^{1}

^{*}

^{2}

^{3}

^{4}

^{1}

Euclidian geometry pertained only to the artificial realities of the first, second and third dimensions. Fractal geometry is a new branch of mathematics that proves useful in representing natural phenomena whose dimensions (fractal dimensions) are non-integer values. Fractal geometry was conceived in the 1970s, and mainly developed by Benoit Mandelbrot. In fractal geometry fractals are normally the results of an iterative or recursive construction using corresponding algorithm. Fractal analysis is a nontraditional mathematical and experimental method derived from Mandelbrot’s Fractal Geometry of Nature, Euclidean geometry and calculus. The main aims of the present study are: 1) to address the dimensional imbalances in some texts on fractal geometry, proving that logarithm of a physical quantity (e.g. length of a segment) is senseless; 2) to define the modified capacity dimension, calculate its value for Koch fractal set and show that such definition satisfies basic demands of physics, before all the dimensional balance; and 3) to calculate theoretically the fractal dimension of a circle of unit radius. A quantitative determination of the similarity using the set of Koch fractals is carried out. An important result is the relationship between the modified capacity dimension and fractal dimension obtained using the log-log method. The text includes some important modifications and advances in fractal theory. It is important to notice that these modifications and quantifications do not affect already known facts in fractal geometry and fractal analysis.

Geometry is a branch of mathematics that concerns itself with questions of shape, size, relative position of figures and properties of space. The foundation of Euclidean geometry is a group of elementary notions and axioms. In Euclidean geometry, which has been posed in modern terms by David Hilbert [

In Fractal geometry the geometrical fractal set should be considered as an infinite ordered series of geometrical objects defined on a metric space. To determine a fractal set we need specify three things [

Basic definitions and laws of fractal planimetry can be demonstrated on some classical fractal models [_{0} (

equilateral triangle whose side is one-third the length of the previous side. The result after the first iteration (the stage of construction z = 1) is shown in ^{th} stage of construction (r_{z}) and the number of segments at the same stage (N_{z}) are respectively,

Similarity is a typical property of fractal sets. To define this concept we introduce a generating element of a generator. A generator is usually made up of straight-line segments (for example, see

Fractal dimension is a quantitative measure of morphological complexity of an object. There are many different definitions of fractal dimension [

Since the logarithm is defined only for dimensionless values (numbers), not for physical quantities like 1/r (where r is a length), we define the modified capacity dimension as:

where r_{0} is a reference scale [_{0} can be the length r_{0} of the scale of an initiator of the Koch set, _{0} (Equation (1)) could be included into the definition of the capacity dimension (Equation (3)) because the ratio r_{0}/r_{z} in Equation (3) is a dimensionless quantity (a number).

If we submit Equation (1) into the last definition (Equation (3)) and put z to tend to infinity, one can see that the expression in Equation (3) gives

Fractal dimension D is the main quantifier to measure complexity of a set of geometrical and natural objects. The larger the D, the higher the complexity of the set is. For example, since the fractal dimension of the Sierpinski set is

Considering Equation (3) it would be interesting to analyze a relation between N_{z} and r_{z} of the Koch fractal set in a log-log coordinate system. If along the horizontal coordinate axis we put the values r_{z} = 1/3^{z} for _{0} is set to 1for visual clarity) and along the vertical coordinate axis the values of

with the coefficient of determination R^{2} = 1. The fitting parameter is −1.262 which is equal to the negative fractal dimension of the Koch fractal set already obtained using the formula for modified capacity dimension (Equation (3)). This procedure can be thought of as the log-log method. This method is also used in many other fractal techniques, particularly in computer-based ones.

Following the fractal methodology, we inscribe an equilateral triangle (as the initiator) in a circle of unit radius (

where r_{n} is the side length of a regular polygon inserted in the circle, n is the number of sides of the polygon and r_{2n} is the side length of the polygon with 2n sides. From this expression it is obvious that the ratio r_{2n}/r_{n} is not constant for every n (e.g., r_{6}/r_{3} = 0.577, r_{12}/r_{6} = 0.518 etc.), so that the class of such regular polygons inscribed in a circle cannot be considered as a set of prefractals. If n tends to infinity, the polygons tend to the circle, which is not a limit fractal. Therefore this set of regular polygons inscribed in a circle cannot be a fractal set.

On the other hand, if we apply the log-log method to the data of the polygonal set obtained, a straight-line graph is again obtained (

which is analogous to Equation (6). From this equation the fitting parameter −1.027 corresponds to the negative value of the fractal dimension D of the nonfractal set considered. This is the fractal dimension of a circle of unit radius (which is the limit object). Using the mass method Smith et al. [

The main properties of geometrical fractal sets are geometrical self-similarity and scale invariance. The object’s

property known as self-similarity was first coined by Mandelbrot [

Scale invariance is a feature of objects or laws that they do not change if scales of length or other variables are multiplied by a common factor. For example, this common factor for the Koch fractal set is 1/3. If r_{0} is 1 cm (

We showed that Koch set is a fractal set because it satisfies the geometrical similarity. It means that the ratio r_{z}_{+1}/r_{z} is the same for every z. If z tends to infinity, we obtained the limit fractal which cannot be drawn or imagined. Nevertheless this fractal exists because we calculated the fractal dimensions of the limit fractal using both the formula for modified capacity dimension (Equation (3)) and log-log method. In both cases we found a real and finite value D = 1.262. We conducted the same procedure using the set of polygons inscribed in a circle. Although the polygonal set is not a fractal set we calculated the fractal dimension of this limit object using the log-log method. In fact, we calculated the fractal dimension of the circle which corresponds to the limit object in which the set of polygons is inscribed. Besides, the initiator in Fractal geometry is included into the fractal iterations as a part of prefractals while the starting object (the circle) in fractal analysis does not change its integrity during iterations.

The circle is a closed line, but the log-log method can be applied to any open line (

The polygonal method needs a group of about ten printed copies of a considered line and in each of them inscribes different regular polygon (

Since the length of a polygon’s segment (

The term fractal was introduced by Mandelbrot based on his paper on self-similarity [

where the value of the exponent D seems to depend upon the shape of the border chosen. Topology fails to discriminate between different borders [

it is obvious that the factor Fr^{−D} corresponds to the number of divider’s steps. For a given r the border of smaller length has smaller number of steps (smaller F) in relation to a longer border of the same shape (the same D) which has larger number of steps (larger F).

To Richardson, the D was a fitting parameter of no particular significance. Having in mind “unearthed” Richardson’s work in which he claimed that his lines’ slopes had no theoretical interpretation [

When the physical or biological problem is stated in mathematical terms, dimensional balance should be a routine part of the solution of any problem. Exponents and logarithms must always be dimensionless [

Gabriele A.Losa,DušanRistanović,DejanRistanović,IvanZaletel,StefanoBeltraminelli, (2016) From Fractal Geometry to Fractal Analysis. Applied Mathematics,07,346-354. doi: 10.4236/am.2016.74032