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FractalsNC July 2020.

Fractals and triangles What is a fractal? A fractal is a pattern created by repeating the sameprocess on a different scale. One of the most famous fractals is the Mandelbrotset, shown below. This was first printed out on dot matrix paper! (askyour teacher ).Created by Wolfgang Beyer with the program Ultra Fractal 3. - Own workNC July 2020.

Koch Curve You will want to draw with pencil so you can erase yourprevious lines.Draw an equilateral triangle. If using isometric paper,the edges should be 9cm.Free isometric paper can be found reFreeGraphPaper.htmNC July 2020.

Koch Curve Erase the middle 3 cm and add an equilateral triangle. Repeat this on the other 2 edges.NC July 2020.

Koch CurveContinue the process. For each 3cm edge, erase themiddle 1cm and draw a 1cm equilateral triangle extendingfrom the edge. The start is shown here.NC July 2020.

Koch Curve You should now have a shape that looks like this. Youcan see why the shape has the name ‘Koch’ssnowflake’. You can continue the process for as long as you wish make sure for each iteration you are dividing each edgein to 3 – it’s very easy to miss some out!NC July 2020.

Koch Curve These are the continuing steps. If you look closely oneach edge, what shape do you see emerging? There is a computer generated zoom herehttps://www.youtube.com/watch?v PKbwrzkupaUNC July 2020.

Koch Snowflake Assuming that the length of each side of the originaltriangle is 1 unit complete the following table:STAGEPERIMETER1323456 NC July 2020.Can you work out a formula for the perimeter at thenth stage?What happens to the perimeter as n increases?

Koch SnowflakeSTAGEPERIMETER13243456NC July 2020.Each edge has1an extra added,3so each newedge has length113

Do you think the area of thesnowflake curve is finite orinfinite?NC July 2020.

Does this picture help you?NC July 2020.

More fractals Before exploring the next shape, you may wishto watch this numberphile video https://www.youtube.com/watch?v kbKtFN71Lfs Geogebra file herehttps://www.geogebra.org/m/yr2XXPmsNC July 2020.

The Sierpinski Triangle The Sierpinski Triangle is made by repeatedlysplitting a equilateral triangle into 4. There is a lovely animation of an infiniteSierpinski Triangle here: ctals/ Or here https://www.youtube.com/watch?v TLxQOTJGt8cNC July 2020.

Can you draw a SierpinskiTriangle?NC July 2020.

Step 1:Split the unshaded equilateraltriangle in to 4 equilateraltriangles. Shade the middletriangle.Each iteration you will half thelength of the triangle, so youneed the triangle to have alength that is a power of 2.The example uses a sidelength 32.NC July 2020.

Iteration 1Step 2:Split the unshaded equilateraltriangles in to 4 equilateraltriangles. Shade the middletriangles.NC July 2020.

Iteration 2Step 3:Split the unshaded equilateraltriangles in to 4 equilateraltriangles. Shade the middletriangles.NC July 2020.

Iteration 3Step 4:Continue NC July 2020.

Iteration 4NC July 2020.

Iteration 5NC July 2020.

Doing some maths Look at the triangle after the first iteration. What fractionof the triangle did you NOT shade? What fraction of the triangle is NOT shaded after thesecond / third iteration?NC July 2020.

Doing some maths Fill in this tableIteration12Unshaded34916Shaded143456 NC July 2020.Can you generalise for the nth iteration?Can you describe what this means as n ?

Continued watching and reading More fractals can be explored from the AMSP fractalsenrichment lesson – some of the material ment-lessons To explore fractal perimeter some more, watch thisexcellent numberphile video on the coastline paradoxhttps://www.youtube.com/watch?v 7dcDuVyzb8YNC July 2020.

Fractals and triangles What is a fractal? A fractal is a pattern created by repeating the same process on a different scale. One of the most famous fractals is the Mandelbrot set, shown below. This was first printed out on dot matrix paper! (ask your teacher ). Created by Wolfgang Beyer with the program Ultra Fractal 3. - Own work

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A commonly asked question is: What are fractals useful for Nature has used fractal designs for at least hundreds of millions of years. Only recently have human engineers begun copying natural fractals for inspiration to build successful devices. Below are just a few examples of fractals being used in engineering and medicine.

Fractals and 3D printing 1. What are fractals? A fractal is a geometric object (like a line or a circle) that is rough or irregular on all scales of length (invariant of scale). A Fractal has a broken dimension. By zooming-in and zooming-out the new object is similar to the original object: fractals have a self-similar structure.

Edge midpoints, ::: Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 38/41. Height Maps: Clouds and Coloring Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 39/41. Other Applications Video Games Fracture - link Ceramic Material - link

Generalities on fractals Many self-similar (fractal) structures in nature and many ways to model them: A random walk in free space or on a periodic lattice etc. Fractals provide a useful testing ground to investigate properties of disordered classical or quantum systems, renormalization group and phase transitions,

Fractals in Max Peter Elsea 1/31/12 1 Fractals in Max and Jitter Simple iterative process Fractal geometry is the study of objects that have a property known as self-similarity – They are made up of smaller copies of the overall shape. One of the most popular is called the Sierpinski triangle: Figure 1.

2.2 Fractals: The Mathematics of Self-Similarity Scale symmetry and the golden spiral both pave the way to understand fractals. Many of the most beautiful fractals require computer generation to appreciate them fully, but there are several that are feasible and enjoyable to construct by

the scaling laws of disorder averages of the con gurational properties of SAWs, and clearly indicate a multifractal spectrum which emerges when two fractals meet each other. 1. INTRODUCTION Self-avoiding walks (SAWs) on regular lattices provide a successful description of the universal con gurational properties of polymer chains in good solvent .

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