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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 or 3-sphere 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 point.
Such processes depend on the geometry of the container holding the spherical grains. [2] When spheres are randomly added to a container and then compressed, they will generally form what is known as an "irregular" or "jammed" packing configuration when they can be compressed no more. This irregular packing will generally have a density of about ...
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.
The hexagonal packing of circles on a 2-dimensional Euclidean plane. These problems are mathematically distinct from the ideas in the circle packing theorem.The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, for instance the plane or a sphere.
Its sides are arcs of great circles—the spherical geometry equivalent of line segments in plane geometry. Such polygons may have any number of sides greater than 1. Two-sided spherical polygons— lunes , also called digons or bi-angles —are bounded by two great-circle arcs: a familiar example is the curved outward-facing surface of a ...
Generalisations can be made to higher dimensions – this is called sphere packing, which usually deals only with identical spheres. The branch of mathematics generally known as "circle packing" is concerned with the geometry and combinatorics of packings of arbitrarily-sized circles: these give rise to discrete analogs of conformal mapping ...
Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle) is called the reference plane (sometimes fundamental plane).
In spherical geometry, the basic concepts are point and great circle. However, two great circles on a plane intersect in two antipodal points, unlike coplanar lines in Elliptic geometry. In the extrinsic 3-dimensional picture, a great circle is the intersection of the sphere with any plane through the center.