Cantor set

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.

The first few steps

Cantor Set Media

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

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  This image shows how points in the Cantor set correspond to binary sequences.

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  Radial plot of the first ten steps

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  Cantor cubes recursion progression towards Cantor dust

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  2D fractal. Vector version of Cantor dust.png

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  Cantors cube a variation of the fractal Cantor set.

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  an image of the 2nd iteration of Cantor dust in two dimensions

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  an image of the 4th iteration of Cantor dust in two dimensions

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  an image of the 6th iteration of Cantor dust in two dimensions

Related pages

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

Other websites

• Barile, Margherita and Weisstein, Eric W. "Cantor Set". Wolfram MathWorld. Retrieved 23 January 2012.{{cite web}}: CS1 maint: multiple names: authors list (link)
• Su, Francis E.; et al. "Cantor Set". Math Fun Facts. Archived from the original on 14 January 2012. Retrieved 23 January 2012.
• Neal Carothers. "The Cantor Set". Archived from the original on 2012-02-04. Retrieved 2012-01-23.
• Cantor set at PRIME Archived 2012-02-23 at the Wayback Machine
• Media related to Cantor sets at Wikimedia Commons