middlethirds

Middle thirds refers to the central portion of a segment obtained by dividing it into three equal parts. For a closed interval [a, b], the middle third is the subinterval [a + (b − a)/3, a + 2(b − a)/3], and its length is (b − a)/3. In many contexts, especially in the construction of fractals, the middle third is removed as an open interval (a + (b − a)/3, a + 2(b − a)/3), leaving the two outer thirds.

A prominent application is the Cantor set construction. Starting with [0,1], one removes the open middle third (1/3, 2/3), leaving [0,1/3] and [2/3,1]. This process is repeated recursively on every remaining interval. After n steps, the set consists of 2^n closed intervals each of length (1/3)^n, and the total length remaining is (2/3)^n. As n grows without bound, the total length removed tends to 1, so the Cantor set has Lebesgue measure zero.

The Cantor set is uncountable, perfect (every point is a limit point), and nowhere dense. It is also a canonical example in fractal geometry, with a Hausdorff dimension of log 2 / log 3. Points in the Cantor set correspond to ternary representations using only the digits 0 and 2, and there is a standard mapping between this set and binary sequences.

In summary, middle thirds describe a simple partition concept that underpins a rich area of analysis and fractal geometry, illustrating how removing central segments can yield surprising and counterintuitive sets.