Definitions

# Sphericity

[sfi-ris-i-tee]
Sphericity is a measure of how spherical (round) an object is. As such, it is a specific example of a compactness measure of a shape. Defined by Wadell in 1935, the sphericity, $Psi$, of a particle is the ratio of the surface area of a sphere (with the same volume as the given particle) to the surface area of the particle:

$Psi = frac\left\{pi^\left\{frac\left\{1\right\}\left\{3\right\}\right\}\left(6V_p\right)^\left\{frac\left\{2\right\}\left\{3\right\}\right\}\right\}\left\{A_p\right\}$

where $V_p$ is volume of the particle and $A_p$ is the surface area of the particle

## Ellipsoidal Objects

The sphericity, $Psi$, of an oblate spheroid (similar to the shape of the planet Earth) is defined as such:

$Psi =$
frac{pi^{frac{1}{3}}(6V_p)^{frac{2}{3}}}{A_p} = frac{2sqrt[3]{ab^2}}{a+frac{b^2}{sqrt{a^2-b^2}}ln{(frac{a+sqrt{a^2-b^2}}b)}}

''(where a, b are the semi-major, semi-minor axes, respectively.

## Derivation

Hakon Wadell defined sphericity as the surface area of a sphere of the same volume as the particle divided by the actual surface area of the particle.

First we need to write surface area of the sphere, $A_s$ in terms of the volume of the particle, $V_p$

$A_\left\{s\right\}^3 = left\left(4 pi r^2right\right)^3 = 4^3 pi^3 r^6 = 4 pi left\left(4^2 pi^2 r^6right\right) = 4 pi cdot 3^2 left\left(frac\left\{4^2 pi^2\right\}\left\{3^2\right\} r^6right\right) = 36 pi left\left(frac\left\{4 pi\right\}\left\{3\right\} r^3right\right)^2 = 36,pi V_\left\{p\right\}^2$

therefore

$A_\left\{s\right\} = left\left(36,pi V_\left\{p\right\}^2right\right)^\left\{frac\left\{1\right\}\left\{3\right\}\right\} = 36^\left\{frac\left\{1\right\}\left\{3\right\}\right\} pi^\left\{frac\left\{1\right\}\left\{3\right\}\right\} V_\left\{p\right\}^\left\{frac\left\{2\right\}\left\{3\right\}\right\} = 6^\left\{frac\left\{2\right\}\left\{3\right\}\right\} pi^\left\{frac\left\{1\right\}\left\{3\right\}\right\} V_\left\{p\right\}^\left\{frac\left\{2\right\}\left\{3\right\}\right\} = pi^\left\{frac\left\{1\right\}\left\{3\right\}\right\} left\left(6V_\left\{p\right\}right\right)^\left\{frac\left\{2\right\}\left\{3\right\}\right\}$

hence we define $Psi$ as:


Psi = frac{A_s}{A_p} = frac{ pi^{frac{1}{3}} left(6V_{p}right)^{frac{2}{3}} }{A_{p}}

## Sphericity of common objects

Name Picture Volume Area Sphericity
Platonic Solids
tetrahedron $frac\left\{sqrt\left\{2\right\}\right\}\left\{12\right\},s^3$ $sqrt\left\{3\right\},s^2$ $left\left(frac\left\{pi\right\}\left\{6sqrt\left\{3\right\}\right\}right\right)^\left\{frac\left\{1\right\}\left\{3\right\}\right\} approx 0.671$
cube (hexahedron) $,s^3$ $6,s^2$ $left\left( frac\left\{pi\right\}\left\{6\right\} right\right)^\left\{frac\left\{1\right\}\left\{3\right\}\right\} approx 0.806$
octahedron $frac\left\{1\right\}\left\{3\right\} sqrt\left\{2\right\}, s^3$ $2 sqrt\left\{3\right\}, s^2$ $left\left( frac\left\{pi\right\}\left\{3sqrt\left\{3\right\}\right\} right\right)^\left\{frac\left\{1\right\}\left\{3\right\}\right\} approx 0.846$
dodecahedron $frac\left\{1\right\}\left\{4\right\} left\left(15 + 7sqrt\left\{5\right\}right\right), s^3$ $3 sqrt\left\{25 + 10sqrt\left\{5\right\}\right\}, s^2$ $left\left( frac\left\{left\left(15 + 7sqrt\left\{5\right\}right\right)^2 pi\right\}\left\{12left\left(25+10sqrt\left\{5\right\}right\right)^\left\{frac\left\{3\right\}\left\{2\right\}\right\}\right\} right\right)^\left\{frac\left\{1\right\}\left\{3\right\}\right\} approx 0.910$
icosahedron $frac\left\{5\right\}\left\{12\right\}left\left(3+sqrt\left\{5\right\}right\right), s^3$ $5sqrt\left\{3\right\},s^2$ $left\left(frac\left\{ left\left(3 + sqrt\left\{5\right\} right\right)^2 pi\right\}\left\{60sqrt\left\{3\right\}\right\} right\right)^\left\{frac\left\{1\right\}\left\{3\right\}\right\} approx 0.939$
Round Shapes
ideal cone
$\left(h=2sqrt\left\{2\right\}r\right)$

$frac\left\{1\right\}\left\{3\right\} pi, r^2 h$
$= frac\left\{2sqrt\left\{2\right\}\right\}\left\{3\right\} pi, r^3$

$pi, r \left(r + sqrt\left\{r^2 + h^2\right\}\right)$
$= 4 pi, r^2$

$left\left(frac\left\{1\right\}\left\{2\right\} right\right)^\left\{frac\left\{1\right\}\left\{3\right\}\right\} approx 0.794$
hemisphere
(half sphere)
$frac\left\{2\right\}\left\{3\right\} pi, r^3$ $3 pi, r^2$ $left\left( frac\left\{16\right\}\left\{27\right\} right\right)^\left\{frac\left\{1\right\}\left\{3\right\}\right\} approx 0.840$
ideal cylinder
$\left(h=2,r\right)$
$pi r^2 h = 2 pi,r^3$ $2 pi r \left(r + h \right) = 6 pi,r^2$ $left\left( frac\left\{2\right\}\left\{3\right\} right\right)^\left\{frac\left\{1\right\}\left\{3\right\}\right\} approx 0.874$
ideal torus
$\left(R=r\right)$
$2 pi^2 R r^2 = 2 pi^2 ,r^3$ $4 pi^2 R r = 4 pi^2,r^2$ $left\left( frac\left\{9\right\}\left\{4 pi\right\} right\right)^\left\{frac\left\{1\right\}\left\{3\right\}\right\} approx 0.894$
sphere $frac\left\{4\right\}\left\{3\right\} pi r^3$ $4 pi,r^2$ $1,$

## Sphericity in statistics

In statistical analyses, sphericity relates to the equality of the variances of the differences between levels of the repeated measures factor. Sphericity requires that the variances for each set of difference scores are equal. This is an assumption of an ANOVA with a repeated measures factor, where violations of this assumption can invalidate the analysis conclusions. Mauchly's sphericity test is the statistical test used to evaluate sphericity.