21st Math'fia

Fractals.

Overview.

The term fractals came from the Latin word “fractus,” which means irregular or fragmented. The term “fractals” was named by Benoit Mandelbrot, a mathematician. In the simplest term, fractals are visual representations of a repeating pattern. The figure starts with a simple pattern and progressively becomes complex as the pattern enters a loop.

Trivia!

Did you know that the application fractals came out just before the term was created? It was Lewis Fry Richardson, an English Mathematician, who discovered the concept of fractals when we were trying to study the length of the English coastline. According to Richardson, the length of the coastline depends on the measuring tool used. When you use a meter stick, yark stick, ruler, etc. you get different values. He concluded that you will end up with an infinitely long coastline in a finite space.

Fractal Properties.

As mentioned above, fractals tend to get more complex as the pattern enters a loop. Hence, fractals have different properties that will help us understand and appreciate their unique qualities.

1. Self-similarity

All fractals contain infinitely complex patterns that have self-similarity across the figure in different segments, scales, dimensions, and sizes. This means that when you zoom in on the tiniest part of the figure, you will observe the same patterns put together within sections.

An example of this is a fern. Take a look at the whole fern, each frond has the same pattern as the leaf, leaflet, and individual fern pinnule. Each fragment contains the same pattern that makes up a bigger fragment.

2. Iteration

All fractals contain infinitely complex patterns that have self-similarity across thFractals have the same repeating pattern through a process of the loop, which we call iteration. This process simply generates the same pattern or equation many times until a random, beautiful, and unique has been created.

3. Recursive

Fractals also possess the property of recursiveness, regardless of their scale. An example of this is when you enter a dressing room and find yourself surrounded by mirrors. Hence, what you are seeing is a recursive image of yourself.

4. Fractal dimension

Fractal dimension has something to deal with Geometry. This property tells us how “complicated” a fractal figure is. Like zero-dimensional points, one-dimensional lines, two-dimensional plane figures (e.g. squares, circles, etc.), and three-dimensional figures (cubes, spheres, etc.), fractals also have dimensions, but not whole numbers. Dimensions of fractals are expressed in terms of fractions like 1.48, which can fall between a line and a plane.

Sierpinski Traingle.

Step 1

Draw an equilateral triangle

Step 2

Connect the midpoints of the three sides of the base and shade the newly formed triangle

Step 3

Apply the motif shade to the newly formed white triangles

Koch Edge.

Step 1

Draw a straight line

Step 2

Divide the straight line with a “crinkled” version at angles 60° and 120°

Step 3

Repeat steps 1 and 2 on each of the line segments

Koch Snowflake.

Step 1

Start drawing a shaded equilateral triangle. This will be the “seed” of the Koch snowflake

Step 2

Add three triangles, ⅓ length of the seed, on each side of the seed

(12 sides)

Step 3

Add two more ⅓ length triangles on each of the 12 sides of the snowflakes in Step 2.

This will result in a 12 x 4 = 48 sides

Cantor Set.

Step 1

Draw a line segment

Step 2

Divide the line segment into thirds

Step 3

Remove the second (mid) line segment

Repeat steps 2 and 3

Sierpinski Carpet.

Step 1

Draw and shade a square

Step 2

Draw nine squares inside the big square drawn on Step 1. Remove the shade of the middle square.

Step 3

Repeat step 2

Menger Sponge.

Step 1

Draw a cube

Step 2

Put 20 cubes together with the center parts removed

Step 3

Repeat Step 2

Mandelbrot Set.

This was discovered after the invention of the personal computer. Fractals can be observed algebraically by calculating a simple equation repeatedly, which is called bifurcation. Mandelbrot is expressed as z = z^2 + C, where the two-fold symmetry branches and becomes four-fold, then doubles to eight-fold, and then into 16-fold symmetry, which then enters into an unending process.

Julia Set.

Julia sets are algebraic fractals. Using the Mandelbrot’s equation, z = z^2 + C, when the exponent is raised to three, the Julia set takes on a three-fold symmetry. The degree of symmetry corresponds to the degree of the exponent.

Fractals in Real life

Trees

Plants

Neurons

Blood Vessels

Lightning

Rock Formations

Snowflakes

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