Christiane's Hair

Abstract : We explore the geometric and measure-theoretic properties of a set built by stacking central Cantor sets with continuously varying scaling factors. By using self-similarity, we are able to describe in a fairly complete way its main features. We show that it is made of an uncountable number of analytic curves, compute the exact areas of the gaps of all sizes, and show that its Hausdor and box counding dimension are both equal to 2. It provides a particularly good example to introduce and showcase these notions because of the beauty and simplicity of the arguments. Our derivation of explicit formulas for the areas of all of the gaps is elementary enough to be explained to rst-year calculus students.
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Submitted on : Monday, October 22, 2012 - 4:17:09 PM
Last modification on : Thursday, April 25, 2019 - 11:12:26 AM
Long-term archiving on: Wednesday, January 23, 2013 - 3:43:28 AM

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Jacques Lévy Véhel, Franklin Mendivil. Christiane's Hair. American Mathematical Monthly, Mathematical Association of America, 2013, 120 (9), pp.771-786. ⟨10.4169/amer.math.monthly.120.09.771 ⟩. ⟨hal-00744268⟩

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