Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith[1][2][3][4] and mentioned by German mathematician Georg Cantor in 1883.[5][6]
Through consideration of this set, Cantor and others helped lay the foundations of modern
More generally, in topology, a Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology). By a theorem of L. E. J. Brouwer, this is equivalent to being perfect, nonempty, compact, metrizable and zero dimensional.[7]
![](http://upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Cantor_Set_Expansion.gif/600px-Cantor_Set_Expansion.gif)
Construction and formula of the ternary set
The Cantor ternary set is created by iteratively deleting the
and
for , so that
The first six steps of this process are illustrated below.
Using the idea of self-similar transformations, and the explicit closed formulas for the Cantor set are[8]
where every middle third is removed as the open interval from the
where the middle third of the foregoing closed interval is removed by intersecting with
This process of removing middle thirds is a simple example of a finite subdivision rule. The complement of the Cantor ternary set is an example of a fractal string.
In arithmetical terms, the Cantor set consists of all real numbers of the unit interval that do not require the digit 1 in order to be expressed as a ternary (base 3) fraction. As the above diagram illustrates, each point in the Cantor set is uniquely located by a path through an infinitely deep binary tree, where the path turns left or right at each level according to which side of a deleted segment the point lies on. Representing each left turn with 0 and each right turn with 2 yields the ternary fraction for a point.
Mandelbrot's construction by "curdling"
In The Fractal Geometry of Nature, mathematician Benoit Mandelbrot provides a whimsical thought experiment to assist non-mathematical readers in imagining the construction of . His narrative begins with imagining a bar, perhaps of lightweight metal, in which the bar's matter "curdles" by iteratively shifting towards its extremities. As the bar's segments become smaller, they become thin, dense slugs that eventually grow too small and faint to see.
CURDLING: The construction of the Cantor bar results from the process I call curdling. It begins with a round bar. It is best to think of it as having a very low density. Then matter "curdles" out of this bar's middle third into the end thirds, so that the positions of the latter remain unchanged. Next matter curdles out of the middle third of each end third into its end thirds, and so on ad infinitum until one is left with an infinitely large number of infinitely thin slugs of infinitely high density. These slugs are spaced along the line in the very specific fashion induced by the generating process. In this illustration, curdling (which eventually requires hammering!) stops when both the printer's press and our eye cease to follow; the last line is indistinguishable from the last but one: each of its ultimate parts is seen as a gray slug rather than two parallel black slugs.[9]
Composition
Since the Cantor set is defined as the set of points not excluded, the proportion (i.e., measure) of the unit interval remaining can be found by total length removed. This total is the geometric progression
So that the proportion left is 1 − 1 = 0.
This calculation suggests that the Cantor set cannot contain any interval of non-zero length. It may seem surprising that there should be anything left—after all, the sum of the lengths of the removed intervals is equal to the length of the original interval. However, a closer look at the process reveals that there must be something left, since removing the "middle third" of each interval involved removing
It may appear that only the endpoints of the construction segments are left, but that is not the case either. The number 1/4, for example, has the unique ternary form 0.020202... = 0.02. It is in the bottom third, and the top third of that third, and the bottom third of that top third, and so on. Since it is never in one of the middle segments, it is never removed. Yet it is also not an endpoint of any middle segment, because it is not a multiple of any power of 1/3.[10] All endpoints of segments are terminating ternary fractions and are contained in the set
which is a
Properties
Cardinality
It can be shown that there are as many points left behind in this process as there were to begin with, and that therefore, the Cantor set is uncountable. To see this, we show that there is a function f from the Cantor set to the closed interval [0,1] that is surjective (i.e. f maps from onto [0,1]) so that the cardinality of is no less than that of [0,1]. Since is a
To construct this function, consider the points in the [0, 1] interval in terms of base 3 (or ternary) notation. Recall that the proper ternary fractions, more precisely: the elements of , admit more than one representation in this notation, as for example 1/3, that can be written as 0.13 = 0.103, but also as 0.0222...3 = 0.023, and 2/3, that can be written as 0.23 = 0.203 but also as 0.1222...3 = 0.123.[11] When we remove the middle third, this contains the numbers with ternary numerals of the form 0.1xxxxx...3 where xxxxx...3 is strictly between 00000...3 and 22222...3. So the numbers remaining after the first step consist of
- Numbers of the form 0.0xxxxx...3 (including 0.022222...3 = 1/3)
- Numbers of the form 0.2xxxxx...3 (including 0.222222...3 = 1)
This can be summarized by saying that those numbers with a ternary representation such that the first digit after the
The second step removes numbers of the form 0.01xxxx...3 and 0.21xxxx...3, and (with appropriate care for the endpoints) it can be concluded that the remaining numbers are those with a ternary numeral where neither of the first two digits is 1.
