Exterior dimension

In geometry, exterior dimension is a type of dimension that can be used to characterize fat fractals.

A fat fractal is a Cantor set with Lebesgue measure (an extension of the classical notions of length and area to more complicated sets) greater than 0.

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 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 whose ternary expansions do not contain 1.

Iterating the process 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, ... (Sloane's A088917) whose nth term is D(n, n) = P_n (mod 3), where D(n, n) is a (central) Delannoy number and P_n(x) is a Legendre polynomial (E. W. Weisstein, Apr. 9, 2006).

References

• Cullen, H. F. (1968). Introduction to General Topology. Heath.