Fractal Dimension

In this video I would like to talk about the

fractal

dimension

and before we talk about the

dimension

of these three

fractal

s here, let's remember what dimension means in the usual sense, for example, here we have a point and we consider points to be zero-dimensional objects. This is a line and lines we consider one-dimensional for an object like a square or something in the plane we consider them two-dimensional and for something in space like a cube this is considered three-dimensional and with fractals They generally do not have these dimensions of integers, in fact their dimensions are usually between these numbers, for example, this here is the Snowflake coke and the dimension of this is approximately one point 2 6. and we have to write it approximately since it is actually an irrational number this comes from a logarithm which we'll look at in more detail as we go, but most of these fractal dimensions will be represented as logarithms and logarithms are generally not integers, they are generally irrational this object right here this fractal we Call this the sponge of the manger and this one here is called the Serpinsky triangle.

Now we'll look at these two fractals in more detail in later videos, but for now the dimension of the manger sponge is approximately 2.73 and for this Serpinsky triangle the dimension of this object is approximately one point five eight five. Now an important question is to ask how these dimensions are actually calculated and to find out we must first start with simpler shapes as remember that fractals are self-similar objects when you zoom in on them. they look similar to the overall image and the simple shapes have the same characteristic if we were to cut the line or a square or a cube into smaller pieces those smaller pieces would look the same as the overall piece and from that idea we can notice a pattern and then once we notice that pattern, we can apply those ideas to fractals and figure out how they are actually calculated, so let's make some space and then we'll basically start by looking at these simple shapes and like I.

I said we're going to try to figure out the pattern so let's take each of these shapes and cut them into smaller pieces and we can cut them into whatever size pieces we want, but let's say we cut them into thirds so with the line we can cut this into three pieces equals where each piece is now one third of the original size line and we have three pieces for a square. We can do the same thing but it's a little different. Each of these pieces is now a third of the original size and we'll do it on each side, but when we divide a square into these pieces, we actually get several more pieces.

In this case, we will end up with nine pieces where each of these pieces has side lengths that are one-third the length of the sides. original size and for a cube it is similar to a square, we will take each of these sides and divide them into a third of the original size, but with a cube we will actually end up with 27 pieces, so let me draw this very quickly and with the cube, you can count them if you want, but you can see that there are three rows of three cubes on top and then you have three rows, which means you have nine on top and three rows of nine, which is 27. and for everyone.

For these shapes, as I mentioned, we simply take the lengths of the original sides and cut them into three equal parts so that each of these new pieces is one third of the length of the original sides and from this idea we can notice a pattern that It relates the scale factor to essentially what we cut each of these pieces and the number of pieces we end up with and the dimension, so let's define some variables, let's say our scale factor that we're going to divide each of these lengths into. original sides we can call it 1 over R where in this case r is equal to 3. we can say that n is the number of pieces that we have left after performing this division process and we can say that D is the dimension that we are dealing with and for In all of these cases we know that our scale factor is 1 over R is one third, which means that r is three.

We know that in this case the number of pieces is also three but the dimension is one since it is a line and for the square. We know that the dimension is two, it has the same scale factor that they all have, but in this case the number of pieces is nine and for the cube we know that the dimension is three and the number of pieces we end up with is 27. and somehow from With this pattern we can generate an equation that relates these ideas to each other. In fact, you can notice that in this first case, if we raise the scale factor to the dimension, we get the number of pieces and this will be true for all three.

In these cases if we take the scale factor we raise it to the dimension we obtain the number of pieces and for this taking that scale factor of 3 raising it to that third dimension we obtain 27 since 3 times 3 times 3 is 27. and from this pattern now we can use these variables to write an equation now we have that R from the scale factor raised to the dimension is equal to n the number of pieces and this is the equation that we will use to determine the dimension of these fractal objects So in the next video we'll look at the coke snowflake dimension, since we're already a little familiar with that, and then in later videos we'll look at the manger sponge dimension and the Sabrinsky triangle, as well as several. other fractals