Fractals

What is a fractal?

A fractal is a rough (rugoso) or fragmented geometric shape that can be subdivided in parts, each of which is (approximately) a

reduced-size copy of the whole shape.

What is a Fractal? • A fractal is a mathematical object that is both self-similar and chaotic.

•self-similar: When you magnify, you see the object over and over again in its parts.

•chaotic: Fractals are infinitely complex.

The most famous of all fractals is the Mandelbrot set.

With the Mandelbrot Set, we will look at self-similarity and infinte

complexity.

Infinite complexity: Zoom and zoom and zoom and the pattern will never end. It goes forever!

Self-similarity: As you zoom in, the shape repeats.

Self-similarity / Infinite Complexity

As we magnify the object, we see the same thing over and over again.....This is

Self Similarity

More fractals

Many fractals can be found in nature. Question: What do you think could be a fractal

in nature? For example, ferns are natural fractals.

Fern fractal

More natural fractals

Other places we see fractals: Fault lines, Mountain Ranges, Craters, Lightning

bolts, Coastlines, Mountain Goat horns, Trees, Animal coloration patterns, Pineapples, Heart rates, Heart sounds, Earthquakes, Snowflakes, Crystals,

Blood vessels and pulmonary vessels, Ocean waves, DNA, Soil pores, Rings of Saturn, and

Proteins

Now, let´s look at dimensions. First, no dimension or 0 dimension.

A point has no dimensions - no length, no width, no height.

. That dot is obviously way too big to really represent a

point. But we'll live with it, if we all agree what a point really is.

One dimension

What has one dimension?

One dimension

A line has one dimension - length. It has no width and no height, but infinite length.

Again, this model of a line is really not very good, but until we learn how to draw a line with 0

width and infinite length, it'll have to do.

2 dimensions

A plane has two dimensions - length and width, no depth.

It's an absolutely flat tabletop extending out both ways to infinity.

3 dimensions

Space, a huge empty box, has three dimensions, length, width, and depth,

extending to infinity in all three directions.

What about the dimension of fractals?!

How many dimensions do you think a fractal has?

Fractals can have fractional (or fractal) dimension. A fractal

might have dimension of 1.6 or 2.4. How could that be?

Let's investigate!

Take a self-similar figure like a line segment, and double its length. Doubling the length gives two copies of the

original segment.

Take another self-similar figure, this time a square 1 unit by 1 unit. Now multiply the length and width by 2. How many copies of the original size square do you

get? Doubling the sides gives four copies.

Take a 1 by 1 by 1 cube and double its length, width, and height. How many copies of the original size cube do

you get? Doubling the side gives eight copies.

Let's look further at what we mean by dimension.

Let´s organize our information. Do you see a pattern? It appears that the dimension is the

exponent - and it is! So when we double the sides and get a similar figure, we write the number of copies as a power of 2

and the exponent will be the dimension.

Figure Dimension Num. Of Copies

Line 1 2 = 2¹ Square 2 4 = 2²

Cube 3 8 = 2³

Doubling similarity

d N = 2^d

The Sierpinski Triangle (the first iteration)

(the second iteration)

Figure Dimension Num. Of Copies

Line 1 2 = 2¹ Square 2 4 = 2²

Cube 3 8 = 2³

Doubling similarity

d N = 2^d

Sierpinski Triangle

? 3 = 2^?

So, fractals are between dimensions!

Sierpinski Triangle

1.6 3 = 2^1.6

Basically, a fractal is any pattern that shows more complexity as it is made bigger. Thus, fractals graphically show the idea of 'worlds within worlds´

Is the entire universe a fractal???