==A comparison between “rod counting” and “box counting” methods in linear fractal dimensions measurements==

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'''C.  R.  Acosta1, F.  Peñuñuri 1, O. Carvente 1, G. Rivadeneyra 1, I. Martín 1'''

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==Abstract==

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In this work, the “rod counting” method is reviewed in comparison with the “box counting” method applied to linear measurements on irregular figures. Both methods are applied in measurements of fractal dimension of some knowing fractals. Results shows that the “rod counting” method is more accurate in linear measurements. Later the method is applied in a Mandelbrot fractal figure to measure its perimeter fractal dimension.

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'''keywords'''

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Rod Counting, Box Counting, Mandelbrot fractal, Fractal dimension, ImageJ

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Facultad de Ingeniería de la Universidad Autónoma de Yucatán

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organization = Applied Physics Department, Facultad de Ingeniería de la Universidad Autónoma de Yucatán             addressline =  Av. Industrias no contaminates por anillo periférico norte S/N              city = Mérida             postcode = 97310              state = Yucatán             country = Mexico

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==1 Introduction==

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When Benoit Mandelbrot initially refers to the method of Lewis Fry Richardson <span id='citeF-1'></span><span id='citeF-2'></span><span id='citeF-3'></span>[[#cite-1|[1,2,3]]] on the measurements of the coast lines (<math display="inline">L</math>) of Britain or borders of Germany, Spain, etc., he does so using the idea of “a compass opening” or rods of the same size (<math display="inline">s</math>) in a first measurement, and that later, changes the size of the rods to measure the coastline or border again (Richardson's method).

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Then, when plotting these results wit <math display="inline">\ln \left(L\right)</math> on the vertical axis and <math display="inline">\ln \left(s\right)</math> on the horizontal axis, it is described a straight line when a linear fit of the measurements is made and the Hausdorff-Besicovitch dimension is related with the slope of the line.

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Mandelbrot also refers to the fact that the measurements made in 2-H, in the case of self-similarity, coincide with the Hausdorff-Besicovitch dimension <span id='citeF-2'></span>[[#cite-2|[2]]]. However, the comparison that Mandelbrot makes in his 1967 article <span id='citeF-1'></span>[[#cite-1|[1]]], for linear measurements he does with variants of the Koch curve and whose statistical self-similarity is the Hausdorff-Besicovitch fractal dimension.

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The fractal dimension method are used in medicine to study breast cancer, but the use of this technique is done by analysing the contour of an image generated in a mammography and periodically reviewing the evolution of the disease through the revision of the perimeter of the updated images <span id='citeF-4'></span><span id='citeF-5'></span>[[#cite-4|[4,5]]]. The same procedure is used for recognition of plants <span id='citeF-6'></span>[[#cite-6|[6]]] through the study of their leaves (analysis of the contour and veins).

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The “box counting” method has been used to measure the fractal dimension of areas as well as lengths, this consists of building a box with side ``<math display="inline">\xi _k</math>'' and counting the points that remain inside the box, the relation of powers for any dimension is (eq. [[#eq-1|1]]):

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<span id="eq-1"></span>

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>N \approx \frac{L}{\xi _k} \;\; {;} \;\; N \approx \frac{A}{\xi _k^2} \;\; {;} \;\; N \approx \frac{V}{\xi _k^3} \;\; \Rightarrow \;\; N  \propto \frac{1}{\xi _k^D} </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (1)

|}

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where <math display="inline">L</math> is the perimeter length <math display="inline">\xi _k</math> is the scale, <math display="inline">A</math> is the area, <math display="inline">V</math> is the volume, <math display="inline">N</math> is the number of “rods”, “boxes” or “cubes” used, and “D” is the Hausdorff-Besicovitch dimension.

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Applying the relationship only for longitudinal measurements, let us compare the results given by the measurement made using boxes (Box Counting) with that made through rods (Rod Counting). The measurements for the counting of boxes will be carried out with the “ImageJ” application <span id='citeF-7'></span>[[#cite-7|[7]]] that is part of a “SciJava OSS” free software project using the “FracLac” plugin and the counting of rods with the “GeoGebra” app <span id='citeF-8'></span>[[#cite-8|[8]]].

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The box counting technique has been reviewed by some authors and an optimization algorithm has been proposed <span id='citeF-9'></span>[[#cite-9|[9]]], however the results obtained when applying the procedure are close to the rod counting method, but the latter being simpler in its implementation.