Continuing in this way, for a number not to be excluded at step n, it must have a ternary representation whose nth digit is not 1. For a number to be in the Cantor set, it must not be excluded at any step, it must admit a numeral representation consisting entirely of 0s and 2s.
It is worth emphasizing that numbers like 1, 1/3 = 0.13 and 7/9 = 0.213 are in the Cantor set, as they have ternary numerals consisting entirely of 0s and 2s: 1 = 0.222...3 = 0.23, 1/3 = 0.0222...3 = 0.023 and 7/9 = 0.20222...3 = 0.2023. All the latter numbers are "endpoints", and these examples are right
This set of endpoints is dense in (but not dense in [0, 1]) and makes up a
The function from to [0,1] is defined by taking the ternary numerals that do consist entirely of 0s and 2s, replacing all the 2s by 1s, and interpreting the sequence as a
- where
For any number y in [0,1], its binary representation can be translated into a ternary representation of a number x in by replacing all the 1s by 2s. With this, f(x) = y so that y is in the range of f. For instance if y = 3/5 = 0.100110011001...2 = 0.1001, we write x = 0.2002 = 0.200220022002...3 = 7/10. Consequently, f is surjective. However, f is not injective — the values for which f(x) coincides are those at opposing ends of one of the middle thirds removed. For instance, take
- 1/3 = 0.023 (which is a right limit point of and a left limit point of the middle third [1/3, 2/3]) and
- 2/3 = 0.203 (which is a left limit point of and a right limit point of the middle third [1/3, 2/3])
so
Thus there are as many points in the Cantor set as there are in the interval [0, 1] (which has the uncountable cardinality ). However, the set of endpoints of the removed intervals is countable, so there must be uncountably many numbers in the Cantor set which are not interval endpoints. As noted above, one example of such a number is 1/4, which can be written as 0.020202...3 = 0.02 in ternary notation. In fact, given any , there exist such that . This was first demonstrated by Steinhaus in 1917, who proved, via a geometric argument, the equivalent assertion that for every .[12] Since this construction provides an injection from to , we have as an immediate corollary. Assuming that for any infinite set (a statement shown to be equivalent to the axiom of choice by Tarski), this provides another demonstration that .
The Cantor set contains as many points as the interval from which it is taken, yet itself contains no interval of nonzero length. The irrational numbers have the same property, but the Cantor set has the additional property of being closed, so it is not even dense in any interval, unlike the irrational numbers which are dense in every interval.
It has been conjectured that all algebraic irrational numbers are normal. Since members of the Cantor set are not normal, this would imply that all members of the Cantor set are either rational or transcendental.
Self-similarity
The Cantor set is the prototype of a
Repeated iteration of and can be visualized as an infinite binary tree. That is, at each node of the tree, one may consider the subtree to the left or to the right. Taking the set together with
The automorphisms of the binary tree are its hyperbolic rotations, and are given by the modular group. Thus, the Cantor set is a homogeneous space in the sense that for any two points and in the Cantor set , there exists a homeomorphism with . An explicit construction of can be described more easily if we see the Cantor set as a product space of countably many copies of the discrete space . Then the map defined by is an involutive homeomorphism exchanging and .
Conservation law
It has been found that some form of conservation law is always responsible behind scaling and self-similarity. In the case of Cantor set it can be seen that the th moment (where is the fractal dimension) of all the surviving intervals at any stage of the construction process is equal to a constant which is one in the case of the Cantor set.[13][14] We know that there are intervals of size present in the system at the th step of its construction. Then if we label the surviving intervals as then the th moment is since .
The Hausdorff dimension of the Cantor set is equal to ln(2)/ln(3) ≈ 0.631.
Topological and analytical properties
Although "the" Cantor set typically refers to the original, middle-thirds Cantor set described above, topologists often talk about "a" Cantor set, which means any
As the above summation argument shows, the Cantor set is uncountable but has
For any point in the Cantor set and any arbitrarily small
Every point of the Cantor set is also an accumulation point of the complement of the Cantor set.
For any two points in the Cantor set, there will be some ternary digit where they differ — one will have 0 and the other 2. By splitting the Cantor set into "halves" depending on the value of this digit, one obtains a partition of the Cantor set into two closed sets that separate the original two points. In the
As a topological space, the Cantor set is naturally homeomorphic to the product of countably many copies of the space , where each copy carries the
which can also be identified with the set of
From the above characterization, the Cantor set is homeomorphic to the
The Cantor set is a subset of the reals, which are a
We have seen above that the Cantor set is a totally disconnected perfect compact metric space. Indeed, in a sense it is the only one: every nonempty totally disconnected perfect compact metric space is homeomorphic to the Cantor set. See Cantor space for more on spaces homeomorphic to the Cantor set.
