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.).[1][2][3] 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
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. Retrieved 23 January 2012.
{{cite web}}: Explicit use of et al. in:|author=(help) - Neal Carothers. "The Cantor Set".
- Cantor set at PRIME
Media related to Cantor sets at Wikimedia Commons
References
- ↑ "Theory of Infinite Sets". Exploratorium.edu. Retrieved 2012-01-27.
- ↑ Erin Chamberlain. "The Cantor Set" (PDF). Retrieved 2012-01-27.
- ↑ Rupert Levene (February 13, 2009). "The Cantor set is uncountable" (PDF). Retrieved 2012-01-27.
