Common volume measurements, geometrical

Consider, for example, the question What is the shape of the most chiral right triangle in R2 The answer to this question depends on the particular function that is chosen as the chirality measure According to measures based on geometric chirality products,151 on symmetry coordinates,152 on common volumes,153 and on Hausdorff distances,131 154 the smallest internal angles are 18.8°, 30.0°, 37.5°, and 35.2°, respectively.147 The function x(0) = sin 40 mentioned earlier achieves its maximum [ (0) = 1] at 0 = 22.5°. Because there are as many answers to this question as there are functions that can measure the chirality of a triangle, and because there is, in principle, no limit to the number of such functions, the question remains, in a deep sense, unanswerable. 

The measurable physical properties of catalyst particles commonly used in geometric models are the total surface area Sg (m /g), pore volume Vg (cm /g), solid density ps (g/cm ), void fraction or porosity Gp, and occasionally pore-volume distribution. 

Fractal Dimension Measure of a geometric object that can have fractional values. It refers to the measure of how fast the length, area, or volume of an object increases with a decrease in scale. Fractal dimension can be calculated by box counting or by evaluating the information dimension of an object. Generator Collection of scaled copies of an initiator. Hausdorff-Besicovitch Dimension Mathematical statement used to obtain a dimension that is not a whole number, commonly written as d = log (N)/ log (r). 

Figures 23.30-23.40 are a collection of common geometric figures. Area (A), volume (V), and other measurable features are indicated.

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