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## IV Dimensions## 4.1 EuclidUp to these days we have been used to think and talk in the words of traditional Euclidean geometry[01] . But many complex objects described and composed by single Euclidean sections[02] do not really reflect the characteristic of the whole real-world object, clouds and mountains respectively do not correspond to simple geometric rules. |

## 4.1.1 Euclidean and Fractal GeometryFractal curves consist of infinite elements which are infinitely small and which are, because of that, not tangible. These infinite elements are the reason why the length increases to infinity at an infinitely small scale and by that makes it impossible to define a point of a fractal curve by co-ordinates or describe its position on the curve exactly after all[03]. That is the main difference to Euclidean geometry: looking at any point of an Euclidean curve, its position can be described by only one parameter, e.g. the distance from a starting point. Whereas if we define a point on a coastline by its distance from another specific starting point, we will see that the position of the resulting point strongly depends on the scale of map we are using for measurement. Measuring a specific distance, e.g. given in kilometers, from a starting point on a map with a scale of 1:50.000 will lead to a new point more distant from the starting position than measuring the same distance by walking along the coastline, which would mean using a scale of 1:1[04] . From this follows that a coastline has to be more than an Euclidean one-dimensional line, but it can not be a two-dimensional object either because it does not fill the entire plane. |

## 4.1.2 Different DimensionsThe concept of dimension used in school mostly deals with Euclidean geometry. In short, in an E-dimensional system of co-ordinates at least E-co-ordinates are needed for defining the position of a point. Consequently a point corresponds to a zero-dimensional system of co-ordinates, points on a line to a one-dimensional, a line on a plane to a two-dimensional and finally a plane on a cube to a three dimensional system of co-ordinates - see picture 22[05]. |

picture 22: Euclidean Geometry The point has no width, no height, no length and therefore no dimension. As well as a line in the Euclidean sense cannot be drawn exactly, because it has no thickness and is characterized by infinity to both sides, something similar is true for fractal curves. First they also have no thickness and second they are unrestricted, which means that they are of infinite length bound between two ends. The thickness in general is no problem, but also infinity does no harm, because the character and attributes of fractals can be shown after only a few iterations - e.g. only a few iterations can produce fern-like, mountain-like or cloud-like fractals. In addition to that natural fractal structures are always bound between certain scales - remember "20 meters rock"-limit at coastlines. |

The "topological dimension", however, proceeds from the fact that each structure can be reduced to a set of points. In this connection the disconnected set of points corresponds to the dimension zero. The dimension is then a rate of how many elements of the set of points are necessary for breaking the set: e.g. a line has the dimension of one because only one point has to be taken out for breaking the line into two pieces. This also means that fractal curves are still curves because intuitively the standard-arch is a connected set that can be separated - turned into an unconnected set - by taking out only one point. In the same sense the topological dimension of the Cantor Set is zero because there are not any two points, which are connected - to separate them no point has to be removed. |

## 4.2 Fractal DimensionThe following pages give a summary of the characteristics and explanations of fractal dimension[06]. Visually the fractal dimension is the expression of the degree of roughness, which means how much texture an object has[07]. It also shows how fast the length of a fractal increases from one iteration to the next. Fractal dimension is not an integer in contrast to the dimension in Euclidean geometry. The complex forms of clouds, blood vessels, coastlines or mountains seem to have an unrestricted complexity, but they nevertheless have a geometric regularity, their scale-independence. That means, if we analyze the structure on different scales, we will always find the same basic elements. Fractal dimension also expresses the connection between these different scales. There are different kinds of measurement-methods for fractal dimension, some of which will be explained on the next pages: e.g. for "true" mathematical fractals the so-called self-similarity dimension "Ds" can be measured by the increase of length from one iteration to the next. The dimension of coastlines and borders as examples of fractal curves can be measured by the structured walk-method. Finally, the box-counting method is suitable for measuring the dimension of elevations of buildings, mountains and other objects. This latter method is then used in chapter " |

## 4.3 ConclusionThe three different dimensions dealt with above - "Ds", "d", "Db" - may produce different values for the dimension of a curve. The reasons are: -) Self-similar structures on all scales are only found in truly mathematical fractals. In the journal " Long-lasting processes, called evolution, produce nature and its objects. This implies that there are many factors that influence an object in its development. The importance of these influences varies from scale to scale, which may lead to different dimensions. Combining different rules, defined as affine transformations, like scaling, translation or rotation, can produce such more natural-looking patterns. If these combinations moreover are produced at random, the effects are even more natural, which means that random is an important factor to get natural-looking fractal patterns. |

## Footnotes[01] Euclid, Greek Eukleides, was a Greek mathematician in the 4th/3rd century B.C. He was the author of "Stoicheia", "elements", which was translated into Latin in late antiquity and is the most important mathematical education manual. He taught at the Platon Academy of Alexandria. |