Fractal surfaces from simple arithmetic operations

by   Vladimir Garcia-Morales, et al.

Fractal surfaces ('patchwork quilts') are shown to arise under most general circumstances involving simple bitwise operations between real numbers. A theory is presented for all deterministic bitwise operations on a finite alphabet. It is shown that these models give rise to a roughness exponent H that shapes the resulting spatial patterns, larger values of the exponent leading to coarser surfaces.



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I Introduction

Fractal surfaces Mandelbrot ; Feder are ubiquitously found in biological and physical systems at all scales, ranging from atoms to galaxies Moriarty ; Richardella ; Martinez ; Baryshev . Mathematically, fractals arise from iterated function systems PeitgenBOOK ; Barnsley , strange attractors Ott , critical phenomena SornetteBOOK , cellular automata Wolfram ; VGM1 , substitution systems Wolfram ; VGM4 and any context where some hierarchical structure is present Yamamoto ; Yamamoto2 . Because of these connections, fractals are also important in some recent approaches to nonequilibrium statistical mechanics of steady states Tsallis ; Tsallis2 ; Hoover ; VGMStat .

Fractality is intimately connected to power laws, real-valued dimensions and exponential growth of details as resulting from an increase in the resolution of a geometric object Indekeu5 . Although the rigorous definition of a fractal requires that the Hausdorff-Besicovitch dimension is strictly greater than the Euclidean dimension of the space in which the fractal object is embedded, there are cases in which surfaces are sufficiently broken at all length scales so as to deserve being named ‘fractals’ Mandelbrot ; Indekeu5 ; MVBerry . Most of these surfaces are random and governed by growth, adsorption and deposition processes on interfaces Indekeu5 ; Vicsek ; Vicsek2 ; StanleyFractals ; Indekeu2 ; Indekeu3 ; Indekeu4 ; Indekeu6 .

The multidisciplinary field of disorderly surface growth has experienced a rapid development StanleyFractals . However, fractal surfaces formed by deterministic processes and hierarchical rules are also interesting and can be useful, e.g., as models for cityscapes Indekeu5 . In this article, by regarding ‘fractals’ in the broad sense mentioned in the previous paragraph, we construct a wide variety of such surfaces and show that they possess fractal self-affine properties. If a surface is self-affine it satisfies Vicsek ; Vicsek2


where is the roughness exponent characterizing the self-affine scaling Mandelbrot Sinha . We prove that the surfaces here constructed obey Eq. (1). Although our construction proceeds abstractly, we illustrate it with specific numerical examples, and we believe that the generality of the method is such that it may find many applications in the modeling of complex physical systems. Although the notation may seem unfamiliar, the mathematics behind is elementary, and is based on generalized bitwise arithmetic on real numbers. To introduce this idea, let us consider two real numbers and (that we may truncate to a finite number of digits after the decimal point). If we expand and in base 10, we see that the ordinary sum of these numbers splits into the ordinary sum of two different parts


here denotes addition modulo and denotes the contribution of the carries to the sum . If and then, it is clear that if . If then . If and/or are larger than 10, the sum is clearly reduced to separately considering the digits of and and adding them, taking care of the carries. Addition modulo 10 means that the carries are neglected. For example, if and we have that . This sum can be seen as the ordinary addition of two different contributions and . The structure of the operators and is very interesting. All positions within the numbers are independent of each other as regards the bitwise action of : Each two digits corresponding to the same power of ten are added modulo ten and no carry is transferred from one position to another. Therefore, under the action of such operator, the positions within the number are ‘uncoupled’ and independent. Since each position within a number in a standard positional number system corresponds to a different power of the base, adding modulo 10 two real numbers means performing the same operation at all scales, the latter being independent of each other. We claim that, as a result of this, the function (where and are real numbers) has fractal features. It is a discontinuous and non-differentiable function that exhibits the same details at all scales, displaying self-similarity. In fact this is generally the case for the function , where , is any base, with denoting addition modulo (VGM2 ; VGM3 ). We shall prove below that such a function obeys Eq. (1). Furthermore, since is not a fractal, the function must be a fractal as well so as to compensate the discontinuities of . In Fig. (1) the functions (left), (right) and are plotted for and . While is just an inclined plane, the functions and exhibit a more complex behavior displaying ‘squares within squares’ at all scales. The behavior of these functions is such that under ordinary addition, all discontinuous jumps disappear at all scales, yielding the smooth function (which is everywhere equal to ).

