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The thing is, each time you do a fold, you don't get any more of the triangle's perimeter covered by the staircase. Not only that, every point that ends up on

The thing is, each time you do a fold, you don't get any more of the triangle's perimeter covered by the staircase. Not only that, every point that ends up on the triangle's line segment is adjacent to a point that isn't: there's always some little epsilon that you can step along, and you are no longer on an overlapping point. You've made no difference to the perimeter. When it comes to the area, this is not true. Folding the triangular segments over the smooth line moves an area on which you can slightly move to one side or another, and you are still in an overlap. Gotta wonder whether the sawtooth is actually a triangle. Each step makes it... not any closer to a triangle. Why would it be a triangle when you keep going to infinity?

JKCalhoun 2y ago
> The thing is, each time you do a fold, you don't get any more of the triangle's perimeter covered by the staircase. Not disagreeing with that. > When it comes to the area, this is not true.… Not following your point. And this part is the stickler for me: I believe the area does in fact approach the area of the triangle as the steps approach infinity. Is this not the case as you see it? Perhaps my "stickler" is not really one. To your first point, it's not really a triangle in the end, even as you approach infinity (though I still believe the area becomes that of said triangle). And there is nothing in geometry that says you can't have a shape with the area of a triangle but with a different perimeter. It's just this "shape" looks an awful lot like a triangle but with a much larger perimeter.
- lordnacho 2y ago
  > Is this not the case as you see it? Yeah I see it your way. Perimeter doesn't change, but the area does. But it's not a triangle. For instance, the internal angles don't add up to 180, and they never tend to either as you fold more and more.
- sfink 2y ago
  Hm, it doesn't seem problematic for the area to be lower than expected for a given perimeter. You could do the same thing by having a triangle (an actual triangle), but then at some random point have the bounding line shoot off in some random outward direction, then come back (increasing the perimeter without enclosing any area at all). It's no more or less a triangle than your fractally jagged triangle, and it has the same perimeter and area. I think the lesson for me is that any part of any line, if you zoomed into it enough, could actually be found to be all wadded up and much longer than you would otherwise expect. A line from point A to point B must be at least as long as the straight line from A to B, but could be any length greater than that. The only thing that makes your jaggy line special is that the distance from any point along it to the nearest point on the straight (triangle's) line is zero.
constantcrying 2y ago
>Why would it be a triangle when you keep going to infinity? Because the difference goes to zero. They are identical, they are equal in the strongest mathematical sense. The sets which make up their shape are the same set. They couldn't be more equal.

JKCalhoun 2y ago

> The thing is, each time you do a fold, you don't get any more of the triangle's perimeter covered by the staircase. Not disagreeing with that. > When it comes to the area, this is not true.… Not following your point. And this part is the stickler for me: I believe the area does in fact approach the area of the triangle as the steps approach infinity. Is this not the case as you see it? Perhaps my "stickler" is not really one. To your first point, it's not really a triangle in the end, even as you approach infinity (though I still believe the area becomes that of said triangle). And there is nothing in geometry that says you can't have a shape with the area of a triangle but with a different perimeter. It's just this "shape" looks an awful lot like a triangle but with a much larger perimeter.

lordnacho 2y ago
> Is this not the case as you see it? Yeah I see it your way. Perimeter doesn't change, but the area does. But it's not a triangle. For instance, the internal angles don't add up to 180, and they never tend to either as you fold more and more.
sfink 2y ago
Hm, it doesn't seem problematic for the area to be lower than expected for a given perimeter. You could do the same thing by having a triangle (an actual triangle), but then at some random point have the bounding line shoot off in some random outward direction, then come back (increasing the perimeter without enclosing any area at all). It's no more or less a triangle than your fractally jagged triangle, and it has the same perimeter and area. I think the lesson for me is that any part of any line, if you zoomed into it enough, could actually be found to be all wadded up and much longer than you would otherwise expect. A line from point A to point B must be at least as long as the straight line from A to B, but could be any length greater than that. The only thing that makes your jaggy line special is that the distance from any point along it to the nearest point on the straight (triangle's) line is zero.

constantcrying 2y ago

>Why would it be a triangle when you keep going to infinity? Because the difference goes to zero. They are identical, they are equal in the strongest mathematical sense. The sets which make up their shape are the same set. They couldn't be more equal.