|WikiProject Metalworking||(Rated C-class, Low-importance)|
The derivations for the volume, area, and sphericity of the dodecahedron are correct, but the numerical value for the sphericity should be about 0.910. I stopped my derivation for its sphericity at an earlier stage and erroneously concluded that the formula given was wrong without using it. If you don't believe the original numerical value given is incorrect, try substituting numerical values for the volume and area into the defining formula.
The discoverer still deserves a full citation: Hakon Wadell, “Volume, Shape and Roundness of Quartz Particles,” Journal of Geology, Vol. 43, 1935, pp. 250-280. I know that 1935 is correct because I copied“Systematic Packing of Spheres,” by L.C. Graton and H.J. Fraser, Journal of Geology, Vol. 43, 1935, from the same volume at a library.
The author of the sphericity entry should also write one for the circularity of closed planar areas, deriving a formula for it in a similar fashion. It would have been logical for Wadell to have done it in his 1935 paper, but I don't know if he did.
The problem with this article is that it is not always possible to derive simple algebraic formulæ for volumes and sphericities. As a result Coxeter and other mathematicians have tended to ignore the Archimedean and other regular-faced polyhedra. Computer science becoming a separate specialty has not helped any. It seems to have made mathematicians have even more difficulty accepting that an algorithm is every bit as good as a formula.
Numerical values for the sphericities of many more polyhedra may be found at http.//geocities.com/freasoner_2000/spheric.htm
Error in formula for a hemisphere?
On this page - http://en.wikipedia.org/wiki/Sphericity...
The equation provided for the volume of a hemisphere produces a value half the value of the equation provided for the volume of a sphere, but the equation for the surface area of a hemisphere produces a value 3/4ths the value of the equation for the surface area of a sphere. Both should be half. This problem can also be seen by thinking of the area formula as the derivative of the volume formula. That's the case for the formulas provided for the sphere but not for the formulas provided for the hemisphere.
I posted the note above, then realized a complication.
The complication is that, in the areas of science in which I work , "the area of a hemisphere" would refer to the area of the curved surface only, not also ncluding the area of the flat surface that bisects the full sphere as part of "the area of a hemisphere". On that view, my post is correct (that there is an error in the wikipedia page). On the other hand, for some purposes one would indeed want to include that flat surface, in which case the wikipedia page entry is correct.
The link at the bottom of the page:
Grain Morphology: Roundness, Surface Features, and Sphericity of Grains http://people.uncw.edu/dockal/gly312/grains/grains.htm