CANTOR SET (Triadic)
| Collection | International Encyclopedia of Systems and Cybernetics |
|---|---|
| Year | 2004 |
| Vol. (num.) | 2(1) |
| ID | ◀ 352 ▶ |
| Object type | Methodology or model |
This set, imagined by Georg CANTOR towards the end of 19th century is probably the first ever abstract representation of a fractal set.
It is constructed, starting from a segment, which is divided in three equal parts, the middle third being removed. This operation is renewed on the two subsisting segments ad infinitum (supposedly!). In this way, we should obtain finally an infinity of points… whose total length dimension, or linear measure, would be zero. The Cantor triadic set has a fractal dimension of 0.63. Another Cantor set can be constructed removing at each stage two fifth segments within the remaining part of the original segment. It has also a zero linear measure and its fractal dimension is O.68.
The Cantor set is self similar.
If we could invert the construction process of the Cantor set (i.e. from an infinity of points to a complete segment) we would have modelled the genesis of a macrolevel system constructed from discrete parts.
Other mathematical objects of the same type are the “snowflake” Koch curve, and the Peano curve, made from an infinitely folded segment which finally would wholly cover a surface. There are also objects of the same kind in three or more dimensions.