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Spheres can be generalized to spaces of any number of dimensions. For any natural number n, an n-sphere, often denoted S n, is the set of points in (n + 1)-dimensional Euclidean space that are at a fixed distance r from a central point of that space, where r is, as before, a positive real number. In particular:
[12] [13] Callippus modified this system, using five spheres for his models of the Sun, Moon, Mercury, Venus, and Mars and retaining four spheres for the models of Jupiter and Saturn, thus making 33 spheres in all. [13] Each planet is attached to the innermost of its own particular set of spheres.
Visualization of a celestial sphere. In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth.All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphere, which may be centered on Earth or the observer.
The innermost spheres are the terrestrial spheres, while the outer are made of aether and contain the celestial bodies. In Plato's Timaeus (58d) speaking about air, Plato mentions that "there is the most translucent kind which is called by the name of aether (αἰθήρ)" [9] but otherwise he adopted the classical system of four elements.
A geodesic polyhedron is a convex polyhedron made from triangles. They usually have icosahedral symmetry, such that they have 6 triangles at a vertex, except 12 vertices which have 5 triangles. They are the dual of corresponding Goldberg polyhedra, of which all but the smallest one (which is a regular dodecahedron) have mostly hexagonal faces.
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If developed as a part of solid geometry, use is made of points, straight lines and planes (in the Euclidean sense) in the surrounding space. In spherical geometry, angles are defined between great circles, resulting in a spherical trigonometry that differs from ordinary trigonometry in many respects; for example, the sum of the interior angles ...
Some candidates proposed for exotic 4-spheres are the Cappell–Shaneson spheres (Sylvain Cappell and Julius Shaneson ) and those derived by Gluck twists . Gluck twist spheres are constructed by cutting out a tubular neighborhood of a 2-sphere S in S 4 and gluing it back in using a diffeomorphism of its boundary S 2 ×S 1.