In 2011, Gilles Baroin presented the Planet-4D model, a new vizualisation system based on graph theory that embeds the traditional Tonnetz on a 4D Hypersphere.
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Ivan scans the local hypersphere force lines and determines that one of them is unmarked on all star charts ( i . e . it does not lead anywhere ).
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In 1953, Wolfgang Pauli generalised the Kaluza-Klein theory to a six-dimensional space, and ( using dimensional reduction ) derived the essentials of an hypersphere.
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For the ideal gas, the accessible phase space is an ( n-1 )-sphere ( also called a hypersphere ) in the n dimensional \ vec v space:
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Similarly, you can compute the Hypersphere # Hyperspherical volume of an " i "-dimensional sphere . Blotwell 02 : 57, 26 October 2006 ( UTC)
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The addition of a point at infinity to the space obviates the distinction between hyperplane and hypersphere; higher dimensional inversive geometry is frequently studied then in the presumed context of an M�bius transformations.
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One intersects the complex hypersurface with a hypersphere, centred at the isolated singular point, and with sufficiently small radius so that it does not enclose, nor encounter, any other singular points.
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This same general statement, when applied to a hypersphere instead of a simplex, gives the Borsuk Ulam theorem, that ? must map two opposite points of the sphere to the same point.
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In geometry of higher dimensions, a "'hypersphere "'is the set of points at a constant distance from a given point called its "'center " '.
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In other words : the 30 vertices of the 600-cell which lie at arc distances of 90 degrees on its circumscribed hypersphere from a pair of opposite vertices, are the vertices of an icosidodecahedron.
