Cantor Set

The Cantor set , sometimes also called the Cantor comb or no middle third set (Cullen 1968, pp. 78-81), is given by taking the interval (set ), removing the open middle third (), removing the open middle third of each of the two remaining pieces (), and continuing this procedure ad infinitum. It is therefore the set of points in the interval that admit a ternary expansion containing no digit 1, illustrated above. Both endpoints remain at every stage. In particular, the endpoint has the ternary expansion .

The th iteration of the Cantor set is implemented in the Wolfram Language as CantorMesh[n].

Iterating the string rewriting system 1 -> 101, 0 -> 000 starting with 1 gives the sequence 1, 101, 101000101, 101000101000000000101000101, .... The sequence of binary bits thus produced is therefore 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, ... (OEIS A088917) whose th term, indexed from , is given by , where is a central Delannoy number and is a Legendre polynomial (E. W. Weisstein, Apr. 9, 2006). The recurrence plot for this sequence is illustrated above.

The Cantor set consists of the real numbers that can be written as

(1)

where may equal 0 or 2 for each . This is an infinite perfect set. For a nonnegative integer , the total length of the line segments in the th iteration is

(2)

and the number of line segments is , so the length of each element is

(3)

and the capacity dimension is

			(4)
			(5)
			(6)
			(7)

(OEIS A102525). The Cantor set is nowhere dense, and has Lebesgue measure 0.

More generally, a nonempty perfect set in the real line that is nowhere dense is uncountable and may have 0 or positive Lebesgue measure. The Cantor set is, up to a homeomorphism, the unique nonempty totally disconnected, compact metric space that is a perfect set (Willard 1970; Calegari 2025, pp. 8-9).