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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 ...
The set of all spheres satisfying this equation is called a pencil of spheres determined by the original two spheres. In this definition a sphere is allowed to be a plane (infinite radius, center at infinity) and if both the original spheres are planes then all the spheres of the pencil are planes, otherwise there is only one plane (the radical ...
In other words, the space of lines in RP 3 is the quadric in P(R 3,3). Although this is not the same as the Lie quadric, a "correspondence" can be defined between lines and spheres using the complex numbers: if x = (x 0,x 1,x 2,x 3,x 4,x 5) is a point on the (complexified) Lie quadric (i.e., the x i are taken to be complex numbers), then
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.
Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world.
The 3-sphere is the boundary of a -ball in four-dimensional space. The ( n − 1 ) {\displaystyle (n-1)} -sphere is the boundary of an n {\displaystyle n} -ball. Given a Cartesian coordinate system , the unit n {\displaystyle n} -sphere of radius 1 {\displaystyle 1} can be defined as:
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The Dali cross is a tile space polyhedron, [50] [51] which can be represented as the net of a tesseract. A tesseract is a cube analogous' four-dimensional space bounded by twenty-four squares, and it is bounded by the eight cubes known as its cells. [52]