The Cantor set is sometimes regarded as "universal" in the
For any
Measure and probability
The Cantor set can be seen as the compact group of binary sequences, and as such, it is endowed with a natural Haar measure. When normalized so that the measure of the set is 1, it is a model of an infinite sequence of coin tosses. Furthermore, one can show that the usual Lebesgue measure on the interval is an image of the Haar measure on the Cantor set, while the natural injection into the ternary set is a canonical example of a singular measure. It can also be shown that the Haar measure is an image of any probability, making the Cantor set a universal probability space in some ways.
In Lebesgue measure theory, the Cantor set is an example of a set which is uncountable and has zero measure.[16] In contrast, the set has a Hausdorff measure of 1 in its dimension of log 2 / log 3.[17]
Cantor numbers
If we define a Cantor number as a member of the Cantor set, then[18]
- Every real number in [0, 2] is the sum of two Cantor numbers.
- Between any two Cantor numbers there is a number that is not a Cantor number.
Descriptive set theory
The Cantor set is a meagre set (or a set of first category) as a subset of [0,1] (although not as a subset of itself, since it is a Baire space). The Cantor set thus demonstrates that notions of "size" in terms of cardinality, measure, and (Baire) category need not coincide. Like the set , the Cantor set is "small" in the sense that it is a null set (a set of measure zero) and it is a meagre subset of [0,1]. However, unlike , which is countable and has a "small" cardinality, , the cardinality of is the same as that of [0,1], the continuum , and is "large" in the sense of cardinality. In fact, it is also possible to construct a subset of [0,1] that is meagre but of positive measure and a subset that is non-meagre but of measure zero:[19] By taking the countable union of "fat" Cantor sets of measure (see Smith–Volterra–Cantor set below for the construction), we obtain a set which has a positive measure (equal to 1) but is meagre in [0,1], since each is nowhere dense. Then consider the set . Since , cannot be meagre, but since , must have measure zero.
Variants
![](http://upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Cantor_set_radial.svg/220px-Cantor_set_radial.svg.png)
Smith–Volterra–Cantor set
Instead of repeatedly removing the middle third of every piece as in the Cantor set, we could also keep removing any other fixed percentage (other than 0% and 100%) from the middle. In the case where the middle 8/10 of the interval is removed, we get a remarkably accessible case — the set consists of all numbers in [0,1] that can be written as a decimal consisting entirely of 0s and 9s. If a fixed percentage is removed at each stage, then the limiting set will have measure zero, since the length of the remainder as for any such that .
On the other hand, "fat Cantor sets" of positive measure can be generated by removal of smaller fractions of the middle of the segment in each iteration. Thus, one can construct sets homeomorphic to the Cantor set that have positive Lebesgue measure while still being nowhere dense. If an interval of length () is removed from the middle of each segment at the nth iteration, then the total length removed is , and the limiting set will have a Lebesgue measure of . Thus, in a sense, the middle-thirds Cantor set is a limiting case with . If , then the remainder will have positive measure with . The case is known as the Smith–Volterra–Cantor set, which has a Lebesgue measure of .
Stochastic Cantor set
One can modify the construction of the Cantor set by dividing randomly instead of equally. Besides, to incorporate time we can divide only one of the available intervals at each step instead of dividing all the available intervals. In the case of stochastic triadic Cantor set the resulting process can be described by the following rate equation[13][14]
and for the stochastic dyadic Cantor set[21]
where is the number of intervals of size between and . In the case of triadic Cantor set the fractal dimension is which is less than its deterministic counterpart . In the case of stochastic dyadic Cantor set the fractal dimension is which is again less than that of its deterministic counterpart . In the case of stochastic dyadic Cantor set the solution for exhibits dynamic scaling as its solution in the long-time limit is where the fractal dimension of the stochastic dyadic Cantor set . In either case, like triadic Cantor set, the th moment () of stochastic triadic and dyadic Cantor set too are conserved quantities.
