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Fractals: Infinitely Complex Patterns in Nature and Beyond
- 1. FRACTALS Ciara O’Keeffe Stephen O’Riordan AM3064 Presentation
- 2. OVERVIEW • Introduction • Examples of Fractals • Cantor Set • Mandelbrot Set • Sierpinksi Triangle • Koch Snowflake • Hausdorff Dimension • Relation to Chaos Theory • Area filling Fractals • Applications of Fractals • How to measure the length of a coastline • Image compression • Fractals and Stocks • Use in creative media
- 3. AN INTRODUCTION What are Fractals? Examples in Nature Formal Definition
- 4. WHAT ARE FRACTALS • Infinitely complex pattern • Each part has the same statistical character as the whole. • They are useful in modelling things that have repeating patterns • Snowflakes • Brownian Motion • Biological systems • Describe chaotic phenomena • Crystal growth • Galaxy Formation
- 6. FORMAL DEFINITION OF A FRACTAL • The fractal dimension needs to exceed the topological dimension • Topological dimension - coordinates needed to describe a point • Fractal dimension - a measure of roughness • Mandelbrot 1975
- 7. DIFFERENT TYPES OF FRACTALS Cantor Set Mandelbrot Set Julia Sets Sierpinksi Triangle Koch Snowflake
- 8. CANTOR SET • Late 1800’s • Infinite segments 𝑁𝑛 = 2𝑛 𝑁 = lim 𝑛 → ∞ 𝑁𝑛 = ∞ • Zero Length 𝐿𝑛 = 2 3 𝑛 𝐿 = lim 𝑛→ ∞ 𝐿𝑛 = 0
- 9. HAUSDORFF DIMENSION • One example of a fractal dimension. • Also called the similarity dimension • If after each step a fractal makes 𝑚 copies of itself scaled down by a factor of 𝑟 then we have: 𝑚 = 𝑟𝑑 𝑑 = ln 𝑚 ln 𝑟
- 10. CANTOR SET • 𝑚 = 2 copies of the previous step • Scaled down by a factor of 𝑟 = 3 so 𝑑 = ln 2 ln 3 ≈ 0.63
- 11. MANDELBROT SET • Created from the logistics function: 𝑓 𝑧 = 𝑧2 + 𝑐 • The Mandelbrot set is the values for c for which this function does not diverge (after infinite steps) starting from 𝑧 = 0. • Not perfectly self-similar • Hausdorff Dimension of boundary 𝑑 = 2
- 12. SECTIONS OF THE MANDELBROT SET • Different areas have different properties • Main Cardioid body – set of values c for which map has attracting fixed point • Period Bulb – values c for attracting period 2 cycle • Smaller bulb- larger cycle
- 13. JULIA SETS • We defined the Mandelbrot by the logistics function. This is one (very famous) example of a Julia Set • The general definition is any complex, non-constant function f(z) that is holomorphic and maps the Riemann sphere onto itself. • This turns out to be all nonconstant complex rational functions such that: 𝑓 𝑧 = 𝑝 𝑧 𝑞(𝑧) where p(z) and q(z) are complex polynomials.
- 14. FUN JULIA SETS 𝑓 𝑧 = 𝑧2 + 𝑐 • We know certain values c that can generate interesting graphs. −0.42 + 0.6𝑖 0.355 + 0.355𝑖 0.37 + 0.1𝑖 −0.54 + 0.54𝑖 • Obviously pictures are a little more engaging • We’ll show you where you can generate your own later
- 15. −0.54 + 0.54𝑖 −0.42 + 0.6𝑖 0.355 + 0.355𝑖 0.37 + 0.1𝑖
- 16. SIERPINSKI TRIANGLE • Zero area 𝐴𝑛 = 3 4 𝑛 𝐴 = lim 𝑛 → ∞ 𝐴𝑛 = 0 • 3 copies of the original, scaled down by 2 𝑑 = ln 3 ln 2 ≈ 1.58 Waclaw Sierpinksi 1915
- 17. SIERPINSKI TRIANGLE • Used in Antenna for internet/frequencies • Compatible with many frequencies • Compact size
- 18. KOCH SNOWFLAKE • Finite Area – fits inside a circle 𝐴 = 8 5 𝑎0 • Same pattern on all sides - Actually 3 fractals - Focus on one side only
- 19. KOCH SNOWFLAKE • Infinite Length 𝐿𝑛 = 4 3 𝑛 𝐿 = lim 𝑛→∞ 𝐿𝑛 = ∞ • Look at one side – contains 4 copies scaled by 3 𝑑 = ln 𝑚 ln 𝑟 ≈ 1.26
- 20. RELATION TO CHAOS THEORY • Mandelbrot’s book - “Fractals and Chaos the Mandelbrot Set and Beyond” • Mandelbrot set is linked to logistics function. • The areas “coloured-in” represent stable periods. The end of the cardioid is the split from one periodic orbit to two. • Chaos is present along the x-axis afterwards.
