The compactness measure of a shape, sometimes called the shape factor, is a numerical quantity representing the degree to which a shape is compact. The meaning of "compact" here is not related to the topological notion of compact space. Various compactness measures are used. However, these measures have the following in common:
- They are applicable to all geometric shapes.
- They are independent of scale and orientation.
- They are dimensionless numbers.
- They are not overly dependent on one or two extreme points in the shape.
- They agree with intuitive notions of what makes a shape compact.
A common compactness measure is the Isoperimetric quotient, the ratio of the area of the shape to the area of a circle (the most compact shape) having the same perimeter.
Compactness measures can be defined for three-dimensional shapes as well, typically as functions of volume and surface area. One example of a compactness measure is sphericity Ψ. Another measure in use is .
A common use of compactness measures is in redistricting. The goal is to maximize the compactness of electoral districts, subject to other constraints, and thereby to avoid gerrymandering. Another use is in zoning, to regulate the manner in which land can be subdivided into building lots. Another use is in pattern classification projects so that you can classify the circle from other shapes.
Famous quotes containing the words measure and/or shape:
“Since we are assured that the all-wise Creator has observed the most exact proportions of number, weight and measure in the make of all things, the most likely way therefore to get any insight into the nature of those parts of the Creation which come within our observation must in all reason be to number, weigh and measure.”
—Stephen Hales (1677–1761)
“Surely among a rich man’s flowering lawns,
Amid the rustle of his planted hills,
Life overflows without ambitious pains;
And rains down life until the basin spills,
And mounts more dizzy high the more it rains
As though to choose whatever shape it wills....”
—William Butler Yeats (1865–1939)