Figure 1: The functions (left), (center) and (right) for and . We have everywhere. Dark blue means a zero value, green a value of 1 and dark red a value of 2.

The purpose of this paper is to construct all possible bitwise operators acting on a finite alphabet of symbols (to which and do belong, and which we explicitly construct as well, as an example) and to elucidate their associated fractal features. We thus find a new kind of surfaces with self-affine properties, that we term ‘patchwork quilts’. We constructively prove the general fact that bitwise operators locally acting on each point in the plane leads to these surfaces, providing explicit mathematical expressions for them. The results can be easily extended to hypersurfaces in . We explicitly construct the generalized bitwise operators involved in this process by making use of the framework of digital calculus, that we have very recently introduced VGM4 ; QUANTUM ; CHAOSOLFRAC . We show that these abstract surfaces give rise to a roughness exponent that shapes the resulting spatial patterns, larger values of the exponent leading to coarser surfaces.

Recently, we have found a new construction for fractals based on a fractal decomposition of a given function CHAOSOLFRAC linking the resulting fractal objects to finite group theory CHAOSOLFRAC . The construction of fractal surfaces given here is new and different to our previous one in CHAOSOLFRAC . Although operators acting on two variables are again considered, these operators are not restricted to have the latin square property as in CHAOSOLFRAC and the construction is different even when the concepts that are used in it are closely related.

In Section II we introduce the main concept of digital calculus, the digit function QUANTUM ; CHAOSOLFRAC and we construct all discrete operators acting on two variables whose values are restricted to be any of the integers in the interval . With the aid of these tools, we then construct local bitwise operators acting on points in and we rigorously prove that all give rise to (hyper)surfaces satisfying the self-affine property meant by Eq. (1). We then also prove that the roughness exponent behaves as in experimental physical systems, larger values of it leading to coarser surfaces Sinha ; Chiarello ; Indekeu . In Section III we illustrate these results with specific numerical examples.

Ii Digital calculus: The digit function, discrete operators and generalized bitwise arithmetic

In this article denotes the set of integers in the interval . The digit function , for , and , is the surjective mapping defined as QUANTUM ; CHAOSOLFRAC ; VGM4


and gives the -th digit of the real number (when it is non-negative) in a positional numeral system in radix . If the digit function satisfies and it does not relate to a positional numeral system. In Eq. (3) denotes the floor function (lower closest integer) of the quantity between the brackets.

With the digit function, we can express any real number as QUANTUM ; CHAOSOLFRAC ; VGM4


The integer part of is given by


Example: In the decimal radix , the number has digits , , , , , etc.

In general, a truncation to digits after the radix point is given by


We shall call , with being an integer number, the coarse graining operator.

Figure 2: The digit function plotted for (from top to bottom) in the interval . arxiv2

Most properties of the digit function are easily understood from its plot, as shown in Fig. 2 for the case and . The digit function is in all cases a periodic staircase that is set to zero each time that is an integer multiple of . The digit function possesses an important scaling property CHAOSOLFRAC that we shall use below. We have

Let be a natural number. Then, the following relationship also holds


This can be easily seen by using the definition, since we have


which proves Eq. (8).

Let us now construct the operators and mentioned in the introduction. We consider and nonnegative real numbers, for simplicity. From the digit function, Eq. (4), we have that, on one hand


and, on the other hand,


Since , we have and we can write this last expression as


Therefore, by using Eq. (8)


Thus, we see that the ordinary sum of two real numbers can be split into the bitwise addition modulo of and


plus a term which provides the total contribution of the carries


Note that if the kth-digits of and are such that there is no contribution to the carries coming from the -th digit since in that case . We thus have proved


Our goal in this paper is to find all possible bitwise operators which, as is the case of , act on two arbitrary real numbers. This goal is attained by first constructing all discrete operators that are laws of composition (magmas) of . We denote such operators by . They act on integers and , both in . The operator can be defined through a table as

where is listed in the columns and in the rows, and the result , is given by the table. Clearly, we have for any that . Since all ’s are integers this table provides the most general description of a law of composition (also called a magma) Lang ; Bruck . The integer number entering in the label of the operator is defined as


Thus, . Therefore, it is clear that, from the above table


since the -th digit of the number is . This last expression in terms of the digit function most concisely expresses any possible magma. Let now and be arbitrary real numbers. The generalized bitwise operator , with acting on these real numbers is defined as


where . This is the expression that we shall analyze in the rest of the article.

