Ball

A ball is a 3-dimensional shape made up of the solid interior of a sphere.

The symmetry group of a ball is O(3), because all rotations and reflections about the center of the ball leave the ball unchanged.

Structure and Sections[]

The ball is made up of a single curved face. Its sections look like a point growing to a disc before shrinking again.

Hypervolumes[]

vertex count = $0$
edge length = $0$
surface area = $\pi l^2$
surcell volume = $\frac{1}{6}\pi l^3$

Subfacets[]

1 sphere (2D)
1 ball (3D)

Equations[]

The surface of a ball of radius r, centered at the origin, can be given by $x^2+y^2+z^2 = r^2$ .

Notations[]

Toratopic notation: $(|)$
Tapertopic notation: $3$

Coordinate System[]

The coordinate system corresponding to the ball is called spherical polar coordinates, with the three coordinates being $\left( r, \theta, \phi \right)$ .

These can be converted to Cartesian coordinates using the formulae

${\displaystyle \begin{align} x &= r \, \sin\theta \, \cos\phi \\ y &= r \, \sin\theta \, \sin\phi \\ z &= r \, \cos\theta \end{align}}$

The position vector is hence given by ${\displaystyle \vec{r} = \begin{pmatrix} r \sin\theta \cos\phi \\ r \sin\theta \sin\phi \\ r \cos\theta \\ \end{pmatrix}}$ .This means the three unit vectors are ${\displaystyle \hat r = \begin{pmatrix} \sin\theta \cos\phi \\ \sin\theta \sin\phi \\ \cos\theta \\ \end{pmatrix}}$ , ${\displaystyle \hat \theta = \begin{pmatrix} \cos\theta \cos\phi \\ \cos\theta \sin\phi \\ -\sin\theta \\ \end{pmatrix}}$ and ${\displaystyle \hat \phi = \begin{pmatrix} -\sin\phi \\ \cos\phi \\ 0 \\ \end{pmatrix}}$ .

Generalizations[]

Generalization of ball for euclidean spaces of arbitrary dimension is hyperball (n-ball).

Speaking even more generally, for a metric space (M,d), where M is a set and d is a metric (distance function), the open ball $B_o$ of radius r and center c is the set:

$B_o=\{x\in M|d(x,c)<r\}$

and the closed ball $B_c$ of radius r and center c is the set:

$B_c=\{x\in M|d(x,c)\le r\}$ .

See Also[]

Dimensionality	Negative One	Zero	One	Two	Three	Four	Five	Six	Seven	Eight	Nine	Ten	Eleven	Twelve	Thirteen	Fourteen	Fifteen	Sixteen	Seventeen	...	Aleph null
Simplex $\{3^{n-1}\}$	Null polytope $)($ $\emptyset$	Point $()$ $\mathbb{B}^0$	Line segment $\{\}$ $\mathbb{B}^1$	Triangle $\{3\}$	Tetrahedron $\{3^2\}$	Pentachoron $\{3^3\}$	Hexateron $\{3^4\}$	Heptapeton $\{3^5\}$	Octaexon $\{3^6\}$	Enneazetton $\{3^7\}$	Decayotton $\{3^8\}$	Hendecaxennon $\{3^9\}$	Dodecadakon $\{3^{10}\}$	Tridecahendon $\{3^{11}\}$	Tetradecadokon $\{3^{12}\}$	Pentadecatradakon $\{3^{13}\}$	Hexadecatedakon $\{3^{14}\}$	Heptadecapedakon $\{3^{15}\}$	Octadecapedakon $\{3^{16}\}$	...	Omegasimplex
Cross $\{3^{n-2},4\}$				Square $\{4\}$	Octahedron $\{3, 4\}$	Hexadecachoron $\{3^2, 4\}$	Pentacross $\{3^3, 4\}$	Hexacross $\{3^4, 4\}$	Heptacross $\{3^5, 4\}$	Octacross $\{3^6, 4\}$	Enneacross $\{3^7, 4\}$	Dekacross $\{3^8, 4\}$	Hendekacross $\{3^9, 4\}$	Dodekacross $\{3^{10}, 4\}$	Tridekacross $\{3^{11}, 4\}$	Tetradekacross $\{3^{12}, 4\}$	Pentadekacross $\{3^{13}, 4\}$	Hexadekacross $\{3^{14}, 4\}$	Heptadekacross $\{3^{15},4\}$	...	Omegacross
Hydrotopes $\{3^{n-2}, 5\}$				Pentagon $\{5\}$	Icosahedron $\{3, 5\}$	Hexacosichoron $\{3^2, 5\}$	Order-5 pentachoric tetracomb $\{3^3, 5\}$	Order-5 hexateric pentacomb $\{3^4, 5\}$	...
Hypercube $\{4, 3^{n-2}\}$				Square $\{4\}$	Cube $\{4,3\}$	Tesseract $\{4, 3^2\}$	Penteract $\{4, 3^3\}$	Hexeract $\{4, 3^4\}$	Hepteract $\{4, 3^5\}$	Octeract $\{4, 3^6\}$	Enneract $\{4, 3^7\}$	Dekeract $\{4, 3^8\}$	Hendekeract $\{4, 3^9\}$	Dodekeract $\{4, 3^{10}\}$	Tridekeract $\{4, 3^{11}\}$	Tetradekeract $\{4, 3^{12}\}$	Pentadekeract $\{4, 3^{13}\}$	Hexadekeract $\{4, 3^{14}\}$	Heptadekeract $\{4,3^{15}\}$	...	Omegeract
Cosmotopes $\{5, 3^{n-2}\}$				Pentagon $\{5\}$	Dodecahedron $\{5, 3\}$	Hecatonicosachoron $\{5, 3^2\}$	Order-3 hecatonicosachoric tetracomb $\{5, 3^3\}$	Order-3-3 hecatonicosachoric pentacomb $\{5, 3^4\}$	...
Hyperball $\mathbb B^n$				Disk $\mathbb B^2$	Ball $\mathbb B^3$	Gongol $\mathbb B^4$	Pentorb $\mathbb B^5$	Hexorb $\mathbb B^6$	Heptorb $\mathbb B^7$	Octorb $\mathbb B^8$	Enneorb $\mathbb B^9$	Dekorb $\mathbb B^{10}$	Hendekorb $\mathbb B^{11}$	Dodekorb $\mathbb B^{12}$	Tridekorb $\mathbb B^{13}$	Tetradekorb $\mathbb B^{14}$	Pentadekorb $\mathbb B^{15}$	Hexadekorb $\mathbb B^{16}$	Heptadekorb $\mathbb {B} ^{17}$	...	Omegaball $\mathbb B^{\aleph_0}$


$\mathbb S^{2}$	$\mathbb{T}^2$	$2\mathbb{T}^2$	$3\mathbb{T}^2$	$4\mathbb{T}^2$
Ball	Solid torus	Solid double torus	Solid triple torus	Solid quadruple torus