Compactness measure of a shape

The compactness measure of a shape 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.

Properties

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.

Examples

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 ${\displaystyle \Psi }$. Another measure in use is ${\displaystyle (\mathrm {surfacearea} )^{1.5}/(\mathrm {volume} )}$,[1] which is proportional to ${\displaystyle \Psi ^{-3/2}}$.

The Polsby-Popper Test is a mathematical measure of compactness that was developed to quantify the degree of gerrymandering of political districts.

Applications

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.[2] Another use is in zoning, to regulate the manner in which land can be subdivided into building lots.[3] Another use is in pattern classification projects so that you can classify the circle from other shapes.[citation needed]