The idea of a sphere-world was constructed by

Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...

who, while pursuing his argument for conventionalism (see philosophy of space and time), offered a thought experiment about a sphere with strange properties.

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

R^2-r^2

. In addition, all bodies have the same

coefficient of dilatation Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions. Temperature is a monotonic function of the average molecular kineti ...

so every body shrinks and expands in similar proportion as they move about the sphere. To finish the story, Poincaré states that the

index of refraction In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...

will also vary with the distance ''r'', in inverse proportion to

R^2-r^2

. 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 rays In optics a ray is an idealized geometrical model of light, obtained by choosing a curve that is perpendicular to the ''wavefronts'' of the actual light, and that points in the direction of energy transfer, energy flow. Rays are used to model t ...

. They would find that a

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

is not a straight line, and that the ratio of a circle’s circumference to its radius is greater than

2\pi

. 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'' and in

Jeremy Gray Jeremy John Gray (born 25 April 1947) is an English mathematician primarily interested in the history of mathematics. Biography Gray studied mathematics at Oxford University from 1966 to 1969, and then at Warwick University, obtaining his P ...

's article "Epistemology of Geometry" in the

Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Eac ...

. This sphere-world is also described in Ian Stewart's book '' Flatterland'' (chapter 10, Platterland).

The concept

Commentary

References

See also