TRANSPORTATION DATA IN A FRACTAL DOMAIN

ABSTRACT

In following with previous papers about international trade technology and transportation, this paper will focus on the challenging issues posed by an overlay of surface topology on transportation traffic in the context of the geography of nations. But, is not only traffic and infrastructure where roads and rail exist but how these Euclidian spaces affect cost. Here is where this paper introduces the theory of applying fractal properties to distances and geographic forms and how mapping of surfaces can be simplified by combining geometric properties with fractal properties over a geographic overlay.

Introduction

One telling advantage of fractals in this context is the property to increase the diameter of a surface without increasing its area. This very property could be use to describe irregular geographic surfaces of nations that can be turned into regular polygons with known properties in Euclidian surfaces and be able to measure variables and collect them in a linear algebra matrix in loadings of: Direction, distance, volume, cost, and so on.

Topology of a surface can be illustrated and understood in the area of a sphere as stated by: http://mathcentral.uregina.ca/index.php

“…In a surface of a sphere of radius R inside a cylinder, with the cylinder just touching the equator, and cut off at the height of the top and bottom of the sphere.

The polygon area of the curved part of the cylinder is 2 Pi R x 2R = 4 Pi R². This is found by slicing the cylinder surface and rolling it out as a rectangle. It is no accident that the cylinder surface is exactly the area of the sphere”.

Surface Analysis of Special Polygons

After establishing the geometry between a sphere and a cylinder this paper will now introduce a common surface--the ellipse to introduce the theory of transportation in a fractal domain. However, this is an ellipse formed by two intersecting circles of equal diameter where geometric distances can be defined and circle centers are mutually shared.

This is the junction where the intersecting Euclidian geometry, topology and fractal geometry are mapped and the properties of this new space are used to expand the idea of transportation geography in a fractal domain. As an example to this idea we’ll use the borderline of France to illustrate its topological and fractal geometrical transformation into a Euclidian space.

This outline of the borderline is a fractal surface. It follows that if this figure is cut and stretched into a string-line to form a new figure in a Euclidian space we can then use plane geometry and its properties. An ellipsis formed by two equal circles overlapping their centers has unique properties.

            Geography of France     Area 643,801 km²        

              Coastline     3,427 km

              Borders        4,176 km 

                   Total Borderline Perimeter 7,603 km

This outline of the borderline is a fractal surface. It follows that if this figure is cut and stretched into a string-line to form a new figure in a Euclidian space we can then use plane geometry and its properties. An ellipsis formed by two equal circles overlapping their centers has unique properties.

Polygon Properties Citations

1.- The topology of two equal spaces is homomorphic. Each circumference of the elliptical surface =3801.5 km.

2.- A country’s area measured by length and width provides a major and a minor axis of distance that can be described by the resulting geometry of the intercepted circles.

3.- The effect of distance on rates is inversely proportional to freight cost because cost is spread over a shorter or longer distance within or between countries.

4.-Cross border rates change at the tangent line of two countries to obtain a between countries rate. At these crossings independent matrices of distance, commodity, handling, weight-volume and a city origin-destination can be developed to reflect freight differentials due to infrastructure and other factors.

5.-Each matrix of (data) n x m (measurements) has an SVD or PCA which can be used to determine the aggregate of the most influential data points. A cross border price is generated and related to each country’s internal economic performance such as GDP, current account balance, GDP/ Debt ratio, inflation rates, interest rates, budget balance and currency exchange.

              

Argument for the Proposed Theory

In a generic (n x m) matrix any set of (n)  items chosen with (m) measurements can be used for decomposing the matrix and calculating the SVD based bythe following equations:

C = UΣVᵀ  followed by  CᵀC = VΣᵀΣVᵀ and CV = UΣ

Where U and V are orthogonal matrices and Σ is a diagonal matrix with corresponding eigenvalues and eigenvectors.

But to state the argument in context we need to revisit the issue of fractal transformations. For instance: squares  can be used to generate fractal polygons with a similarity to the coast of France. Adding a unit square to each side respectively a square with dimension one third of the squares results from the previous iteration and both the length of the perimeter and the total area are determined by a geometric progression. Example for this case:

  





G

 This area converges to 2 while the progression for the perimeter diverges to infinity, as is the case for the Koch snowflake, where a finite area is bounded by an infinite fractal polygon.  The resulting area fills a square with the same center as the original, but twice the area, and rotated by π/4 radians, the perimeter touching but never overlapping itself.

http://mathworld.wolfram.com/KochSnowflake.html

Another iteration that can be shown with a couple lines of a programming loop in Matlab is:   =

   = [1 2 3 4 5 6];

for i = 1:length( )

       disp( (i))

end

Description of the Polygons Referenced

1.-  The circumference of circle of radius one = 3.14 which means that a unit Vesica Piscis is formed by two half circumferences of 1.57x 2= 3.14 or two half circles areas A=πr²/2.

                                                       

2.- The Sum of the sides of a square inscribed in circle of area 3.14 = a diameter of   {\displaystyle A_{n}={\frac {1}{5}}+{\frac {4}{5}}\sum _{k=0}^{n}\left({\frac {5}{9}}\right)^{k}\quad {\mbox{giving}}\quad \lim _{n\rightarrow \infty }A_{n}=2\,,}=1.41 x 4= 5.65 which is a ratio of circle to square 6.32/5.65=1.12 to infinity.

3.- Euclid defined the equilateral triangle inscribed in the Vesica Piscis using the properties of this polygon. More recent names include the Reuleaux triangles and the Euler spherical triangles which include the chords of a circle.

                                                                           

           

4.- Also defined by the two circles is a triangle with sides 2 and 3 and a hypotenuse of C²=√13. A triangle with concurrent points is also derived.

5.- The argument made is that maximum and minimum distances within a country can be defined with these polygons inside and outside the Vesica Piscis.

Topological and Fractal Geometry and the Transformation Matrix.

In this context n values used for a within commerce matrix can be defined for each country’s boundaries, distances, economy and financials. At the same time, m values can be used for measures of commodities traded, volume, price and origin-destination with a daily or any other number of n samples and m measurement for each point of the data matrix in multiple dimensions.

What the mathematical tools of SVD and PCA allow us to do is process large amounts of data in multidimensional space and to reduce its dimensionality to a more manageable space in order to discover its data pattern and correlation using the Eigen decomposition of the covariance matrix.

Here is an exercised published by MIT

 MIT Paper Singular Value Decomposition (SVD) BE.400 / 7.548

Closing Remarks

To summarize the ideas presented herein are based on a geographical boundary of fractal space which is then transformed to a Euclidian space of known quantities. An unique data matrix for any country can be used to measure international trade with as many data points and measurements as needed which is then decomposed by its SVD of (3 matrices to obtain a sigma diagonal matrix) which provides principal values in its diagonal responsible for the most variation of the variance matrix entered in the initial set up matrix. It follows that any matrix of any dimensional size can be decomposed into its SVD components.

Needless to say the SVD is a powerful mathematical tool to assess the prominent number of factors that have the most relevance in commerce within a country and of trade between countries. Furthermore, the SVD result can be used by a national government as indicator of inefficiencies in both its financial or commercial markets and take corrective action.