Cantor set facts for kids

The Cantor set is a subset of real numbers with certain properties that are interesting to mathematicians. These properties relate to topology, measurement, geometry, as well as set theory. Some of them are:

• When represented geometrically, the set is a fractal, it has a Hausdorff dimension which is not an integer
• It has the same cardinality as the set of real numbers
• It is self-similar

The set is named after Georg Cantor. Henry John Stephen Smith discovered it in 1875, and Cantor first described it in 1883.

The set is made by starting with a line segment and repeatedly removing the middle third. The Cantor set is the (infinite) set of points left over. The Cantor set is "more infinite" than the set of natural numbers (1, 2, 3, 4, etc.). This property is called uncountability. It is related to the Smith–Volterra–Cantor set and the Menger Sponge. The Cantor set is self-similar.

Related pages

• Fractal
• Set theory
• Smith–Volterra–Cantor set
• Cantor dust
• Menger Sponge

Images for kids

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  Zoom in Cantor set. Each point in the set is represented here by a vertical line.

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  Cantor cubes recursion progression towards Cantor dust

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  Column capital with pattern evocative of the Cantor set, but expressed in binary rather than ternary. Engraving of Île de Philae from Description d'Égypte by Jean-Baptiste Prosper Jollois and Édouard Devilliers, Imprimerie Impériale, Paris, 1809-1828

See also

In Spanish: Conjunto de Cantor para niños