## Saturday, January 11, 2014

### Fractal dimensions

Let's talk about fractal dimensions. Just for fun.

1. As many know, fractals are a type of geometric object which consists of self-similar patterns on all scales of magnification.

2. What's not as often discussed is fractal dimensions.

a. We know a point has zero dimensions. A line has one dimension (length). A square has two dimensions (length, width). A cube has three dimensions (length, width, depth). Likewise, fractals have fractal dimension.

b. An example that's sometimes used to illustrate fractal dimension is the Sierpinski triangle. What is a Sierpinski triangle?

Cut out a central triangle as follows:

Next, cut out three smaller triangles as follows:

And then repeat for each of the triangles that haven't been cut out as follows:

If we continue, the result will be a Sierpinski triangle like this one:

c. Normally, if we double the lengths of a triangle, its area will be quadrupled. Say we have a triangle with base 2 and height 3. Its area would be (2 x 3) / 2 = 3. If we double its base and height, then we would have (4 x 6) / 2 = 12. And 12 is 4x greater than 3.

d. Strangely, though, if we double the lengths of a Sierpinski triangle, its area will not be quadrupled. Instead, its area will only be tripled.

This is due to the fact that when we take three smaller Sierpinski triangles to form a larger Sierpinski triangle, the larger Sierpinski triangle has doubled in size.

Thus, the Sierpinski triangle has a dimension = log 3 / log 2. That is, approximately 1.585 dimensions (if we round up).

How can we have 1.585 dimensions though? Since we cut out triangles within triangles over and over again, and in fact we could do so an infinite number of times, the resulting Sierpinski triangle is more like an assembly of points (with zero dimensions) or a grid of lines (with 1 dimension). Hence overall the Sierpinski triangle isn't 2 dimensional like a normal triangle would be, but it has 1.585 dimensions.

e. BTW, Sierpinski triangles can also be Sierpinski pyramids:

3. A few further notes:

a. In reality, a dimension either is or isn't. We can't have partial dimensions. Thus, 1.585 (or whatever) dimensions is not reality, per se, but a mathematical abstraction.

b. That said, say we have a piece of string. Say we assume this string is 1 dimensional. (Of course, all physical objects in our universe are actually at least three dimensional.) Say it's a long piece of string. Say we roll this string up so it becomes a ball. As such, the originally 1 dimensional string has become in a sense 3 dimensional.

Take another example. Say we assume a piece of paper is 2 dimensional. Say we crumple up this piece of paper. As such, the originally 2 dimensional piece of paper is 3 dimensional.

Fractals can likewise be "crumpled," so to speak.

c. Of course, the real universe in which we live is 3+1 dimensions. That is, 3 space dimensions + 1 time dimension.

We can use fractals and fractal dimensions to better understand our own world. There's a lifetime of study in this alone.

However, perhaps we can likewise use fractals and fractal dimensions to help us imagine worlds beyond our own? That is, like Edwin Abbott did in Flatland, or Ian Stewart did in Flatterland, it's possible to imagine that we live in 4+1 dimensions, for example. Perhaps fractals and fractal dimensions can be an aid here.

## Saturday, January 11, 2014

### Fractal dimensions

Let's talk about fractal dimensions. Just for fun.

1. As many know, fractals are a type of geometric object which consists of self-similar patterns on all scales of magnification.

2. What's not as often discussed is fractal dimensions.

a. We know a point has zero dimensions. A line has one dimension (length). A square has two dimensions (length, width). A cube has three dimensions (length, width, depth). Likewise, fractals have fractal dimension.

b. An example that's sometimes used to illustrate fractal dimension is the Sierpinski triangle. What is a Sierpinski triangle?

Cut out a central triangle as follows:

Next, cut out three smaller triangles as follows:

And then repeat for each of the triangles that haven't been cut out as follows:

If we continue, the result will be a Sierpinski triangle like this one:

c. Normally, if we double the lengths of a triangle, its area will be quadrupled. Say we have a triangle with base 2 and height 3. Its area would be (2 x 3) / 2 = 3. If we double its base and height, then we would have (4 x 6) / 2 = 12. And 12 is 4x greater than 3.

d. Strangely, though, if we double the lengths of a Sierpinski triangle, its area will not be quadrupled. Instead, its area will only be tripled.

This is due to the fact that when we take three smaller Sierpinski triangles to form a larger Sierpinski triangle, the larger Sierpinski triangle has doubled in size.

Thus, the Sierpinski triangle has a dimension = log 3 / log 2. That is, approximately 1.585 dimensions (if we round up).

How can we have 1.585 dimensions though? Since we cut out triangles within triangles over and over again, and in fact we could do so an infinite number of times, the resulting Sierpinski triangle is more like an assembly of points (with zero dimensions) or a grid of lines (with 1 dimension). Hence overall the Sierpinski triangle isn't 2 dimensional like a normal triangle would be, but it has 1.585 dimensions.

e. BTW, Sierpinski triangles can also be Sierpinski pyramids:

3. A few further notes:

a. In reality, a dimension either is or isn't. We can't have partial dimensions. Thus, 1.585 (or whatever) dimensions is not reality, per se, but a mathematical abstraction.

b. That said, say we have a piece of string. Say we assume this string is 1 dimensional. (Of course, all physical objects in our universe are actually at least three dimensional.) Say it's a long piece of string. Say we roll this string up so it becomes a ball. As such, the originally 1 dimensional string has become in a sense 3 dimensional.

Take another example. Say we assume a piece of paper is 2 dimensional. Say we crumple up this piece of paper. As such, the originally 2 dimensional piece of paper is 3 dimensional.

Fractals can likewise be "crumpled," so to speak.

c. Of course, the real universe in which we live is 3+1 dimensions. That is, 3 space dimensions + 1 time dimension.

We can use fractals and fractal dimensions to better understand our own world. There's a lifetime of study in this alone.

However, perhaps we can likewise use fractals and fractal dimensions to help us imagine worlds beyond our own? That is, like Edwin Abbott did in Flatland, or Ian Stewart did in Flatterland, it's possible to imagine that we live in 4+1 dimensions, for example. Perhaps fractals and fractal dimensions can be an aid here.