Fractals, Geometry of Nature and Logistic Model
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Fractals are geometric shapes that exhibit self-similarity and complex patterns at every scale. Koch's snowflake is a famous fractal where the perimeter tends towards infinity as more iterations are done, even as the area approaches a limit. The Mandelbrot set is another well-known fractal that maps the behavior of values of c under a complex iterative function, resulting in diverse patterns when zooming in. Fractals are found throughout nature in shapes like clouds, coastlines, and Romanesco broccoli. They can also be generated through computer programs and used in applications like diagnosing skin cancer.
Fractals, Geometry of Nature and Logistic Model
- 1. Fractals by Osman Villanueva García
- 2. Can you describe how the animation is progressing? What is a fractal? What is happening to the perimeter and area of the shape as more iterations are done? It can be shown that the area tends towards 160% of the original triangle’s area, but the perimeter tends towards infinity The shape, known as Koch’s snowflake, also has some ‘self-similarity’, meaning that zooming into a small section will reveal the original structure These properties – of infinite iterations and self-similarity – are what make a fractal
- 3. & The most famous fractal shows the Mandelbrot set, which investigates the effect of the iterative rule below for any complex number c: The Mandelbrot set what happens as the number of iterations gets very large Example: The Mandelbrot set Investigate for yourself what happens when c varies using the spreadsheet... It does this for all possible values of c (ie any complex number!) The value of zn behaves in an unpredictable way!
- 4. To turn the outcomes of this investigation into a fractal, the following rule is applied: If a value of c doesn’t diverge then it is shown as a black point on the Argand diagram If a value of c does diverge then it is shown as a white point on the Argand diagram For any complex number you start with c: This leads to: This fractal was named after the man who discovered it – Benoit Mandelbrot Argand diagram refers to a geometric plot of complex numbers as points z = x + iy using the horizontal x-axis as the real axis and the vertical y-axis as the imaginary axis.
- 5. To obtain a more striking fractal, vary the colour of a point depending on how quickly it diverges You’re probably still thinking <<OK these look nice, but what’s the point?>> It turns out this strange image also has some amazing properties...
- 6. Zooming in - Fractals Zooming into different parts of the edge of a Mandelbrot set reveals strange and beautiful patterns...: You can zoom further and further... https://math.hws.edu/eck/js/mandelbrot/MB.html
- 7. Multibrot sets vary the power used in the iterative rule: Changing the pattern
- 8. Julia sets are generated by keeping c fixed and making the point you are considering zo
- 9. Newton fractals use the rule for different functions f zn+1 = zn - f(zn ) f’(zn )
- 10. Fractals in art Here are a small selection of the fractals created by artists – investigate for yourself what can be made!
- 11. Fractals in nature The mathematical fractals we have seen are infinitely complex, but have been built from an extremely simple equation repeated endlessly. In the same way, natural fractal forms really are built up by simple rules - ultimately, the interactions between atoms. The Earth from space A fern leaf Romanesco brocoli Ice crystals
- 12. The Geometry of Nature ⚫ “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.” (Mandelbrot, 1983). ⚫ And here is a quote by Thomasina, from Arcadia: “Each week I plot your equations dot for dot, and every week they draw themselves as commonplace geometry, as if the world of forms were nothing but arcs and angles. God's truth, Septimus, if there is an equation for a curve like a bell, there must be an equation for one like a bluebell, and if a bluebell, why not a rose? Do we believe nature is written in numbers?” Arcadia is a 1993 stage play written by English playwright Tom Stoppard, which explores the relationship between past and present, order and disorder, certainty and uncertainty. It has been praised by many critics as the finest play from “one of the most significant contemporary playwrights” in the English language. It is widely believed that the character of Thomasina Coverly in Arcadia is loosely based on Ada Lovelace, an English mathematician in the 1800’s. Many people actually regard her as the first computer programmer!
- 13. Landscapes ⚫ Can you determine which images are real and which are computer generated?
- 14. Clouds
- 16. A Medical Application ⚫ Fractals are used in the diagnosis of skin cancer and liver diseases. ⚫ There is a notion of fractal dimension. ⚫ This is applied to images of the affected area and its boundary (they are both fractal).
- 17. Fractals ⚫ Choose some similarities (with contracting scaling). ⚫ Let Fractalina play the chaos game with those similarities.
- 18. More Examples
- 21. Add inversion in the design…
- 24. Add Reflections to this mix…
- 26. Design - Mandelbrot Sets
- 29. Logistic Model - Origin of caos https://www.geogebra.org/classroom/c8kgxsq8#tasks/c6mu8mh5 P{n+1} = r Pn (1 - Pn )