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Fractals and dimensions

Octopus-like fractal. Source: https://michaelheasell.com/ galleries/fmandel/mandelbrot-deep.png

Octopus-like fractal. Source: https://michaelheasell.com/ galleries/fmandel/mandelbrot-deep.png

“Clouds are not spheres, mountains are not cones, coastlines are not circles and bark is not smooth nor does lightning travel in straight line.” - B. Mandelbrot in The Fractal Geometry of Nature (1982).

Fern-like fractal. Source: https:// commons. wikimedia.org/w/ index.php?curid=8933735Fern-like fractal. Source: https:// commons. wikimedia.org/w/ index.php?curid=8933735

When we studied Euclidean Geometry at school, we made approximations that made our calculations simple and left out shapes that were too complex to study. In 1977, Benoit Mandelbrot coined the word ‘fractal’ to describe these complex shapes from the Latin word ‘fractus’, which besides meaning ‘fragmented’ also means ‘irregular’. Some fractal sets are curves or surfaces. Others are disconnected ‘dusts’ and yet others are so oddly shaped that there are no good terms for them in either the sciences or the arts. But the best way to understand what we mean by fractals is to look at some mathematically-produced fractal images which, besides striking us with their beautiful designs, can draw our attention to the similarity to Nature’s own way of making its complex shapes (see the article images).

We define fractals to be complex geometric shapes with fine structures at arbitrary small scales, usually having some degree of self-similarity. The process that produces fractal patterns is called a ‘Random Iteration Algorithm’.

Fractal dimensions

Fractal Geometry is different from Euclidean Geometry in that the objects studied not only have complex shapes but also have a ‘fractal dimension’. There are various definitions of dimension but one way of looking at it is to say that if a cube of side 1 unit has its sides doubled, we get 8 = 23 cubes of similar size and shape, so we say the dimension of the new cube is 3. We can generalize by saying that if the side length is measured by a ruler whose units are scaled down to a fraction r of 1, then we get N similar shapes of volume r3 each, so Nr3 = 1 (the volume of the unit cube). Thus, for the unit cube, we have 8 x (1/2)3 = 1. In each case, the dimension D is given by NrD = 1.

The Koch Snowflake

The following is an explanation of the Koch Snowflake construction demonstrated in one of the images.

We start with an equilateral triangle whose side lengths are 1 unit each. For each side, we insert in the middle third a kink in the shape of two sides of an equilateral triangle.

This is called the ‘generator’. We repeat this procedure indefinitely. Each step involves scaling a line (a side) by a factor of 1/3 and producing 4 copies of the ‘scaled-down’ line (2 of which are 2 sides of an equilateral triangle), that is, 4 lines also of length 1/3 unit (replacing the original line of length 1 unit). Thus, by the formula NrD = 1 defining the dimension D, we get 4(1/3)D = 1, giving D = 1.26.

Did you know?

The coastline and its ruggedness. The Koch curve can be used as a model of a coastline whose perimeter increases if we decrease the scale to measure it. The better the resolution of an aerial photograph the more indented the coast looks. If we could have unlimited resolution, would the length of the coastline be considered infinite?

• In 1961, Lewis F. Richardson found that to approximate a coastline by a broken line one needs roughly F/xD intervals of length x so that the length of the coastline is given by L(x) = xF/xD, where F and D can be determined by drawing a suitable straight line graph. Mandelbrot called the number D the dimension and found that Richardson’s equation is satisfied for the Koch curve.

• The dimension can be a measure of the ruggedness of the coast, lying between one and two. The dimension of the west coast of Malta can be worked out by using two or more maps of different scales and different values of x. The coast of Dingli Cliffs gives a dimension of 1.15.

For more trivia, see www.um.edu.mt/think.

Sound bites

• Prof. Richard Taylor is the head of the Department of Physics of the University of Oregon (UoO). He is also a painter and a photographer with an advanced art degree. When still a boy, he was fascinated by paintings of nature by Jackson Pollock. When he became a physicist at UoO, he investigated why scenes in nature evoke emotions in us and reduce stress just like music. ‘Bioinspiration’ is his main focus – exploring the favourable properties that make fractals so prevalent in nature and applying them to both art and science. He wanted to know if the fractals in the paintings of Pollock might explain why people were drawn to them as well as to things such as pulsating screen savers and music. He measured people’s skin conductance to observe the nervous system activity and found that they recovered better from stress when viewing computer images with a fractal dimension between 1.3 and 1.5. The more complex the image, the higher the dimension is. To find out what dimension induced a particular mental state, Taylor together with Caroline Hagerhall, used EEG to measure people’s brain waves while viewing geometric fractal images for one minute. Again they found that the subject’s frontal lobes easily produced the feel-good alpha brainwaves when the dimension of the images was between 1.3 and 1.5. Among other projects, Taylor is now using ‘bioinspiration’ to design better solar panels. Is there a reason why nature’s solar panels – trees and plants – are branched? The cause of our relaxation could be the images and patterns we look at, just like the music we listen to. We need nature to produce these fractal patterns. So is it wise to keep on diminishing the natural environment to substitute it with straight-lined buildings? Reference: ‘Why Fractals are so Soothing’ by Florence Williams and Aeon (January 26, 2017).

For more science news, listen to Radio Mocha on Radju Malta 2 every Saturday at 11.05am.

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