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==2 Calculation of the fractal dimension with the Richardson method.==

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To carry out this comparison we will use the Koch curve, whose Hausdorff-Besicovitch fractal dimension is known (<math display="inline">D=1.26186</math>), <span id='citeF-10'></span><span id='citeF-11'></span>[[#cite-10|[10,11]]]. Figure ([[#img-1|1]]) shows five moments of construction of the Koch curve, which was generated in GeoGebra <span id='citeF-8'></span>[[#cite-8|[8]]].

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<div id='img-1'></div>

{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"

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|[[Image:Review_986628909362-Fig1-Chaos.png|364px|Koch curve generated in GeoGebra.]]

|- style="text-align: center; font-size: 75%;"

| colspan="1" | '''Figure 1:''' Koch curve generated in GeoGebra.

|}

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Arranging these construction moments in table [[#table-1|1]] following the methodology proposed by Richardson, we obtain the straight line whose slope is the Hausdorff-Besicovitch fractal dimension <span id='citeF-1'></span>[[#cite-1|[1]]], <span id='citeF-12'></span>[[#cite-12|[12]]]-<span id='citeF-18'></span>[[#cite-18|[18]]].

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{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"

|+ style="font-size: 75%;" |<span id='table-1'></span>Table. 1 Richardson's method used to calculate the statistical self-similar fractal dimension of Koch's curve.

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  L. Rod (<math display="inline">a_k</math>) 

| style="border-left: 2px solid;border-right: 2px solid;" | Scale (<math display="inline">\xi _k = a_k/a_1</math>) 

| style="border-left: 2px solid;border-right: 2px solid;" | N. Rods <math display="inline">N\left(\xi _k\right)</math> 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_k\left(\ln \left(1/\xi _k\right), \ln \left(N\right)\right)</math>

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  3 

| style="border-left: 2px solid;border-right: 2px solid;" | 1 

| style="border-left: 2px solid;border-right: 2px solid;" | 4 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_1\left(0, 1.38629\right)</math>

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  1 

| style="border-left: 2px solid;border-right: 2px solid;" | 1/3 

| style="border-left: 2px solid;border-right: 2px solid;" | 16 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_2\left(1.09861, 2.77259\right)</math>

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  1/3 

| style="border-left: 2px solid;border-right: 2px solid;" | 1/9 

| style="border-left: 2px solid;border-right: 2px solid;" | 64 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_3\left(2.19732, 4.15888\right)</math>

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  1/9 

| style="border-left: 2px solid;border-right: 2px solid;" | 1/27 

| style="border-left: 2px solid;border-right: 2px solid;" | 256 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_4\left(3.2954, 5.54518\right)</math>

|- style="border-top: 2px solid;border-bottom: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  1/27 

| style="border-left: 2px solid;border-right: 2px solid;" | 1/81 

| style="border-left: 2px solid;border-right: 2px solid;" | 1024 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_5\left(4.39445, 6.93147\right)</math>

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|}

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Linear fit of the data in the last column of table [[#table-1|1]] leads us to the linear function (eq. [[#eq-2|2]]):

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<span id="eq-2"></span>

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>f\left(x\right)=1.26186x+1.38629 </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (2)

|}

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Where it is observed that the slope of this function is the fractal dimension calculated based on the Hausdorff-Besicovitch description for Koch's curve, but obtained with Richardson's procedure (fig. [[#img-2|2]]).

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<div id='img-2'></div>

{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"

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|[[Image:Review_986628909362-Fig2-Chaos.png|215px|Fractal dimension of Koch's curve measured following Richardson's method with the “rod” sizes equals to the original size used in every step of construction.]]

|- style="text-align: center; font-size: 75%;"

| colspan="1" | '''Figure 2:''' Fractal dimension of Koch's curve measured following Richardson's method with the “rod” sizes equals to the original size used in every step of construction.

|}

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Geometric fractals of a) Hilbert's curve, b) Koch 85's curve and c) the Cantor middle-thirds set were also generated in GeoGebra <span id='citeF-8'></span>[[#cite-8|[8]]], whose fractal dimensions were calculated with the rod counting method, which are shown in table [[#table-2|2]]. These figures were generated and measured since their fractal dimension are calculated with the perimeter or contour procedure. There are other fractal figures that can be generated, such as the Sierpinsky's triangle and carpet and some others, but since areas are generated in these elements, it is more convenient to use the box counting method there.

