Block Fractals

Background - Perimeter and Area of the Gasket

First, we compute the perimeter of the gasket and find the gasket has infinite perimeter.

Next, we compute the area of the gasket and find the gasket has zero area.

Recall the gasket has dimension = Log(3)/Log(2) = 1.585, so 1 < dim(gasket) < 2.

Perimeter is a 1-dimensional measure, and the gasket has infinite perimeter.

Area is a 2-dimensional measure, and the gasket has zero area.

These are examples of the general principle that

measured below its dimension, a shape has infinite measure,

measured above its dimension, a shape has zero measure.

Return to Background.