Shailesh shirali is director of sahyadri school kfi, pune, and also head of the community mathematics centre in rishi valley school ap. General article fractal dimension and the cantor set. Hausdor measure turns out to be lebesgue measure when s is an integer. The fractal dimension is, as usual, given by the relation 4 where nr is the number of boxes of radius r needed to cover the fractal dust. Strictly speaking, this is usually called the upper box dimension and the box dimension is usually said to exist when the limit exists and is thus equal to the limsup. Recall that the box dimension of the cantor set is log3 2. In this lecture we construct selfsimilar sets of fractional dimension. The golden ratio is at the core of our fractal universe of infinite dimensions that looks and feels 4dimensional. Introduction to fractals and ifs is an introduction to some basic geometry of fractal sets, with emphasis on the iterated function system ifs formalism for generating fractals. Fractal dimension and the cantor set shailesh a shirali keywords dimension, topological dimension, hausdorffbesicovitch dimension, fractal dimension, fractal, cantor set, sierpinski triangle, koch curve.
If we go up to 2 dimensions and do something similar, we find a collection of shapes whose limit is the sierpinski carpet. The triadic cantor set a has box dimension dim b a log 2log 3. Pdf generalized cantor set and its fractal dimension. Fractal geometry by kenneth falconer encounters with chaos by denny gulick 1 a fractal is a subset of rn with non integer dimension. Similar results follow for a standard class of cantor sets of positive measure and their corresponding cantor functions. Since members of the cantor set are not normal, this would imply that all members of the cantor set are either rational or transcendental. Shirali and others published fractal dimension and the cantor set find, read and cite all the research you need on researchgate. Totally disconnected sets like single points, finite sets, or notably the cantor set have dimension zero. A cantor set is defined as a compact, perfect, totally disconnected subset in remark 1.
Dimensions of some generalized cantor sets yale university. The magnitude of fractal shapes and structures, when measured. Hence the fractal dimension of the cantor set is d log2 log3. The most basic fractal is themiddle thirds cantor set. The cantor set is constructed recursively by dividing. In the case of the cantor set, this amounts to keeping track of the lengths of the intervals that were removed in the construction of the cantor set, along with their multiplicities. The curve is a base motif fractal which uses a square segment as base. In fact, it has a fractional dimension, making it by definition a fractal. Points in the cantor set do not have 1 in the base.
Mandelbrot and nature clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. The fractal dimension of the cantor dust is equal to log4log3, what is about 1. By the lemma, the dimension of hte corresponding fractal increases. The cantor set is a fractal that is obtained by repeatedly removing the middle third of a segment. Fractal geometry by kenneth falconer encounters with chaos by denny gulick 1 a fractal is a subset of. The principal aim of this paper is to introduce a generator of finite subsets of the basic cantor. The highly intricate structure of the julia set illustrated in. The golden ratio is at the core of our fractal universe of infinite dimensions that looks and feels 4. These generalized dimensions are defined via a measure theoretic analysis in real space.
Further, the fractal dimension alone is not sufficient to characterize a fractal set and is only one of a number of generalized dimensions that can be considered. The cantor set is selfsimilar, consisting of n 2 congruent subsets, each when magni. Benoit mandelbrot has stated that a fractal is by definition a set for which the hausdorffbesicovitch dimension strictly exceeds the topological dimension. Lectures on fractals and dimension theory contents 0. Here is some java software to investiate properties of the boxcounting dimension. Pdf in this article, we discuss the cantor set and its fractal dimension. George cantor 18451918 constructed the cantor dust. The motif is to divide the square into nine equal parts and let only the outer four remain. Fractal dimension of the cantor middle set the set 3 is contained in for each.
So the cantor set can be decomposed into two disjoint pieces, each of which is itself a cantor set. Selfsimilarity and fractal dimension math 198, spring 20 background fractal geometry is one of the most important developments in mathematics in the second half of. In the year 18791884, george cantor coined few problems and consequences in the field of set theory. The term fractal had been introduced by benoit mandelbrot in the late 70ies. Since the cantor set is totally disconnected, it has topological dimension 0. We also prove that the similarity dimension is the same as the hausdor dimension of a fractal if the iterated function system associated to the fractal satis es the open set condition. See section 3 below for more details on these sets. Shirali and others published fractal dimension and the cantor set find, read and cite all the research you.
There is an inherent connection between entropy and fractal dimension 15. Logan axon notre dame random fractals march 29, 2010 19 36. One can easily calculate the fractal dimension of the cantor set of singularities by using box counting methods. Fractal geometry is a new way of looking at the world. On the shape of cantor sets 205 and so the subset of the cantor set contained in a is just a linearly scaled down copy of c. This means that small parts of the fractal contain all of the information of the entire fractal, no matter how small the viewing window on the fractal is.
Correlation of measured fractal dimensions with lacunarities. Golden geometry of einfinity fractal spacetime story of phi part 5. Functions, graphs, and limits the cantor set and fractals. Mandelbrot 2004 defined a fractal as a set for which the hausdorffbesicovitch dimension strictly exceeds the topological dimension, although an informal definition attributed to him is a fractal is a shape made of parts similar to the whole in some way feder, 1988. Correlation of measured fractal dimensions with lacunarities in computergenerated threedimensional images of cantor sets and those of fractional brownian motion. Remove the middle thirds of those two segments, and you end up with four disjoint segments. The cantor set, produced by the iterated process of removing the middle third from the previous segments. Last, we apply the theory we have developed to four examples. If we go up yet another dimension,we discover the dizzying idea of the menger sponge. Molter, and ronald shonkwiler we estimate the hausdor. By proposition 1, we can create a cantor set fractal with any given dimension r.
