In effect, the shorter the ruler, the longer the measured border; the Spanish and Portuguese geographers were simply using different-length rulers. The result most astounding to Richardson is that, under certain circumstances, as ℓ approaches zero, the length of the coastline approaches infinity. Richardson had believed, based on Euclidean geometry, that a coastline would approach a fixed length, as do similar estimations of regular geometric figures. For example, the perimeter of a regular polygon inscribed in a circle approaches the circumference with increasing numbers of sides (and decrease in the length of one side). In geometric measure theory such a smooth curve as the circle that can be approximated by small straight segments with a definite limit is termed a rectifiable curve. Measuring a coastline More than a decade after Richardson completed his work, Benoit Mandelbrot developed a new branch of mathematics, fractal geometry, to describe just such non-rectifiable complexes in nature as the infinite coastline. His own definition of the new figure serving as the basis for his study is: A key property of some fractals is self-similarity; that is, at any scale the same general configuration appears. A coastline is perceived as bays alternating with promontories. In the hypothetical situation that a given coastline has this property of self-similarity, then no matter how great any one small section of coastline is magnified, a similar pattern of smaller bays and promontories superimposed on larger bays and promontories appears, right down to the grains of sand. At that scale the coastline appears as a momentarily shifting, potentially infinitely long thread with a stochastic arrangement of bays and promontories formed from the small objects at hand. In such an environment (as opposed to smooth curves) Mandelbrot asserts "coastline length turns out to be an elusive notion that slips between the fingers of those who want to grasp it". There are different kinds of fractals. A coastline with the stated property is in "a first category of fractals, namely curves whose fractal dimension is greater than 1". That last statement represents an extension by Mandelbrot of Richardson's thought. Mandelbrot's statement of the Richardson effect is: where L, coastline length, a function of the measurement unit ε, is approximated by the expression. F is a constant, and D is a parameter that Richardson found depended on the coastline approximated by L. He gave no theoretical explanation, but Mandelbrot identified D with a non-integer