Sphericity

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For sphericity in statistics, see Mauchly's sphericity test.
Schematic representation of difference in grain shape. Two parameters are shown: sphericity (vertical) and rounding (horizontal).

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,[1] 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{\pi^{\frac{1}{3}}(6V_p)^{\frac{2}{3}}}{A_p}

where V_p is volume of the particle and A_p is the surface area of the particle. The sphericity of a sphere is 1 and, by the isoperimetric inequality, any particle which is not a sphere will have sphericity less than 1.

Ellipsoidal objects[edit]

See also: Earth radius

The sphericity, \Psi , of an oblate spheroid (similar to the shape of the planet Earth) is:

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

where a and b are the semi-major and semi-minor axes respectively.

Derivation[edit]

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

therefore

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

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[edit]

Name Picture Volume Area Sphericity
Platonic Solids
tetrahedron Tetrahedron \frac{\sqrt{2}}{12}\,s^3 \sqrt{3}\,s^2 \left(\frac{\pi}{6\sqrt{3}}\right)^{\frac{1}{3}} \approx 0.671
cube (hexahedron) Hexahedron (cube) \,s^3 6\,s^2

\left(
\frac{\pi}{6}
\right)^{\frac{1}{3}} \approx 0.806

octahedron Octahedron  \frac{1}{3} \sqrt{2}\, s^3  2 \sqrt{3}\, s^2

\left(
\frac{\pi}{3\sqrt{3}}
\right)^{\frac{1}{3}} \approx 0.846

dodecahedron Dodecahedron  \frac{1}{4} \left(15 + 7\sqrt{5}\right)\, s^3  3 \sqrt{25 + 10\sqrt{5}}\, s^2

\left(
\frac{\left(15 + 7\sqrt{5}\right)^2 \pi}{12\left(25+10\sqrt{5}\right)^{\frac{3}{2}}}
\right)^{\frac{1}{3}} \approx 0.910

icosahedron Icosahedron \frac{5}{12}\left(3+\sqrt{5}\right)\, s^3 5\sqrt{3}\,s^2 \left(
\frac{ \left(3 + \sqrt{5} \right)^2 \pi}{60\sqrt{3}}
\right)^{\frac{1}{3}} \approx 0.939
Round Shapes
ideal cone
(h=2\sqrt{2}r)
\frac{1}{3} \pi\, r^2 h

= \frac{2\sqrt{2}}{3} \pi\, r^3

\pi\, r (r + \sqrt{r^2 + h^2})

= 4 \pi\, r^2

\left(
\frac{1}{2}
\right)^{\frac{1}{3}} \approx 0.794
hemisphere
(half sphere)
\frac{2}{3} \pi\, r^3 3 \pi\, r^2

\left(
\frac{16}{27}
\right)^{\frac{1}{3}} \approx 0.840

ideal cylinder
(h=2\,r)
\pi r^2 h = 2 \pi\,r^3 2 \pi r ( r + h ) = 6 \pi\,r^2

\left(
\frac{2}{3}
\right)^{\frac{1}{3}} \approx 0.874

ideal torus
(R=r)
2 \pi^2 R r^2 = 2 \pi^2 \,r^3 4 \pi^2 R r = 4 \pi^2\,r^2

\left(
\frac{9}{4 \pi}
\right)^{\frac{1}{3}} \approx 0.894

sphere \frac{4}{3} \pi r^3 4 \pi\,r^2


1\,

See also[edit]

References[edit]

  1. ^ Wadell, Hakon (1935). "Volume, Shape and Roundness of Quartz Particles". Journal of Geology 43 (3): 250–280. doi:10.1086/624298. 

External links[edit]