29 June 1999

The pizza principle describes how areas and volumes change when an object's size is scaled up or down: Areas change by the square of the scale, volumes by the cube. Why is this so? It's easy to see when the object is a rectangle. Its area (A) is, by definition, its length (l) times its width (w). Scalingthat is, changing its sizeamounts to multiplying its dimensions l and w by the same positive value, say x. The new rectangle therefore has area A(x) = (x*l) * (x*w) = x^{2} * l * w = x^{2} * A. That's why the square (second power of x) appears. Likewise, the threedimensional analogue of a rectangle has a volume equal to the product of its three dimensions. The scale factor appears three times in the expression for the volume, whence the cube law. As for an arbitrary object, its area can be approximated to arbitrary accuracy by a collection of rectangles (or rectangular holes). Each one of these rectangles scales by the pizza principle, so the entire object does, too. The same argument holds for the volume. The pizza principle is useful for estimating areas and volumes. It applies to determining storage space for raster data sets: doubling the dimensions of the region covered necessarily requires four times as many cells (if each cell is to remain the same size). It is also useful for symbolizing data using areas (such as pies) or volumes (as in some 3D visualizations): if the area is to be proportional to an attribute, then the diameter of the figure must be proportional to the square root of the attribute. 