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, such as
equidimensional schemes. More specifically, it's also used to characterize the
shape
A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type.
A pl ...
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 synonym for equant. Deviations from equidimensional are used to classify the shape of convex objects like rocks or particles. For instance, 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:
[Th. Zingg (1935).]
Beitrag zur Schotteranalyse
. ''Schweizerische Mineralogische und Petrographische Mitteilungen'' 15, 39–140.
Table 1: Zingg's convex object shape classes
For Zingg's applications, ''R'' was set equal to . 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
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of any solid object. Perfectly equidimensional
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
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 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''=. 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
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Aspect ratio between long and short
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Equant
Equant (or punctum aequans) is a mathematical concept developed by Claudius Ptolemy in the 2nd century AD to account for the observed motion of the planets. The equant is used to explain the observed speed change in different stages of the plane ...
as a noun used in astronomy
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Oblate spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circ ...
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Prolate spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circu ...
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shape analysis
Footnotes
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Sedimentology
Geometric shapes