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{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"

|+ style="font-size: 75%;" |<span id='table-2'></span>Table. 2 Richardson's method used to calculate the statistical self-similar fractal dimension of a) Hilbrts's curve, b) Koch 85's curve and c) Cantor middle-thirds set.

|- style="border-top: 2px solid;"

| rowspan='10' style="border-left: 2px solid;border-right: 2px solid;" |

{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"

|-

|[[Image:Review_986628909362-Hilbert.png|100px|]]

|}

[[Image:Review_986628909362-Hilbert.png|300px|Hilbert.png]]

| colspan='4' style="border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | a) Hilbert's curve fractal <math>h\left(x\right)= 2.041x+1.2403</math>.

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| style="border-left: 2px solid;border-right: 2px solid;" | L. Rod (<math display="inline">a_k</math>) 

| style="border-left: 2px solid;border-right: 2px solid;" | Scale (<math display="inline">\xi _k = a_k/a_1</math>) 

| style="border-left: 2px solid;border-right: 2px solid;" | N. Rods <math display="inline">N\left(\xi _k\right)</math> 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_k\left(\ln \left(1/\xi _k\right), \ln \left(N\right)\right)</math>

|-

| style="border-left: 2px solid;border-right: 2px solid;" | 4 

| style="border-left: 2px solid;border-right: 2px solid;" | 1 

| style="border-left: 2px solid;border-right: 2px solid;" | 3 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_1\left(0, 1.0986\right)</math>

|-

| style="border-left: 2px solid;border-right: 2px solid;" | 2 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.5 

| style="border-left: 2px solid;border-right: 2px solid;" | 15 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_2\left(0.6931, 2.7081\right)</math>

|-

| style="border-left: 2px solid;border-right: 2px solid;" | 1 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.25 

| style="border-left: 2px solid;border-right: 2px solid;" | 63 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_3\left(1.3863, 4.1431\right)</math>

|-

| style="border-left: 2px solid;border-right: 2px solid;" | 0.5 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.125 

| style="border-left: 2px solid;border-right: 2px solid;" | 255 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_4\left(2.0794, 5.5413\right)</math>

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| style="border-left: 2px solid;border-right: 2px solid;" | 0.25 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.625 

| style="border-left: 2px solid;border-right: 2px solid;" | 1023 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_5\left(2.7726, 6.9305\right)</math>

|-

| style="border-left: 2px solid;border-right: 2px solid;" | 0.125 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.0313 

| style="border-left: 2px solid;border-right: 2px solid;" | 4095 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_6\left(3.4657, 8.3175\right)</math>

|-

| style="border-left: 2px solid;border-right: 2px solid;" | 0.0625 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.0156 

| style="border-left: 2px solid;border-right: 2px solid;" | 16383 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_7\left(4.1589, 9.7040\right)</math>

|-

| style="border-left: 2px solid;border-right: 2px solid;" | 0.0313 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.0078 

| style="border-left: 2px solid;border-right: 2px solid;" | 65535 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_8\left(4.8520, 11.0903\right)</math>

|- style="border-top: 2px solid;"

| rowspan='8' style="border-left: 2px solid;border-right: 2px solid;" |

{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"

|-

|[[Image:Review_986628909362-Koch85.png|100px|]]

|}

[[Image:Review_986628909362-Koch85.png|300px|Koch85.png]]

| colspan='4' style="border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | b) Koch 85's curve fractal <math>m\left(x\right)=1.785x+1.386 </math>.

|-

| style="border-left: 2px solid;border-right: 2px solid;" | 4.1392 

| style="border-left: 2px solid;border-right: 2px solid;" | 1 

| style="border-left: 2px solid;border-right: 2px solid;" | 4 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_1\left(0, 1.3863\right)</math>

|-

| style="border-left: 2px solid;border-right: 2px solid;" | 1.9037 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.4599 

| style="border-left: 2px solid;border-right: 2px solid;" | 16 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_2\left(0.7767, 2.7726\right)</math>

|-

| style="border-left: 2px solid;border-right: 2px solid;" | 0.8755 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.2115 

| style="border-left: 2px solid;border-right: 2px solid;" | 64 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_3\left(1.5535, 4.1589\right)</math>

