Sphericity (geology)

In geology, sphericity is a measure of how spherical a body is.

definition

The term sphericity was defined in 1935 by the geologist Hakon Wadell. The sphericity of a body K is the ratio of the surface area of ​​a sphere of equal volume to the surface area of ​​the body: ${\displaystyle \Psi }$

${\displaystyle \Psi ={\frac {\pi ^{\frac {1}{3}}(6V_{p})^{\frac {2}{3}}}{A_{p}}}}$,

where denotes the volume of the body and its surface area. ${\displaystyle V_{p}}$ ${\displaystyle A_{p}}$

application

In sedimentology and soil micromorphology , sphericity is used as an approximation for the grain shape . Since a calculation would be too time-consuming, it is usually estimated using comparison tables , which also enable the grain rounding to be determined. The sphericity is then not given as a numerical value, but by classification (e.g. prismoidal , subprismoidal, spherical , subdiscoidal, discoid ).

Sphericity of known bodies

Surname	image	volume	surface	Sphericity
Platonic solids
Tetrahedron		${\displaystyle {\frac {\sqrt {2}}{12}}\,s^{3}}$	${\displaystyle {\sqrt {3}}\,s^{2}}$	${\displaystyle {\sqrt[{3}]{\frac {\pi }{6{\sqrt {3}}}}}\approx 0{,}671}$
Cube (hexahedron)		${\displaystyle \,s^{3}}$	${\displaystyle 6\,s^{2}}$	${\displaystyle {\sqrt[{3}]{\frac {\pi }{6}}}\approx 0{,}806}$
octahedron		${\displaystyle {\frac {1}{3}}{\sqrt {2}}\,s^{3}}$	${\displaystyle 2{\sqrt {3}}\,s^{2}}$	${\displaystyle {\sqrt[{3}]{\frac {\pi }{3{\sqrt {3}}}}}\approx 0{,}846}$
Dodecahedron		${\displaystyle {\frac {1}{4}}\left(15+7{\sqrt {5}}\right)\,s^{3}}$	${\displaystyle 3{\sqrt {25+10{\sqrt {5}}}}\,s^{2}}$	${\displaystyle {\sqrt[{3}]{\frac {\left(15+7{\sqrt {5}}\right)^{2}\pi }{12\left(25+10{\sqrt {5}}\right)^{\frac {3}{2}}}}}\approx 0{,}910}$
Icosahedron		${\displaystyle {\frac {5}{12}}\left(3+{\sqrt {5}}\right)\,s^{3}}$	${\displaystyle 5{\sqrt {3}}\,s^{2}}$	${\displaystyle {\sqrt[{3}]{\frac {\left(3+{\sqrt {5}}\right)^{2}\pi }{60{\sqrt {3}}}}}\approx 0{,}939}$
Bodies with nonplanar faces
ideal cone ${\displaystyle (h=2{\sqrt {2}}r)}$		${\displaystyle {\frac {1}{3}}\pi \,r^{2}h}$ ${\displaystyle ={\frac {2{\sqrt {2}}}{3}}\pi \,r^{3}}$	${\displaystyle \pi \,r(r+{\sqrt {r^{2}+h^{2}}})}$ ${\displaystyle =4\pi \,r^{2}}$	${\displaystyle {\sqrt[{3}]{\frac {1}{2}}}\approx 0{,}794}$
Hemisphere		${\displaystyle {\frac {2}{3}}\pi \,r^{3}}$	${\displaystyle 3\pi \,r^{2}}$	${\displaystyle {\sqrt[{3}]{\frac {16}{27}}}\approx 0{,}840}$
ideal cylinder ${\displaystyle (h=2\,r)}$		${\displaystyle \pi r^{2}h=2\pi \,r^{3}}$	${\displaystyle 2\pi r(r+h)=6\pi \,r^{2}}$	${\displaystyle {\sqrt[{3}]{\frac {2}{3}}}\approx 0{,}874}$
ideal torus ${\displaystyle (R=r)}$		${\displaystyle 2\pi ^{2}Rr^{2}=2\pi ^{2}\,r^{3}}$	${\displaystyle 4\pi ^{2}Rr=4\pi ^{2}\,r^{2}}$	${\displaystyle {\sqrt[{3}]{\frac {9}{4\pi }}}\approx 0{,}894}$
Bullet		${\displaystyle {\frac {4}{3}}\pi r^{3}}$	${\displaystyle 4\pi \,r^{2}}$	${\displaystyle 1}$

Web links

• University of North Carolina at Wilmington: Grain Morphology: Roundness, Surface Features, and Sphericity of Grains

References

1. ↑ Hakon Wadell: Volume, Shape and Roundness of Quartz Particles . In: Journal of Geology . 43, 1935, pp. 250-280.