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Old 05-24-2022, 11:49 AM   #69
whswhs
 
Join Date: Jun 2005
Location: Lawrence, KS
Default Re: Gaming philosophy conundra

Quote:
Originally Posted by Anthony View Post
No, a sphere is a sphere. Increasing the number of sides on a polyhedron can come arbitrarily close to a sphere, but fundamentally a sphere has a single curved surface, a polyhedron has multiple flat surfaces.
At the very least, a solid body with C sides all equidistant from its center seems to be a sphere. Whether a body with aleph sub null sides is a sphere depends on whether you require a sphere to have a continuous surface.

But you can define a plane that is tangent to a sphere at a single point; that single point is a limiting case for a flat surface.

Addendum: What's sort of a dual case to this may help with the intuition. Say you toss a sphere onto a table. Let the surface of the sphere be printed with colors, in such a way that every point is a different color. For any throw of the "die," the surface of the table must be tangent to one point of the sphere, distinguishable as being one particular color. That is equivalent to a polyhedral die landing on one side, except that the "side" of the sphere is a single point. Now, how many sides does the sphere have? It clearly is larger than any finite integer.
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Last edited by whswhs; 05-24-2022 at 12:37 PM.
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