|-

| style="border-left: 2px solid;border-right: 2px solid;" | 0.4027 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.0973 

| style="border-left: 2px solid;border-right: 2px solid;" | 256 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_4\left(2.3301, 5.5452\right)</math>

|-

| style="border-left: 2px solid;border-right: 2px solid;" | 0.1852 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.0447 

| style="border-left: 2px solid;border-right: 2px solid;" | 1024 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_5\left(3.1068, 6.9315\right)</math>

|-

| style="border-left: 2px solid;border-right: 2px solid;" | 0.0852 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.0206 

| style="border-left: 2px solid;border-right: 2px solid;" | 4096 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_6\left(3.8833, 8.3178\right)</math>

|-

| style="border-left: 2px solid;border-right: 2px solid;" | 0.0392 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.0095 

| style="border-left: 2px solid;border-right: 2px solid;" | 16384 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_7\left(4.6596, 9.7041\right)</math>

|- style="border-top: 2px solid;"

| rowspan='6' style="border-bottom: 2px solid;border-left: 2px solid;border-right: 2px solid;" |

{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"

|-

|[[Image:Review_986628909362-CantorSet.png|100px|]]

|}

[[Image:Review_986628909362-CantorSet.png|300px|CantorSet.png]]

| colspan='4' style="border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | c) Cantor middle-thirds set fractal <math>n\left(x\right)=0.6309x+0.6931</math>.

|-

| style="border-left: 2px solid;border-right: 2px solid;" | 0.3333 

| style="border-left: 2px solid;border-right: 2px solid;" | 1 

| style="border-left: 2px solid;border-right: 2px solid;" | 2 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_1\left(0, 0.6931\right)</math>

|-

| style="border-left: 2px solid;border-right: 2px solid;" | 0.1111 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.3333 

| style="border-left: 2px solid;border-right: 2px solid;" | 4 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_2\left(1.0986, 1.3863\right)</math>

|-

| style="border-left: 2px solid;border-right: 2px solid;" | 0.037 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.1111 

| style="border-left: 2px solid;border-right: 2px solid;" | 8 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_3\left(2.1972, 2.0794\right)</math>

|-

| style="border-left: 2px solid;border-right: 2px solid;" | 0.0123 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.037 

| style="border-left: 2px solid;border-right: 2px solid;" | 16 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_4\left(3.2958, 2.7726\right)</math>

|- style="border-bottom: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" | 0.0041 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.0123 

| style="border-left: 2px solid;border-right: 2px solid;" | 32 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_5\left(4.3944, 3.4657\right)</math>

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|}

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==3 Application of the box counting method.==

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Let's calculate the fractal dimension again for figure ([[#img-1|1]]e), but using the “ImageJ” application <span id='citeF-7'></span>[[#cite-7|[7]]] through the “FracLac” plugin. For this was generated an image in jpg, png or bmp format with only the Koch's curve, to be able to carry out the procedure in the aforementioned software. Figure ([[#img-3|3]]) shows the result obtained <span id='citeF-4'></span>[[#cite-4|[4]]]-<span id='citeF-12'></span>[[#cite-12|[12]]].

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<div id='img-3'></div>

{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"

|-

|[[Image:Review_986628909362-Fig3-Chaos.png|420px|Fractal dimension D<sub>B</sub>= 1.2848 calculated using the “FracLac” plugin of “ImageJ” app.]]

|- style="text-align: center; font-size: 75%;"

| colspan="1" | '''Figure 3:''' Fractal dimension <math>D_B = 1.2848</math> calculated using the “FracLac” plugin of “ImageJ” app.

|}

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The image shown has different sizes of the “boxes” and in the smaller one (fig. [[#img-3|3]]f) the fractal dimension that was obtained and that shows a difference with the one calculated directly from figure ([[#img-1|1]]e) with the procedure carried out in table [[#table-1|1]], this difference of the fractal dimensions is <math display="inline">\Delta D_B = 0.02294</math>.

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==4 Application of the counting method with non-ideal rod sizes.==

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If it is carry out the same procedure of “counting rods”, but with dimensions (scales) of each rod different from those known for its construction, this is to have a generalized method applicable to any figure, the following measurements shown in the figure ([[#img-4|4]]) and calculations are made in the table [[#table-3|3]].