Example: Let us find what is the numerical value of . We have that 5 and 11 are given respectively in base as ’0101’ and ’1011’. Now the operator

has vector

which means that it returns one when the digit of or or both is one and zero otherwise. Then, the resulting string is and, therefore, . Thus .

Let us now also see how the operator is a particular instance of Eq. (19). We note that if we take so that


we then have, from Eq. (18)

Therefore, by taking in Eq. (19) and replacing this expression we obtain


as can be seen from Eq. (14). Thus, the addition modulo of two real numbers , is a particular instance of a generalized bitwise arithmetic operator with and with the magma having code given by Eq. (20).

We note that our expression Eq. (18) for any magma in terms of the digit function can be generalized to an arbitrary -ary operator as


where now is an integer . This operator can be used to define a generalized bitwise operator acting on real numbers , , , as


This function acts in the following way: it takes the digit of each number ,, in radix and introduces the N-ary operator given by Eq. (22) on the digits. Then, the resulting string of digits is interpreted as a number in radix (if ) or simply, in general, as a polynomial in powers of . Thus, is a non-negative real number.

We now prove one of the main results of this article.

Theorem 2.1 (Self-affinity of the generalized bitwise function.) The function defined by Eq. (23) has a self-affine property, so that


so that defines a ’roughness exponent’ of a hypersurface in dimensions, cf. Eq. (1).

Proof: We have


where we have used the scaling property, Eq. (LABEL:scal) in noting that


This proves the result and that is the roughness exponent of the self-affine surface by comparing this latter expression with Eq. (1).

The crucial property established in general by the above theorem proves that the operator in Fig. 1 has indeed a self-affine property and deserves to be called a fractal in its own right.

We also have the following result.

Theorem 2.2 (Coarse-graining theorem). Let be a N-variable discrete operator in radix acting bitwise on the digits of the reals , , , . Then, the function defined in Eq. (23) is smoothened as is increased. Furthermore, if the following expression is asymptotically satisfied


where .

Proof: Since is non-negative by definition, from Eq. (4) we have . All coefficients accompanying powers of in Eq. (19) are non-negative integers. Thus increases monotonically with . Since all coefficients are bounded (and all are independent of ) and their dependence on the coordinates , is governed by it is clear then that, as increases, terms with larger become more significant and the sum displays a longer spatial variability. As a consequence of this, details are gradually lost and coarser surfaces are obtained. Now, note also that . When is large only the most significant digit of becomes relevant and we have


Now, since from Eq. (23), for we have


and the digits of are the same as those of and, hence, belong to the set . Then, from Eq. (28) and, by using the definition of the digit function, Eq. (3) and noting that , we finally obtain


which proves the result.

The coarse graining theorem implies that asymptotically behaves like the most significant digit of for , after a trivial rescaling (i.e. dividing by ). Since all information contained in the less significant digits of is lost in , the latter exhibits no fine details compared to the former and if one takes the numbers , , , as ’coordinates’ only the broadest patches in the space spanned along these coordinates survive: Only the operations on the most significant digit of the numbers , , , become relevant.

Iii Patchwork quilts: Self-affine surfaces and the roughness exponent

Figure 3: Two views of the function in a portion of the plane with and


Figure 4: From top to bottom and left to right, the functions , , and , with and


To better substantiate these results and study particular examples, it is better to fix in Eq. (23) i.e., to work with Eq. (19), so that we can represent such operators as functions taking values on the plane given by coordinates and .