- 22. PEANO CURVE • The first area filling curves • “Found” by Giuseppe Peano in 1890
- 23. HILBERT CURVE • 1 Year later • Simplest area fitting curve • Maps every point in 2D space to a line • Benefit is the points converge to one part
- 24. WHY ARE AREA FILLING CURVES IMPORTANT? Can express N-D space in 1-D space. One line that passes through every point Can be described by a function such that 𝑓: 𝑅 → 𝑅𝑁
- 25. APPLICATIONS OF FRACTALS Measuring the Coastline Image Compression Fractals & Stocks Creative Works
- 26. MEASURING THE LENGTH OF A COASTLINE
- 27. HOW TO MEASURE A COASTLINE • Should be easy? • Length given as: • 3,171km from the OSI • 6,226km from Heritage Council • 6,347km from US Defense Mapping Agency • Who is right? • Depends on the line you draw
- 28. COASTLINE OF BRITAIN • Smaller the line the more accurate the picture • But no part is perfectly straight. How small do you go? • This is essentially a fractal
- 29. BOX- COUNTING DIMENSION • Also called Minkowski-Bouligand dimension. • Defined by: dim𝑏𝑜𝑥(𝑆) ≡ lim 𝜖→0 log 𝑁(𝜖) log 1 𝜖 • Length given by: 𝐿 = 𝑁 ∗ 𝜖𝑑 • Taking Natural log: ln 𝐿 = 𝑑 ln 𝜖 + ln(𝑁)
- 30. DIFFERENT METHODS • There are a few different ways • Manually • Python • We’ll start with the manual method. Programmes used were: • Google Earth • GIMP • Mouse Clickr
- 32. RESULTS
- 34. VALUE OF THE DIMENSION • This gives us a slope of m = -0.21 ± 0.06. Our dimension is then: • d = 1 - m = 1.21 ± 0.06 • Convert to grey image and then count the boxes by the difference in two pixels. This becomes dependent on the threshold you set.
- 35. RESULT OF GREY-SCALE METHOD • What we found was the result of our dimension was entirely based on the threshold we set. We plotted the result against the threshold and obtained this graph. • Whilst this is interesting data, it shows that we can’t be reliant on this image for getting the dimension.
- 36. BASEMAP MODULE • Allows you to plot areas of a map. This is what we obtained with a Mercator projection. • Useful because it’s already black and white. Threshold shouldn’t matter anymore.
- 37. BASEMAP MODULE • If we plot the same thing as before we get: • Value obtained is 1.35 • Hutzler, S. (2013). Fractal Ireland. Science Spin, 58, 19-20 • McCartney M., Abernethy G., and Gault L. (2010). The Divider Dimension of the Irish Coast. Irish Geography, 43, 277-284.
- 38. WHAT DOES THIS ACTUALLY MEAN? The dimension is a measure of how rough a coastline is.
- 40. IMAGE COMPRESSION • A way of reducing file size of digital images • Lossy v Lossless • Fractal image compression involves splitting an image into grids • Create a function that goes from compression to final image • Associate small grids in the range with larger grids in the domain
- 43. FRACTALS AND THE STOCK MARKET • Efficient Market Hypothesis (EMH) • Says that stocks always trade at their fair value on exchanges • Flaw is it doesn’t predict anomalies • Fractal Market Hypothesis (FMH) • Attempts to incorporate volatility and chaos into the model • In markets, stock prices act like fractals. They appear to move in replicating geometric patterns • https://www.investopedia.com/terms/f/fractal- markets-hypothesis-fmh.asp
- 44. CREATIVE WORKS • Have been used on Jackson Pollock’s artwork to identify real works from imitations with a 93% success rate. • David Foster Wallace admitted that the first draft of Infinite Jest was based on a fractal design. • Alexis Wajsbrot has made great use of fractals in both Dr. Strange and Mary Poppins Returns. • Italian Mosaics 13th Century • Medieval Churches
- 45. RECAP General Fractals • Cantor Set • Mandelbrot Set • Julia Sets • Sierpinksi Triangle • Koch Snowflake Area Filling Fractals • Peano Curve • Hilbert Curve Applications • Coastline • Image Compression • Stock Trading • Creative Works
- 46. IMAGE SOURCES • Coastline Lake Mead (ukga.com) • Romanesco Broccoli Scientific American - PDPhoto.org • Fern Wikipedia Commons - António Miguel de Campos • Lightning GeoGebra • Mandelbrot Period https://fractal.institute/hidden-numbers-and-basic- mathematics-in-the-mandelbrot-set/ • Sierpinksi Antenna V. A. Sankar Ponnapalli and P. V. Y. Jayasree • Mandelbrot Chaos Georg-Johann Lay. Public Domain • Gorilla Cincinnati Zoo
Editor's Notes
- https://fractal.institute/hidden-numbers-and-basic-mathematics-in-the-mandelbrot-set/
- V. A. Sankar Ponnapalli and P. V. Y. Jayasree (March 31st 2018). Fractal Array Antennas and Applications, Emerging Microwave Technologies in Industrial, Agricultural, Medical and Food Processing, Kok Yeow You, IntechOpen
- Take time on this slide and try to explain the concept slowly. Make sure to reinforce what was said before to allow people to make the connection.
