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A construction ahead of its time
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Long before Benoît Mandelbrot was born, Georg Cantor described the first fractal set in 1883, some twenty years before von Koch introduced his famous snowflake. His construction is typical of fractals, even if the result has almost no visual appeal because the set lies on a line. We begin with the line segment C0 = \[0, 1\]. We remove its middle open interval, whose length is one-third that of C0, namely (1/3, 2/3). This gives C1 = \[0, 1/3\] \[2/3, 1\]. We repeat the process on each of the two line segments making up C1. This gives C2, the union of four line segments of length 1/9: \[0, 1/9\], \[2/9, 3/9\], \[6/9, 7/9\], and \[8/9, 9/9\]. Continuing in this way yields Cn, the union of 2n disjoint line segments of length 1/3n, with total length ln equal to (2/3)n. The ternary Cantor set C is obtained in the limit as the intersection of all the Cn. Because of its appearance, it is also called Cantor dust.
The length of Cn tends to 0, so in the limit C has length zero. Likewise, C cannot contain any open interval, however small: it has empty interior. Finally, every convergent sequence of elements of C has its limit in C: the ternary Cantor set is closed.
