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The three spheres can be sandwiched uniquely between two planes. Each pair of spheres defines a cone that is externally tangent to both spheres, and the apex of this cone corresponds to the intersection point of the two external tangents, i.e., the external homothetic center. Since one line of the cone lies in each plane, the apex of each cone ...
Steiner described how the three spheres had been growing independent over thousands of years, evolving from ancient theocracies which governed all aspects of society; then, gradually separating out the purely political and legal life (beginning in Ancient Greece and Rome); then again, the purely economic life (beginning with the Industrial ...
Direct projection of 3-sphere into 3D space and covered with surface grid, showing structure as stack of 3D spheres (2-spheres) In mathematics, a hypersphere, 3-sphere, or glome is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central ...
In the extrinsic 3-dimensional picture, a great circle is the intersection of the sphere with any plane through the center. In the intrinsic approach, a great circle is a geodesic; a shortest path between any two of its points provided they are close enough. Or, in the (also intrinsic) axiomatic approach analogous to Euclid's axioms of plane ...
There are two possibilities: if =, the spheres coincide, and the intersection is the entire sphere; if , the spheres are disjoint and the intersection is empty. When a is nonzero, the intersection lies in a vertical plane with this x-coordinate, which may intersect both of the spheres, be tangent to both spheres, or external to both spheres.
The number of circles is n 3-(n-1) 3 = 3n 2-3n+1 = 3n(n-1)+1. These overlapping circles can also be seen as a projection of an n-unit cube of spheres in 3-dimensional space, viewed on the diagonal axis. There are more spheres than circles because some are overlapping in 2 dimensions.
Experts discuss the health benefits of cooking and baking. Plus, tips for getting the most mental health benefits when cooking.
The disk bounded by a great circle is called a great disk: it is the intersection of a ball and a plane passing through its center. In higher dimensions, the great circles on the n -sphere are the intersection of the n -sphere with 2-planes that pass through the origin in the Euclidean space R n + 1 .