- Equidimensional
Equidimensional is an adjective applied to objects that have nearly the same size or spread in multiple directions. As a mathematical concept, it may be applied to objects that extend across any number of dimensions. More specifically, it's also used to characterize the
shape of three-dimensional solids.In geology
The word "equidimensional" is sometimes used by geologists to describe the shape of three-dimensional objects. In that case it is a synomym for equant [American Geological Institute "Dictionary of Geological Terms" (1976, Anchor Books, New York) p.147] . Deviations from equidimensional are used to classify the shape of convex objects like rocks or particles [C. F. Royse (1970) "An introduction to sediment analysis" (Arizona State University Press, Tempe) 169pp.] . For instance Th. Zingg in 1935 pointed out [Th. Zingg (1935). "Beitrag zur Schotteranalyse". "Schweizerische Mineralogische und Petrographische Mitteilungen" 15, 39–140.] that if "a", "b" and "c" are the long, intermediate, and short axes of a convex structure, and "R" is a number greater than one, then four "mutually exclusive" shape classes may be defined by:
Table 1: Zingg's convex object shape classes
For Zingg's applications, "R" was set equal to frac|3|2. Perhaps this is an intuitively reasonable setting in general for the point at which something's dimensions become significantly unequal.
The relationship between the four categories is illustrated in the figure at right, which allows one to plot long and short axis dimensions for the
convex envelope of any solid object. Perfectly equidimensionalsphere s plot in the lower right corner. Objects with equal short and intermediate axes lie on the upper bound, while objects with equal long and intermediate axes plot on the lower bound. The dotted gray and black lines correspond to integer frac|"a"|"c" values ranging from 2 up to 10.The point of intersection for all four classes on this plot occurs when the object's axes "a":"b":"c" have ratios of "R"2:"R":1, or 9:6:4 when "R"=frac|3|2. Make axis "b" any shorter and the object becomes "prolate". Make axis "b" any longer and it becomes "oblate". Bring "a" and "c" closer to "b" and the object becomes "equidimensional". Separate "a" and "c" further from "b" and it becomes "bladed".
For example, the convex envelope for some humans might plot near the black dot in the upper left of the figure.
See Also
*
aspect ratio between long and short
*equant as a noun used in astronomy
*oblate spheroid
*prolate spheroid
*shape analysis Footnotes
External links
Theodor Zingg PhD thesis: http://e-collection.ethbib.ethz.ch/view/eth:21472
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