The idea of a sphere-world was constructed by French mathematician

Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...

who, while pursuing his argument for conventionalism (see

philosophy of space and time Philosophy of space and time is a branch of philosophy concerned with ideas about knowledge and understanding within space and time. Such ideas have been central to philosophy from its inception. The philosophy of space and time was both an inspi ...

), offered a

thought experiment A thought experiment is an imaginary scenario that is meant to elucidate or test an argument or theory. It is often an experiment that would be hard, impossible, or unethical to actually perform. It can also be an abstract hypothetical that is ...

about a

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

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 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 the ratio of the apparent speed of light in the air or vacuum to the speed in the medium. The refractive index determines how much the path of light is bent, or refrac ...

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 Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...

with light rays. They would find that a

geodesic In geometry, a geodesic () is a curve representing in some sense the locally 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 conn ...

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 Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, but instead by

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...

Commentary

This

is discussed in Roberto Torretti's book ''Philosophy of Geometry from Riemann to Poincaré'' and in Jeremy Gray's article "Epistemology of Geometry" in the

Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') is a freely available online philosophy resource published and maintained by Stanford University, encompassing both an online encyclopedia of philosophy and peer-reviewed original publication ...

. This sphere-world is also described in Ian Stewart's book ''

Flatterland ''Flatterland'' is a 2001 book written by mathematician and science popularizer Ian Stewart (mathematician), Ian Stewart about non-Euclidean geometry. It was written as a sequel to ''Flatland'', an 1884 novel that discussed different dimensions ...

'' (chapter 10, Platterland).

The concept

Commentary

References

See also