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S 3: a 3-sphere is a sphere in 4-dimensional Euclidean space. Spheres for n > 2 are sometimes called hyperspheres. The n-sphere of unit radius centered at the origin is denoted S n and is often referred to as "the" n-sphere. The ordinary sphere is a 2-sphere, because it is a 2-dimensional surface which is embedded in 3-dimensional space.
In Greek antiquity the ideas of celestial spheres and rings first appeared in the cosmology of Anaximander in the early 6th century BC. [7] In his cosmology both the Sun and Moon are circular open vents in tubular rings of fire enclosed in tubes of condensed air; these rings constitute the rims of rotating chariot-like wheels pivoting on the Earth at their centre.
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.
In considering the physics of the celestial spheres, scholars followed two different views about the material composition of the celestial spheres. For Plato , the celestial regions were made "mostly out of fire" [ 1 ] [ 2 ] on account of fire's mobility. [ 3 ]
The Middle Ages broadly inherited the concept of the four elements of earth, water, air and fire arranged in concentric spheres about the earth as centre: [3] as the purest of the four elements, fire - and the sphere of fire - stood highest in the ascending sequence of the scala naturae, and closest to the superlunary world of the aether. [4]
The seven heavens and the sublunar spheres, from an engraving of Albertus Magnus' Philosophia naturalis.. Plato and Aristotle helped to formulate the original theory of a sublunary sphere in antiquity, [4] the idea usually going hand in hand with geocentrism and the concept of a spherical Earth.
The -spheres admit several other topological descriptions: for example, they can be constructed by gluing two -dimensional spaces together, by identifying the boundary of an -cube with a point, or (inductively) by forming the suspension of an -sphere.