Katana VentraIP

Ball (mathematics)

In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere.[1] It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).

These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment.


In other contexts, such as in Euclidean geometry and informal use, sphere is sometimes used to mean ball. In the field of topology the closed -dimensional ball is often denoted as or while the open -dimensional ball is or .

In topological spaces[edit]

One may talk about balls in any topological space X, not necessarily induced by a metric. An (open or closed) n-dimensional topological ball of X is any subset of X which is homeomorphic to an (open or closed) Euclidean n-ball. Topological n-balls are important in combinatorial topology, as the building blocks of cell complexes.


Any open topological n-ball is homeomorphic to the Cartesian space Rn and to the open unit n-cube (hypercube) (0, 1)n ⊆ Rn. Any closed topological n-ball is homeomorphic to the closed n-cube [0, 1]n.


An n-ball is homeomorphic to an m-ball if and only if n = m. The homeomorphisms between an open n-ball B and Rn can be classified in two classes, that can be identified with the two possible topological orientations of B.


A topological n-ball need not be smooth; if it is smooth, it need not be diffeomorphic to a Euclidean n-ball.

Katana VentraIP

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, bounded by one plane

cap

, bounded by a conical boundary with apex at the center of the sphere

sector

, bounded by a pair of parallel planes

segment

, bounded by two concentric spheres of differing radii

shell

, bounded by two planes passing through a sphere center and the surface of the sphere

wedge

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A number of special regions can be defined for a ball:

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Smith, D. J.; Vamanamurthy, M. K. (1989). "How small is a unit ball?". . 62 (2): 101–107. doi:10.1080/0025570x.1989.11977419. JSTOR 2690391.

Mathematics Magazine

Dowker, J. S. (1996). "Robin Conditions on the Euclidean ball". . 13 (4): 585–610. arXiv:hep-th/9506042. Bibcode:1996CQGra..13..585D. doi:10.1088/0264-9381/13/4/003. S2CID 119438515.

Classical and Quantum Gravity

Gruber, Peter M. (1982). "Isometries of the space of convex bodies contained in a Euclidean ball". . 42 (4): 277–283. doi:10.1007/BF02761407. S2CID 119483499.

Israel Journal of Mathematics

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