Topological dimension is a trickier thing, actually. Fractal dimension and lower bounds for geometric problems. The principal aim of this paper is to introduce a generator of finite subsets of the basic cantor ternary set and its generalization to the cantor nary set. How would you calculate the fractal dimension of this asymmetric cantor set. It is selfsimilar, because it is equal to two copies of itself, if each copy is shrunk by a factor of 3 and translated. The cantor set and the cantor function tma4225 foundations of analysis clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. What happens when we measure an object in the wrong dimension. The motif is to divide the line segment into three equal parts and replace the middle. Now we compute the boxcounting dimension of the cantor middle thirds set. It makes sense that the fractal dimension of the cantor set is between 0 and 1. The cantor set had properties \between those of a point and a line. A fractal that is constructed in a way similar to the cantor set. Pdf fractal dimension and the cantor set researchgate.
The fractal dimension is about the spacing of the elements of the iterates of the fractal as the number of iterates tends to infinity. Also, the set is regular enough that any reasonable definition of fractal dimension agrees with that computation. Proposition 3for any real r, 0 r 1, there is a continuum of fractals with dimension d r. Presented here is a list of fractals ordered by increasing hausdorff dimension, with the purpose of visualizing what it means for a fractal to have a low or a high dimension. All structured data from the file and property namespaces is available under the creative commons cc0 license. An introduction to the theory of complex dimensions and. Can a continuous function on r have a periodic point of prime period 48 and not one of prime period 56. Then, in its generator we can shift the removed middle part by h, 0 h d, for some d 0. This can be done by removing the central interval of length 20. Hausdorff dimension of sets of nondifferentiability. Cantor set as a fractal and its application in detecting. Fractals and the cantor set fractals are remarkable designs noted for their infinite selfsimilarity. We are familiar with the notion ofdimension of an object, though it can be challenging to give a precise definition.
We have used the limit supremum to avoid problems with convergence. Cantor set, sierpinski carpet or gasket, peanocurve, koch. Fractals and the cantor set 1952 words 123 help me. This cantor set contains a 14scale copy of itself on the left and a 12scale copy of itself on the right. Can a continuous function on r have a periodic point of prime period 176 but not one of prime period 96. If the set of centers has fractal dimension 1, solvable in time nok 1 logn sidiropoulos, sridhar17. Cantor triadic set, 4adictype cantorlike set, and 5adictype cantorlike set, respectively.
This page was last edited on 7 january 2019, at 19. The results provide the relation between the decay rate of this sequence and the dimension of the associated cantor set. In the following the cantor set defined below is taken as a paradigmatic example because of its apparent. With the purpose of researching the changing regularities of the cantor sets multifractal spectrums and generalized fractal dimensions under different probability factors, from statistical physics, the cantor set is given a mass distribution, when the mass is given with different probability ratios, the different multifractal spectrums and the generalized fractal dimensions will be. Fractal geometry mathematical foundations and applications. The fractal dimension of the cantor set is equal to log2log3, what is about 0. We show the generalized cantor sets and we also show the. Fractal dimensions an overview sciencedirect topics.
Cantor set, sierpinski triangle, koch snowflake, fractal dimension. Benoit mandelbrot has stated that a fractal is by definition a set for which the. The curve is a base motif fractal which uses a line segment as base. A new algorithm is used to determine fractal dimensions by box counting for dynamic and iterated function systems. The middle third cantor set f consists of the numbers that are in e k for all k. Cantor set the cantor set was first described by german. Since the cantor set is the prototypical fractal, we would like to study the generalized cantor set in measure space, which is defined by an algorithm and also defined by the shrinking process 9. Loosely speaking, it is the number of independent quantities needed to specify the positions of points. Hausdor dimension is the most widely used notion of fractional dimension, but there are several others. One of them was the cantor ternary set as a classical example of fractals.
The cantor set f may be thought of as the limit of the sequence of sets e k as k tends to in. The hausdorff dimension of the set of points n at which the cantor function is not differentiable is ln2ln32. In principle, there are an unlimited number of fractal or generalized dimensions. How would you calculate the fractal dimension of this. Fractals and fractal dimensions an alternative method for computing box dimension john a. In this paper, 5adic cantor onefifth set as an example of fractal string have been introduced. Kenneth falconer, fractal geometry 2nd, wiley, 2003. The cantor set let us recall the definition of a cantor set in. The cantor set has zero length, and noninteger dimension. Developing ahigherdimensional theory of complex fractal. Such objects are called fractals, and the cantor set is one of the earliest examples of such an object. It is obviously impossible to draw the set f itself, with its in.
The cantor set is an interesting example of an uncountable set of measure zero and has many interesting properties and consequences in the fields of set theory, topology, and fractal theory. Files are available under licenses specified on their description page. The fractals are constructed using a fixed geometric replacement rule. Examples of this type are the julia and mandelbrot sets gaston julia, pierre fatou, benoit mandelbrot, and newton fractal. Hausdorff dimension, packing dimension, cantor set. Any such set is homeomorphic to the cantor middle third set and therefore deserve the name of cantor set. The cantor set is an example of an uncountable set with measure zero and has potential applications in various branches of mathematics such as topology, measure theory, dynamical systems and fractal geometry. The main tool of fractal geometry is dimension in its many forms. A plane analogue of the cantor set, a cantor dust, is illustrated in. Twodimensional displays at z 0, z 1 and z 2 of a 3dimensional triadic cantor set with n 9, of which generator is shown in fig.