Cantor dust
Cantor dust is a multi-dimensional version of the Cantor set. It can be formed by taking a finite
![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Cantorcubes.gif/250px-Cantorcubes.gif)
![]() |
![]() |
A different 2D analogue of the Cantor set is the
Historical remarks
![](http://upload.wikimedia.org/wikipedia/commons/thumb/8/88/Cantor_dust_in_two_dimensions_iteration_2.svg/220px-Cantor_dust_in_two_dimensions_iteration_2.svg.png)
![an image of the 4th iteration of Cantor dust in two dimensions](http://upload.wikimedia.org/wikipedia/commons/thumb/5/50/Cantor_dust_in_two_dimensions_iteration_4.svg/220px-Cantor_dust_in_two_dimensions_iteration_4.svg.png)
Cantor introduced what we call today the Cantor ternary set as an example "of a perfect point-set, which is not everywhere-dense in any interval, however small."[24][25] Cantor described in terms of ternary expansions, as "the set of all real numbers given by the formula: where the coefficients arbitrarily take the two values 0 and 2, and the series can consist of a finite number or an infinite number of elements."[24]
A topological space is perfect if all its points are limit points or, equivalently, if it coincides with its derived set . Subsets of the real line, like , can be seen as topological spaces under the induced subspace topology.[7]
Cantor was led to the study of derived sets by his results on uniqueness of
See also
![an image of the 6th iteration of Cantor dust in two dimensions](http://upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Cantor_dust_in_two_dimensions_iteration_6.svg/220px-Cantor_dust_in_two_dimensions_iteration_6.svg.png)
- The indicator function of the Cantor set
- Smith–Volterra–Cantor set
- Cantor function
- Cantor cube
- Antoine's necklace
- Koch snowflake
- Knaster–Kuratowski fan
- List of fractals by Hausdorff dimension
- Moser–de Bruijn sequence
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
Notes
- ^ Smith, Henry J.S. (1874). "On the integration of discontinuous functions". Proceedings of the London Mathematical Society. First series. 6: 140–153.
- ^ The "Cantor set" was also discovered by Paul du Bois-Reymond (1831–1889). See du Bois-Reymond, Paul (1880), "Der Beweis des Fundamentalsatzes der Integralrechnung", Mathematische Annalen (in German), 16, footnote on p. 128. The "Cantor set" was also discovered in 1881 by Vito Volterra (1860–1940). See: Volterra, Vito (1881), "Alcune osservazioni sulle funzioni punteggiate discontinue" [Some observations on point-wise discontinuous function], Giornale di Matematiche (in Italian), 19: 76–86.
- ISBN 9783034850513.
- ISBN 0140256024.
- ^ S2CID 121930608. Archived from the originalon 2015-09-24. Retrieved 2011-01-10.
- ISBN 978-1-4684-9396-2.
- ^ ISBN 978-0-387-94374-9.
- .
- ^ )
- ^ a b Belcastro, Sarah-Marie; Green, Michael (January 2001), "The Cantor set contains ? Really?", The College Mathematics Journal, 32 (1): 55, JSTOR 2687224
- ^ This alternative recurring representation of a number with a terminating numeral occurs in any positional system with Archimedean absolute value.
- ISBN 978-0-521-69624-1.
- ^ .
- ^ .
- ASIN B0000EG7Q0.
- ^ Irvine, Laura. "Theorem 36: the Cantor set is an uncountable set with zero measure". Theorem of the week. Archived from the original on 2016-03-15. Retrieved 2012-09-27.
- ^
Falconer, K. J. (July 24, 1986). The Geometry of Fractal Sets (PDF). Cambridge University Press. pp. 14–15. ISBN 9780521337052.
- ISBN 0486472043.
- OCLC 527671.
- ^ "Radial Cantor Set".
- S2CID 14494072.
- ISBN 978-3-11-019092-2.
- ISBN 978-3-11-019092-2.
- ^ a b Cantor, Georg (2021). ""Foundations of a general theory of sets: A mathematical-philosophical investigation into the theory of the infinite", English translation by James R Meyer". www.jamesrmeyer.com. Footnote 22 in Section 10. Retrieved 2022-05-16.
- ^ JSTOR 2690689.
References
- MR 0507446.
- Wise, Gary L.; Hall, Eric B. (1993). Counterexamples in Probability and Real Analysis. New York: ISBN 0-19-507068-2.
- ISBN 0521337054.
- Mattila, Pertti (25 February 1999). Geometry of Sets and Measures in Euclidean Space: Fractals and rectifiability. Cambridge studies in advanced mathematics. Cambridge University Press. ISBN 0521655951.
- Mattila, Pertti (2015). Fourier Analysis and Hausdorff Dimension. Cambridge studies in advanced mathematics. Cambridge University Press. ISBN 9781316227619..
- Zygmund, A. (1958). Trigonometric Series, Vols. I and II. Cambridge University Press.
External links
- "Cantor set", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Cantor Sets and Cantor Set and Function at cut-the-knot
- Cantor Set at Platonic Realms