- Sphere-world
The idea of a sphere-world was constructed by
Henri Poincaré while pursuing his argument forconventionalism (seephilosophy of space and time ), offered athought experiment about asphere with strange properties.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 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 withlight ray s. 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 2pi.These inhabitants would in fact determine that their universe is not ruled by
Euclidean geometry , but instead byhyperbolic geometry .See also
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sphere world - a mathematical concept in robotics
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