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In figures ([[#img-4|4]]e) and ([[#img-4|4]]f) approximations of the fit were taken for rod sizes of 0.1 and 0.05 respectively and thus show that this is not exact, given the differences between the magnitudes of the sides of the “ideal”  Koch's curve and the magnitude of the rod used.

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<div id='img-4'></div>

{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"

|-

|[[Image:Review_986628909362-Fig4-Chaos.png|340px|Rod sizes used were 0.5 in a), 0.4 in b), 0.3 in c), 0.2 in d), 0.1 in e) and 0.05 in f).]]

|- style="text-align: center; font-size: 75%;"

| colspan="1" | '''Figure 4:''' Rod sizes used were 0.5 in a), 0.4 in b), 0.3 in c), 0.2 in d), 0.1 in e) and 0.05 in f).

|}

​

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{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"

|+ style="font-size: 75%;" |<span id='table-3'></span>Table. 3 The <math>P_i</math> points are obtained with rod lengths different from those used in the construction of the Koch curve.

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  L. Rod (<math display="inline">a_k</math>) 

| style="border-left: 2px solid;border-right: 2px solid;" | Scale (<math display="inline">\xi _k = a_k/a_1</math>) 

| style="border-left: 2px solid;border-right: 2px solid;" | N. Rods <math display="inline">N\left(\xi _k\right)</math> 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_k\left(\ln \left(1/\xi _k\right), \ln \left(N\right)\right)</math>

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  0.5 

| style="border-left: 2px solid;border-right: 2px solid;" | 1 

| style="border-left: 2px solid;border-right: 2px solid;" | 30 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_1\left(0, 3.4012\right)</math>

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  0.4 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.8 

| style="border-left: 2px solid;border-right: 2px solid;" | 37 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_2\left(0.22314, 3.61092\right)</math>

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  0.3 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.6 

| style="border-left: 2px solid;border-right: 2px solid;" | 64 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_3\left(0.51083, 4.15888\right)</math>

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  0.2 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.4 

| style="border-left: 2px solid;border-right: 2px solid;" | 128 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_4\left(0.91629, 4.85203\right)</math>

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  0.1 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.2 

| style="border-left: 2px solid;border-right: 2px solid;" | 256 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_5\left(1.60944, 5.54518\right)</math>

|- style="border-top: 2px solid;border-bottom: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  0.05 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.1 

| style="border-left: 2px solid;border-right: 2px solid;" | 515 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_6\left(2.30259, 6.24417\right)</math>

​

|}

​

The linear fit for the set of points <math display="inline">P=\{ P_1,P_2,P_3,P_4,P_5,P_6 \} </math> of Table [[#table-3|3]] which was obtained from figure ([[#img-4|4]]) is given by equation (eq. [[#eq-3|3]]) with a fractal dimension of <math display="inline">D_B = 1.25966</math>.

​

<span id="eq-3"></span>

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>g\left(x\right)=1.25966x+3.46763 </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (3)

|}

​

This measurement generates a difference with respect to the original measurement of <math display="inline">\Delta D_B = 0.0022</math>, that is, an order of magnitude with respect to the difference that exists between the “box counting” and the original measurement. This shows that the “rod counting” procedure is a better approximation than the “box counting” in longitudinal measurements of the fractal dimension. Figure ([[#img-4|4]]) shows approaches to the six moments of construction of the adjustment to the Koch curve.

​

It can also be seen that even when both equations (eq. [[#eq-2|2]], eq. [[#eq-3|3]]) do not have the same slopes, those straight lines obtained are “relatively close” to parallelism.

​

==5 Application of the rod counting method to the Mandelbrot fractal.==

​

Using the described method of counting rods on a Mandelbrot fractal generated in GeoGebra and whose profile with the different sizes of rods used is shown in figure ([[#img-5|5]]) and the counting results are described in table [[#table-4|4]].

​

With the results of table [[#table-4|4]], the linear approximation is made, following the Richardson method, which gives us the function of the straight line that best fits the set of points and which is (eq. [[#eq-4|4]]):

​

<span id="eq-4"></span>

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>h\left(x\right)=1.15397x+3.49872 \;\; \Rightarrow \;\; D_{Br} = 1.15397 </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (4)

|}

​

​

{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"

|+ style="font-size: 75%;" |<span id='table-4'></span>Table. 4 Results of rod counting measurements on a Mandelbrot fractal.

