High School What is the Area of a Sierpinski Triangle?

[Cheesycheese213]

TL;DR

I got a bit confused on how they were supposed to be measured (maybe fractal dimensions too?)

I was trying to find some sort of pattern in the triangle (below) to graph it or find some equation, and I thought maybe measuring something would be a good idea.

I was okay just calculating the area for the first few iterations, but then I got confused on how I was supposed to represent like an infinite term? Because the ones that have a fixed number of little triangles all have (I think) area since they stop subdividing, so I could get those numbers.

But from what I think I read the actual fractal that goes on forever wouldn't have an area since it would just be never ending lines? Also, there's the Hausdorff/fractal dimension which is saying it's between 1D and 2D, so then is there no way to "measure" it (like as area/length/volume) when it is the actual infinite fractal?

If so, would the graph just be always approaching zero, or would it eventually become zero since it's only made up of lines?
Thanks!

 

[fresh_42]

I have trouble to understand you-

But from what I think I read the actual fractal that goes on forever wouldn't have an area since it would just be never ending lines?

It doesn't have an area, since the the sequence of area left tends to zero. This doesn't have to do with lines.

Also, there's the Hausdorff/fractal dimension which is saying it's between 1D and 2D, so then is there no way to "measure" it (like as area/length/volume) when it is the actual infinite fractal?

The dimension is $\log_23$. What do you mean by measure it? How do you measure the dimension of a line?

If so,...

If what?

... would the graph just be always approaching zero, or would it eventually become zero since it's only made up of lines?
Thanks!

Maybe you can read the Wikipedia article and point out what you don't understand.

 

[Mark44]

I was trying to find some sort of pattern in the triangle (below) to graph it or find some equation, and I thought maybe measuring something would be a good idea.

View attachment 242377

I was okay just calculating the area for the first few iterations, but then I got confused on how I was supposed to represent like an infinite term? Because the ones that have a fixed number of little triangles all have (I think) area since they stop subdividing, so I could get those numbers.

View attachment 242379

But from what I think I read the actual fractal that goes on forever wouldn't have an area since it would just be never ending lines? Also, there's the Hausdorff/fractal dimension which is saying it's between 1D and 2D, so then is there no way to "measure" it (like as area/length/volume) when it is the actual infinite fractal?

If so, would the graph just be always approaching zero, or would it eventually become zero since it's only made up of lines?
Thanks!

The point about a Sierpinski Triangle (or Gasket) isn't about its area -- it's about the sum of the perimeters of the remaining triangles. In your first figure (upper left), the perimeter is 3, assuming the triangle is 1 unit on each side. In the second figure, four triangles are formed, with the middle one removed. The three remaining triangles are 1/2 unit on a side, and there are three of them, so the sum of the perimeters is 3*3*1/2 = 9/4 = 3 * 3/2.
In the next step, each of the three triangles of the second step has its middle triangle removed, resulting in a sum of perimeters of 27/4 = 9/2 * 3/2.
Each step produces a sum of perimeters that it 3/2 times the sum of the previous step.

As more triangles are formed, the limit of the areas of all the remaining triangles approaches zero, but the sum of perimeters increases without bound.

 

[Cheesycheese213]

Maybe you can read the Wikipedia article and point out what you don't understand.

Sorry for like making no sense I got myself super confused!

On Wikipedia it says
For integer number of dimensions d, when doubling a side of an object, 2^d copies of it are created, i.e. 2 copies for 1-dimensional object, 4 copies for 2-dimensional object and 8 copies for 3-dimensional object. For the Sierpinski triangle, doubling its side creates 3 copies of itself. Thus the Sierpinski triangle has Hausdorff dimension log(3)/log(2) = log₂3 ≈ 1.585, which follows from solving 2^d = 3 for d.

The area of a Sierpinski triangle is zero (in Lebesgue measure). The area remaining after each iteration is 3/4 of the area from the previous iteration, and an infinite number of iterations results in an area approaching zero.

I'm really sorry if it's like dumb, but I think I'm confused about whether the dimension and the area is related or are they totally different things? Sorry and thanks again!

 

[Cheesycheese213]

The point about a Sierpinski Triangle (or Gasket) isn't about its area -- it's about the sum of the perimeters of the remaining triangles. In your first figure (upper left), the perimeter is 3, assuming the triangle is 1 unit on each side. In the second figure, four triangles are formed, with the middle one removed. The three remaining triangles are 1/2 unit on a side, and there are three of them, so the sum of the perimeters is 3*3*1/2 = 9/4 = 3 * 3/2.
In the next step, each of the three triangles of the second step has its middle triangle removed, resulting in a sum of perimeters of 27/4 = 9/2 * 3/2.
Each step produces a sum of perimeters that it 3/2 times the sum of the previous step.

As more triangles are formed, the limit of the areas of all the remaining triangles approaches zero, but the sum of perimeters increases without bound.

Thank you so much!

 

[fresh_42]

I'm really sorry if it's like dumb, but I think I'm confused about whether the dimension and the area is related or are they totally different things?

They are different things. For example the area of a square can be one, but if part of a three dimensional space, its volume is zero. The area is the volume in two dimension, the length the volume in one dimension. The area of the Sierpinski triangle is zero, it has no measurable content. The dimension is a mathematically defined quantity which turns out to be $\log_23$. It is a bit artificial to cover those special objects which fractals are. The triangle lives in the plane, but its area vanishes at infinity. What's left is more than a line which would have had a certain finite length, but less than an area; somewhere in between the two.

 

[Cheesycheese213]

Oh thanks that makes so much more sense!

 

[WWGD]

The setup is that length is seen as a 1d measure , are is a 2d measure. In order to have D-Day n-dimensional measure, an object must have dimension d or higher.Edit: in these types of constructions; see also fat Cantor sets which have empty interior but non-zero length. You can do similar with a triangle.