- Should be easy? Just draw a line. Well not quite. It depends on who you ask. Who’s right? The answer is that is depends on how large of a line you draw and how far you zoom in.
- We can see that when you vary the size of the line you get a more accurate reflection of the length of the coastline The problem is that no stretch of coastline is going to be perfectly straight so you can continue doing this endlessly. In this way, coastlines act like fractals and we can find the rate at which the length grows in comparison to the scale we are working at. We need some figure to describe this rate.
- There are a few different ways of achieving this. We can manually count the number of boxes on a picture with successively smaller grids or we can use python to automate some/all the work. We did both. First let’s take a look at the manual method. This involved getting as high a res image from Google Earth as possible. It was then imported into GIMP and a grid was put over the image. The boxes were marked if the coastline crossed through them and a was used to count mouse clicks as we clicked through all the boxes.
- Now let’s look at another way we can do it. We can convert an image to grey-scale and then count the boxes based on how different two pixels are in colour. This is dependent on the threshold you set. We used the same image for consistency and the converted it to grey like so:
- But of course, there is a python module for everything. The one we utilised was the basemap module. It allows you to plot areas of a map. This is what we obtained with a Mercator projection. What was useful about this is that we could set it to transparent and the “grey” version would just be either fully black or fully white. What we’d expect then is that the dimension we calculate is independent of the threshold
- This is exactly what we observe. If we do the same method as before and plot the dimension against the threshold value, we see we get the same dimension value for each threshold. This turns out to be 1.35 which is higher than what we had before but not too bad There was a group in Trinity that did it the same manual method we did at got 1.20 and there was a group in University of Ulster who used dividers and got 1.23
- Okay, so we have a value and it conforms with previous values. Well, what’s the point? Aside from the absolute riveting fun that is staring at maps for 2 hours, why do we do it. The dimension is a measure of how rough a coastline is. We can compare this against other countries. We can see the slope of the individual countries
- West coast of Ireland v East Coast South Africa smooth Norway not very smooth
- A lossy compression for digital images Uses an iterated function system. Basically a set of functions that will create the image. You separate the image into Ri blocks of size s x s Then for each Ri search for blocks Di that are size 2s x 2s that are similar to Ri Pick too little and it won’t look like your image. Pick too many and it’s computationally intensive Now if people remember back to the first few lectures we were told to keep this entertaining so…
- So, did you notice the Gorilla? Also, bonus marks for anyone that recognises this particular gorilla. First thing we do is reduce the image by just averaging neighbouring pixels. This reduces the amount of computation that we need to do The right hand image is the reduced image after being compressed and decompressed. We can see visual artificing on it. The way it’s being done is quite crude as we’re compressing and decompressing each individual RGB channel separately.
- So if we convert the image to greyscale and then reduce it we can do the same process for different levels of compression and see how close it is to the original image.
- The current “Main” market hypothesis is the efficient market hypothesis which says that stocks always trade at their fair value of exchanges Gamestop and Bitcoin would probably beg to differ Clearly this isn’t a perfect model but it’s main flaw is predicting anomalies Alternative hypothesis is the fractal market hypothesis. This attempt to incorporate volatility and chaos into the model In markets, stock prices act like fractals. They appear to move in replicating geometric patterns. The issue is you don’t know what time frame they’re repeating at. There’s more information at that link.
- Read Clearly the ideas of fractals have been something that humans have found great fascination in for a lot of human history even if it’s only been mathematically rigorous for the past few decades What we have found visually appealing is in fact mathematically complex and can used in a variety of situations. We now invite people who have any questions and we’ll do a brief recap of what we’ve covered in case it jogs your mind.