$fractal dimension$

$fractal dimension$

, where is not an integer dimension, but rather a 'fractional' or 'partial'įactor the exponent out of the scale factor. Here we will carry out numeric calculations, for many different values of and Recall that the basic equation for calculating dimensionĪlthough there is no rule that dimension has to have an integer value, this has been the convention in traditional Refer to such in-between dimensions as the Hausdorff-Besicovitch dimension. Has a dimension of and a square a dimension ofĭimension related to the varying amounts of information they contain. Mathematics refer to reality, they are not certain and as far as they areĪnd Abram Besicovitch(1891-1970) revolutionized mathematics by proposingĭimensions with non-integer values. The Hausdorff-Besicovitch Dimension (top) The calculation of fractal dimension as seen in the following sections. The most common values are andĬenturies later these logarithms would be crucial in Here is a list of logarithm transformations:įigure 4.9 Table of Logarithm Transformations.Ī logarithm's base can have any positive value. Method for multiplying large numbers, along with solving division problems,Ĭalculating powers and roots. Before computers, logarithms were an invaluable The process, he showed how a close approximate product of two numbers couldīe relatively easily calculated. To their corresponding log values, adding them together, and then reversing (represented as log) to simplify multiplication. John Napier (1550-1617) originally invented logarithms Here with logarithms and with iterated function systems in Chapter 5. Mathematical equations too can be transformed. Infinite lengths and triangles within triangles.Īlong with geometrical transformations seen in scaling, Resembles the sea creature, nautilus seen in Figure 3.89 in Chapter 3. Well known spiral with infinite length is the golden mean spiral derivedįrom the ancient Greek's golden ratio see Figure 4.8. Of which the logarithmic spiral is the most famous see Figure 4.7. This idea to show that some spirals could be drawn with an infinite length, Later the mathematician Jacob Bernoulli (1654-1705) expanded Archimedes (287-212 B.C.) used spirals to illustrate repeating He also used self-similar triangles to show the congruency of triangles With infinite lengths to illustrate the concept of parallel lines, there Infinite Lengths and Scale Ability a prelude to fractals.įor example scale ability and line lengths have long been associated with Tools developed over the centuries that help us to understand fractals. Level of the table (and that is how we see him in Flatland) your would see Or all but on the level of the table and if your eye were quite on the The Tradesman, as your would see him if your eye were close to the level, Fig.ġ represents the Tradesman as you would see him while you were bending over Take for example an equilateral Triangle - who represents with usĪ Tradesman of the respectable class. The same way a Triangle, or Square, or any other figure cut out of pasteboard.Īs soon as you look at it with your eye on theĮdge on the table, you will find that it ceases to appear to you a gure,Īnd that it becomes in appearance a straight line.

$fractal dimension$

The same thing would happen if you were to treat in The penny will then have ceased to appear oval at all, and will have become, On the edge of the table (so that you are, as it were, actually a Flatlander) Of the inhabitants of Flatland), and you will find the penny becoming moreĪnd more oval to your view and at last when you have placed you eye exactly Lower your eye (thus bringing yourself more and more into the condition It will appear a circle.īut now, drawing back to the edge of the table, gradually In Space and leaning over it, look down upon it. Place a penny on the middles of one of your tables Nor could be visible, to us, except Straight Lines and the necessity of We could see nothing of the kind, not at least so a to distinguish one figure Kind but I dare say you will suppose that we could at least distinguishīy sight the Triangles, Squares, and other figures, moving about as I have Is impossible that there should be anything of what you call a 'solid' In such a country, you will perceive at once that it Alas, a few years ago, I should have said 'my universe': but now my mind has been opened to higher views Will then have a pretty correct notion of my country and countrymen. It, very much like shadows - only hard and with luminous edges - and you

$fractal dimension$

On or in the surface, but without the power of rising above or sinking below Which straight Lines, Triangles, Squares, Pentagons, Hexagons, and otherįigures, instead of remaining fixed in their places, move freely about, I call our world Flatland, not because we call it so,īut to make its nature clearer to you, my happy readers, who are privileged

$fractal dimension$

Fractal dimension