Polygons of infinite perimeter but finite area

[bonanova]

It is possible to increase the area of a regular triangle by placing smaller regular triangles on the middle thirds of its three sides. By so doing, you obtain a six-pointed star. The process can continue indefinitely. At each step, a smaller regular triangle is placed on the middle third of all the line segmens on the perimeter of the figure obtained from the previous step. Sketching the shapes obtained for the first few steps of this process is an interesting way to spend a few moments.

The perhaps surprising result is that this process converges to a fractal-like figure of infinite perimeter but of finite area. Can you determine the area limit?

A more interesting question arises. Can some similar process converge to a fractal-like figure of infinite perimeter but of zero area?

[James33]

I get (5*sqrt(3)*x^2)/14 where x is the length of the original side.

For the next bit

Just set x=0, then the shape will have infinite perimiter but 0 area.

Edited June 4, 2012 by James33

[witzar]

1. The area of a fractal you have described (known as a

Koch snowflake) equals 8/5 of the original triangle area.

2. Sierpinski carpet is an example of a fractal of infinite perimeter and of zero area.

Coastline paradox is also related to finite areas enclosed by infinite perimeter.

[bonanova]

Hi James33, and welcome to the Den.

I get (5*sqrt(3)*x^2)/14 where x is the length of the original side.

Close. You calculated about .62 x². It's closer to .69 x², or exactly 1.6 times the original area.

For the next bit

Just set x=0, then the shape will have infinite perimiter but 0 area.

Nice idea, but you get into a zero times infinity quandry.

If x is set to 0, then everything collapses to a point, and you lose the infinite length.

What you can say is that the area can be made arbitrarily small by letting x become arbitrarily small.

But the area will remain nonzero.

1. The area of a fractal you have described (known as a

Koch snowflake) equals 8/5 of the original triangle area.

2. Sierpinski carpet is an example of a fractal of infinite perimeter and of zero area.

Coastline paradox is also related to finite areas enclosed by infinite perimeter.

Correct about the figure described in the OP.

Can you imagine a process similar to that described in the OP that gives zero area?

If the snowflake triangles were placed on the interior sides of the perimeter,

you would subtract, instead of adding, .6 of the original area.

[Yoruichi-san]

Hi James33, and welcome to the Den.

Correct about the figure described in the OP.

Can you imagine a process similar to that described in the OP that gives zero area?

If the snowflake triangles were placed on the interior sides of the perimeter,

you would subtract, instead of adding, .6 of the original area.

How "similar" does the process have to be to the OP? Do they have to be regular triangles? Can they overlap, and does that count as 'positive' area? If you can't overlap, you can't really go any bigger with regular triangles... Can you use other shapes, or points, or lines? Can you put multiple shapes on a side? Is generating a sierpinski triangle too cheap... Interesting exercise ;P

[bonanova]

How "similar" does the process have to be to the OP? Do they have to be regular triangles? Can they overlap, and does that count as 'positive' area? If you can't overlap, you can't really go any bigger with regular triangles... Can you use other shapes, or points, or lines? Can you put multiple shapes on a side? Is generating a sierpinski triangle too cheap... Interesting exercise ;P

Let's eliminate overlap, and we've exhausted triangles. You can only add or subtract 60% of their original area. So let's just keep using regular shapes, and push that as far as needed.

[Yoruichi-san]

Let's eliminate overlap, and we've exhausted triangles. You can only add or subtract 60% of their original area. So let's just keep using regular shapes, and push that as far as needed.

What are the rules for the 'corner' pieces of the original triangle? For any non-triangular shape you're going to be left with 3 of that shape to construct more of that shape and then triangular corners. Do you add the shape to the corners like you did the original triangle?

[bonanova]

What are the rules for the 'corner' pieces of the original triangle? For any non-triangular shape you're going to be left with 3 of that shape to construct more of that shape and then triangular corners. Do you add the shape to the corners like you did the original triangle?

The only way is to add smaller versions of the original shape onto its sides. So you wouldn't add triangles to the sides of a pentagon, for example, or vice versa. And the rule for the triangles on triangles construction was to add similar shapes with 1/3 of the perimeter to the middle third of each side.

[Yoruichi-san]

The only way is to add smaller versions of the original shape onto its sides. So you wouldn't add triangles to the sides of a pentagon, for example, or vice versa. And the rule for the triangles on triangles construction was to add similar shapes with 1/3 of the perimeter to the middle third of each side.

Ohh...well, I would add triangles to like, anything, particularly on the inside, but that just might be the graphene growth analysis talking...;P

Squares, each layer is just dividing the previous layer's squares into ninths, effectively forming something like a sierpinski square.

[bonanova]

Ohh...well, I would add triangles to like, anything, particularly on the inside, but that just might be the graphene growth analysis talking...;P

Squares, each layer is just dividing the previous layer's squares into ninths, effectively forming something like a sierpinski square.

Yes. Triangles and squares are the only regular figures that can go inside themselves without overlap. Triangle add or subtract .6 of the starting area; squares add or subtract an amount equal to the total starting area. Placed out side, you get another square, circumscribed at 45 degrees; placed inside, the entire area disappears in fascinating fashion.