The idea of asphere-world was constructed by French mathematicianHenri Poincaré who, while pursuing his argument forconventionalism (seephilosophy of space and time), offered athought experiment about asphere with strange properties.^[1]

Poincaré asks us to imagine a sphere of radiusR. 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 distancer from the center is proportional to${\displaystyle R^2-r^2}$.

In addition, all bodies have the samecoefficient of dilatation so every body shrinks and expands in similar proportion as they move about the sphere. To finish the story, Poincaré states that theindex of refraction will also vary with the distancer, ininverse proportion to${\displaystyle R^2-r^2}$.

How will this world look to inhabitants of this sphere?

In many ways it will looknormal. 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 measuredistance withlight rays. They would find that ageodesic is not a straight line, and that the ratio of a circle’s circumference to its radius is greater than${\displaystyle 2\pi }$.

These inhabitants would in fact determine that their universe is not ruled byEuclidean geometry, but instead byhyperbolic geometry.

Thisthought experiment is discussed inRoberto Torretti's bookPhilosophy of Geometry from Riemann to Poincaré^[2] and inJeremy Gray's article "Epistemology of Geometry" in theStanford Encyclopedia of Philosophy.^[3] This sphere-world is also described inIan Stewart's bookFlatterland (chapter 10, Platterland).

• Flatland

• Thought experiments
• Hyperbolic geometry
• Philosophy of physics
• Henri Poincaré

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