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  L. Rod (<math display="inline">a_k</math>) 

| style="border-left: 2px solid;border-right: 2px solid;" | Scale (<math display="inline">\xi _k = a_k/a_1</math>) 

| style="border-left: 2px solid;border-right: 2px solid;" | N. Rods <math display="inline">N\left(\xi _k\right)</math> 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_k\left(\ln \left(1/\xi _k\right), \ln \left(N\right)\right)</math>

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  0.2 

| style="border-left: 2px solid;border-right: 2px solid;" | 1 

| style="border-left: 2px solid;border-right: 2px solid;" | 33 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_1\left(0, 3.49651\right)</math>

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  0.15 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.8 

| style="border-left: 2px solid;border-right: 2px solid;" | 47 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_2\left(0.28768, 3.85015\right)</math>

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  0.1 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.6 

| style="border-left: 2px solid;border-right: 2px solid;" | 71 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_3\left(0.69315, 4.26268\right)</math>

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  0.08 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.4 

| style="border-left: 2px solid;border-right: 2px solid;" | 97 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_4\left(0.91629, 4.57471\right)</math>

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  0.06 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.2 

| style="border-left: 2px solid;border-right: 2px solid;" | 132 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_5\left(1.20397, 4.88280\right)</math>

|- style="border-top: 2px solid;border-bottom: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  0.04 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.1 

| style="border-left: 2px solid;border-right: 2px solid;" | 213 

| style="border-left: 2px solid;border-right: 2px solid;" | <math>P_5\left(1.60944, 5.36129\right)</math>

​

|}

​

<div id='img-5'></div>

{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"

|-

|[[Image:Review_986628909362-Fig5-Chaos.png|357px|The different sizes of rods used to enclose the Mandelbrot fractal are shown, in a perimeter measurement using the rod counting method. The different lengths of the rods were (in cm): a) 0.2, b) 0.15, c) 0.1, d) 0.08, d) 0.06 and e) 0.04.]]

|- style="text-align: center; font-size: 75%;"

| colspan="1" | '''Figure 5:''' The different sizes of rods used to enclose the Mandelbrot fractal are shown, in a perimeter measurement using the rod counting method. The different lengths of the rods were (in cm): a) 0.2, b) 0.15, c) 0.1, d) 0.08, d) 0.06 and e) 0.04.

|}

​

For this same fractal, the box counting procedure is used, in the FracLac plugin of the ImageJ application <span id='citeF-4'></span>[[#cite-4|[4]]], whose result is shown in figure ([[#img-6|6]]) with a (slope of the straight line), which, as can be seen, has a difference with respect to the calculated with the counting of rods.

​

<div id='img-6'></div>

{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"

|-

|[[Image:Review_986628909362-Fig6-Chaos.png|197px|Results of processing perimeter of Mandelbrot's fractal with the box counting method, in the “FracLac” plugin of “ImageJ” app, D<sub>Bb</sub> = 1.1965]]

|- style="text-align: center; font-size: 75%;"

| colspan="1" | '''Figure 6:''' Results of processing perimeter of Mandelbrot's fractal with the box counting method, in the “FracLac” plugin of “ImageJ” app, <math>D_{Bb} = 1.1965</math>

|}

​

==6 Discussion about rod counting method.==

​

In table [[#table-5|5]] results of calculation by rod counting and box counting are shown and compared with the the theoretical calculations reported <span id='citeF-10'></span>[[#cite-10|[10]]]-<span id='citeF-12'></span>[[#cite-12|[12]]], also results obtained by <span id='citeF-9'></span>[[#cite-9|[9]]] are shown in the last column of table [[#table-5|5]] for comparison. The penultimate row of the table [[#table-5|5]] refers to the calculated Koch's snowflake fractal with non-regular sizes.

​

​

{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"

|+ style="font-size: 75%;" |<span id='table-5'></span>Table. 5 Comparison of measurements made on different fractal figures using different procedures. The last column takes into a count results given by <span id='citeF-9'></span>[[#cite-9|[9]]] .

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  Geometry 

| style="border-left: 2px solid;border-right: 2px solid;" | Theoretical 

| style="border-left: 2px solid;border-right: 2px solid;" | Rod C. 

| style="border-left: 2px solid;border-right: 2px solid;" | B. Count. 

| style="border-left: 2px solid;border-right: 2px solid;" | E. B. Count. 

