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