Fractals Assignment - Fractals: Exploring Patterns and Applications

This assignment explores the concept of fractals, focusing on their mathematical properties and real-world applications. The assignment begins with an introduction to fractals, explaining their self-similar nature and how they differ from regular geometric shapes. It then describes an activity where students explored fractals using triangle grids, observing how smaller triangles are created and maintaining the same shape. The assignment highlights student discoveries, including the concept of iteration and the ratio of scale increase. It further mentions the historical context of fractals, attributing their formal study to Benoit Mandelbrot. The assignment concludes by discussing the general pattern of fractals represented by a mathematical formula (zn+1 = zn2 + c) and provides examples of how fractals are used in scientific research, such as in biology and engineering, and in natural phenomena like snowflakes. References are included for further reading.

Running head: Fractals
Fractals.
Fractals are patterns that to not have an end, which are very complex and they are the same across
different scales at a specific part. Fractals explain the objects that occur naturally. The fractals
that are artificially made show same shapes at increasingly smaller patterns. Unlike normal
shapes like rectangle and circle that are termed as smooth, Fractals are normally rough and
complex. According to Benoit Mandelbrot (Jun, 2008), he said that Fractals are shapes that are
repeated if divided into small parts.
Modifications made to present the activity.
In this case the “students” are teenagers that were presented to. The teenage group are friends in
my neighborhood and they have learnt in school about geometry. During the activities with the
triangle grid, we drew the triangle with different colors making them smaller and smaller, and
coloring each step triangles with different colors to differentiate the triangles made after every
step done. We made sure that all new triangles are divided into equal shapes.
Discoveries made during the activity
My students discovered;
1. The smaller triangle shapes being made from the bigger triangle are the same in shape
and all are equals when divided at the center. Following this discovery, the idea of
Fractals shows that large structures are made by small equal shapes to make one big
complex structure. The nature also have many fractal shapes, example the fern plant.
2. They further discovered that the scale of each new iteration is increasing at the ratio of
one is to two in every new iteration. This shows that in each iteration, there is a complete
recurrence of the first shape.
3. During the activity, the students discovered that the idea of Fractals was discovered by
Benoit Mandelbrot in the 1970’s, although they were in existence for a long time.
The General pattern.
The general fractal pattern is set to a certain n sequence whereby, a particular pattern is followed
as shown below (zn), where zn+1 = zn2 + c. Where c is a constant that is not changing and n is the n
column showing the new pattern that has been created after several iterations. (Sarah, 2014)
Additional information.
The study of Fractal has helped a lot especially in scientific research for example the study in
biology of how bacteria’s multiplies.
Engineers also are not left behind because they are using the idea of fractals to do practical
engineering problems in their field. Therefore the study of fractals generally has helped a lot
Snowflakes use the idea of Fractals as they form in water.
Generally, the idea of Fractals are all over around us.

Fractals
References
Jun, K. (2008). Analysis on Fractals. Cambridge: University Press.
Sarah, C., Richard, P., and Boyds Mills (2014). Mysterious: Finding Fractals in nature. Honesdale,
Pennsylvania: Boyds Mills press.