In Figs. 3 and 4 is plotted as a function of and the operator indicated in each case. Patchwork quilts are obtained in every case when projecting the 3D object on the plane (the figures just only show a finite portion of the plane). Such representations on the plane are useful to get a glimpse of the behavior of the operator within . For example that is commutative can be clearly seen from the reflection axis on the southwest-northeast diagonal exhibited by (see Fig. 4, top left). The different patches give information on how organizes in the plane. The tridimensional aspect of these surfaces, an example is shown in Fig. 4, reminds the towers of the ’exotic’ fractal surfaces reported by Indekeu et al. obtained through random deposition Indekeu5 .

The roughness exponent obtained above has some of the nice properties reported in experimental studies on fractal self-affine surfaces Sinha Chiarello Indekeu : A larger value of the exponent corresponds to a coarser (i.e. ’smoother’) surface. This is a result of the coarse graining theorem in the previous section, which we illustrate in Fig. 5 for the case (the case follows a similar trend although not that markedly). In Fig. 5, the function is plotted for the values of shown in the Figure. We observe that for increasingly large, details are gradually lost and only the coarser structures survive. In the Figure it is observed that for the asymptotic limit is already being approached. Remarkably, for prime details sometimes reemerge (but in a coarser scale). This rather subtle phenomenon poses an interesting number theoretical problem for which we have not yet a solution, and which shall be addressed elsewhere.

Figure 5: Gradual spatial coarsening of the bitwise operation function as the parameter is increased. Shown is the function for all integer values of as indicated besides each panel ( and ).

Increasing leads to coarser structures. If is not asymptotically large the structures at different scales overlap creating complex patterns that are coarser on the average. We can then ask the question whether is it possible to sharply discriminate the coarser structures without significantly changing the value of the function (beyond, of course, the change induced by neglecting the fine details). This is accomplished by the coarse graining operator acting on , i.e. . By applying this operator to Eq. (19) we obtain


The error committed in approximating by such truncation is thus strictly smaller than . Since each term accompanying a power of contributes a spatial structure (that is coarser for larger), making more negative selects the coarser structures while making the value of to slightly decrease.

Figure 6: Action of the coarse graining operator on a self-affine surface, as described by Eq. (31). Shown is the quantity for the values (left), (center) and (right) with and .

The action of the coarse graining operator, as described by Eq. (31), is shown in Fig. 6 for the function for the values (left), (center) and (right). We see that the coarse graining operator merely selects broader patches, eliminating the fine details.

The structure of the coarse graining operator is also interesting in the sense that it reveals that the operation of multipliying by an scalar, i.e. , and taking the floor operator do not generally commute: . This lack of commutativity is behind the loss of information implied by the coarse graining. In QUANTUM we have discussed the implications that, in our view, this fact has for the foundations of quantum mechanics.

Iv Conclusions

In this article a new general method to mathematically design fractal surfaces with self-affine properties has been presented. We have discovered these structures hidden within operations as simple as the ordinary sum of real numbers. Thus, we speculate that such structures may be of general interest in the mathematical modeling of complex physical systems. Our approach is based on digital calculus, which we have introduced to describe rule-based dynamical systems, as cellular automata VGM1 and substitution systems VGM4 . This approach is also the cornerstone of a recent formulation of quantum mechanics QUANTUM and of our method to derive fractal decompositions of conserved physical quantities CHAOSOLFRAC . We have rigorously proved that bitwise operators acting on real scalar fields on each point in the plane lead to surfaces with self-affine properties. We have obtained the roughness exponent for these surfaces and shown that it has the characteristic nice properties expected for such an exponent, as widely reported in experimental systems: The larger this exponent, the coarser the resulting surface Sinha ; Chiarello ; Indekeu .

The fractal surfaces here obtained come under the general name of ’patchwork quilts’ because we are following the acknowledgment made by Donald Knuth in Knuth to one pattern designed by D. Sleator in 1976 (unpublished result, displayed on p. 136 of Knuth ) resembling the ones obtained here. However, no relationship of such pattern to fractals and self-affinity seems to have ever been attempted and no systematic construction of such patterns is found in the literature. In this article, Eq. (23) constitutes a universal form embodying all conceivable bitwise operators in any radices and of standard numeral systems and any number of real scalars , the different digits representing independent bits of information at different scales.

We speculate that our surfaces can be related to free energy landscapes of clusters and biomolecules Wales obtained under the basin hopping scheme WalesDoye . An analysis of this connection shall be investigated elsewhere.


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