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  Koch Snowflake 

| style="border-left: 2px solid;border-right: 2px solid;" | 1.2620 

| style="border-left: 2px solid;border-right: 2px solid;" | 1.26186 

| style="border-left: 2px solid;border-right: 2px solid;" | 1.2848 

| style="border-left: 2px solid;border-right: 2px solid;" | 1.267 

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  Hilbert´s curve 

| style="border-left: 2px solid;border-right: 2px solid;" | 2.0000 

| style="border-left: 2px solid;border-right: 2px solid;" | 2.04097 

| style="border-left: 2px solid;border-right: 2px solid;" | 1.8984 

| style="border-left: 2px solid;border-right: 2px solid;" | 1.974 

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  Koch 85' curve 

| style="border-left: 2px solid;border-right: 2px solid;" | 1.7850 

| style="border-left: 2px solid;border-right: 2px solid;" | 1.78505 

| style="border-left: 2px solid;border-right: 2px solid;" | 1.7032 

| style="border-left: 2px solid;border-right: 2px solid;" | 1.788 

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  Cantor Set 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.6310 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.63093 

| style="border-left: 2px solid;border-right: 2px solid;" | 0.8082 

| style="border-left: 2px solid;border-right: 2px solid;" | - - 

|- style="border-top: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  Koch NIS 

| style="border-left: 2px solid;border-right: 2px solid;" | 1.2620 

| style="border-left: 2px solid;border-right: 2px solid;" | 1.25966 

| style="border-left: 2px solid;border-right: 2px solid;" | 1.2848 

| style="border-left: 2px solid;border-right: 2px solid;" | 1.267 

|- style="border-top: 2px solid;border-bottom: 2px solid;"

| style="border-left: 2px solid;border-right: 2px solid;" |  Mandelbrot 

| style="border-left: 2px solid;border-right: 2px solid;" | - - 

| style="border-left: 2px solid;border-right: 2px solid;" | 1.15397 

| style="border-left: 2px solid;border-right: 2px solid;" | 1.1965 

| style="border-left: 2px solid;border-right: 2px solid;" | - - 

​

|}

​

The application of the “box counting” method implies the measurement of the area of an object, however this same procedure, without any change, is used in the measurement of lengths, but there are differences in the results obtained, as shown in table [[#table-5|5]], when compared with the theoretical results.

​

In column 4 of table [[#table-5|5]] an area method is used and even when this area collapse to a line, this procedure shows a numerical difference with respect to a perimeter method.

​

Based on these differences, it is proposed to use the “rod counting” for longitudinal measurements, the “box counting” method is used to measure areas, as well as for the measurement of 3D distributions, the “cube counting” is used.

​

Taking into account the existing software, the ImageJ application gives us the possibility to efficiently measure the fractal dimension of areas, but it does not have a plugin to perform linear measurements. This is why GeoGebra, even though it is not designed for that purpose, has tools that enable the application of the “rod counting” method in an orderly manner.

​

==7 Conclusions.==

​

* It was possible to show that there is a numerically evaluable difference between the counting of boxes and the counting of rods for linear (perimeter) measurements.

​

* The method of counting rods and boxes is applied to the fractal of the Koch curve and whose differences with respect to the Hausdorff-Besicovitch result give us <math display="inline">\Delta D_{Bb} = 0.02294</math> and <math display="inline">\Delta D_{Br} = 0.0022</math>, where it refers to the counting of boxes and rods respectively.

​

* Both measurement methods are applied to the Mandelbrot fractal, in a perimeter measurement, and it is established that <math display="inline">D_{Bb} = 1.1965</math> (box count), and that <math display="inline">D_{Br} = 1.15397</math> (rod count) which is considered as the correct one, because it is a linear measurement.

​

Therefore, it is established that in the linear measurements made on fractals, the counting of rods provides a better approximation to the result of applying the Hausdorff-Besicovitch definition than those obtained with the counting of boxes, even with “non-standardized” rod sizes.

​

Thus, if the objective is to have a better description of the surrounding reality, the method used to describe the fractal geometry that is generated must be in accordance with the measurement made, to measure lines, the counting of rods, to measure areas, the counting of boxes and to measure volumes, the counting of cubes.

​

===BIBLIOGRAPHY===

​

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​