Partially inspired by Weierstrass’ discovery, George Cantor (1845 – 1918) published an example of an unusual ternary set in 1883 in “Grundlagen einer allgemeinen Mannigfaltigkeitslehre” (“Foundations of a General Theory of Aggregates”). It is a fractal with fractal dimension D = 2 + ln(0.5)/ln(5) = 1.5693. This continuous function is nowhere differentiable. Riemann had asked whether the functionįig. Karl Weierstrass (1815 – 1897) was studying convergence properties of infinite power series in 1872 when he began with a problem that Bernhard Riemann had given to his students some years earlier. This blog page presents the history through a set of publications that successively altered how mathematicians thought about curves in spaces, beginning with Karl Weierstrass in 1872. Here is a short history of fractal dimension, partially excerpted from my history of dynamics in Galileo Unbound (Oxford University Press, 2018) pg. From then onward the concept of dimension had to be rebuilt from the ground up, leading ultimately to fractals. Child’s play!īut how do you think of fractional dimensions? What is a fractional dimension? For that matter, what is a dimension? Even the integer dimensions began to unravel when George Cantor showed in 1877 that the line and the plane, which clearly had different “dimensionalities”, both had the same cardinality and could be put into a one-to-one correspondence. But first, we will explore the ancient and timeless appeal of theįibonacci fractals, including the universal ratio, the Golden Mean.And so on to 5 and 6 dimensions and on. We'll see more examples in the final chapter, on Fractal Applications. Powerful new tool of the fractal dimension. Why are some rivers twistier than others? How does the branchiness of given species of tree change in different environmental conditions? HowĬan you tell a healthy lung from a sick one in an MRI image? Which kind of soil will a plant grow best in? These are just a few of the countless kinds of questions that scientists can ask - and answer - using the We now have a tool that can allow us toĪsk powerful questions, and test hypotheses about the underlying nature of. Examining the fractal dimension can help shed light on the processes. Fractal patterns tell a storyĪbout the repetitive processes that created them. We can now quantitatively describe a pattern, and that allows us to study it. Previously been outside the reach of mathematics. The fractal dimension is a tool that allows us to characterize patterns and shapes in nature that have Why do we care about the fractal dimension? How is this useful? Is your computed value greater than or less than the values stated on the previous page? Use the applet above to determine the fractal dimension of the coast of Britain: One must be careful about what features of the image to include or to filter out. So the way you convert a real picture into a black and white image can affect the dimension. Gray, (and even colors!) But the dimension analyzer works with black and white images only. THe world is not all black and white, but rather filled with shades of (There are much more complicated fractal dimension analyzers that study volume-fillingįractals and use a cubic grid of 3D boxes.)Īnother thing to be careful of is the method used to prepare the images for analysis. Nonetheless, we may still be able to use this information. So whereas a real tree has a fractal dimension somewhere between 2 and 3, the flattened image of a tree can only have aĭimension between 1 and 2. Rather, we'reĪnalyzing the flattened image of a tree. There are many things to be cautious about when interpreting the results.įor one thing, remember that we're not really analyzing the fractal dimension of a tree when we plug a picture of a tree into the analyzer above. BUT - we must be careful not to place too much trust in a number just because it comes out of a computer with a fancyįractal dimension analysis program. Which of these measurements is closest to the theoretical fractal dimension of the koch curve, derived a few pages ago? (a,b,c, or d) įractal Dimensions can be very useful. Use the applet above to determine the fractal dimension of:Ī) The Koch snowflake boundary curve - level 1: ī) The Koch snowflake boundary curve - level 1: Ĭ) The Koch snowflake boundary curve - level 1: ĭ) The Koch snowflake boundary curve - level 1: The slope of the line that these points fall upon. As you click, watch the points appear on the graph, and The number on the button refers to the box size, which is the inverse of the magnification factor. Click on the buttons labelled from 128 to 1 to cover the image with a grid of the corresponding Choose different fractals from the pulldown menu. Fractal Dimension analyzer applet courtesy of Joel Castellanos.
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