Remix.run Logo
Remix clone Hacker News
new | show | ask | jobsGithub
▲ ethmarks 5 hours ago

> Note also that the triangle has infinite perimeter but finite area.

How common is this property in geometry? I know that fractals like the Koch Snowflake also have infinite perimeter over finite area, but I don't know what else does.

▲IgorPartola 4 hours ago | parent | next [-]

Any function that infinitely slowly converges to a finite number will have this property. Discretely, think of 1/2 + 1/4 + 1/8 and so on. The sequence goes on forever but adds up to 1.

▲eru 4 hours ago | parent | next [-]

A continuous function with that property is f(x) := 2^-x (when summed over the non-negative part of the x-axis). Another example is g(x) := 1/x^2.

▲ 4 hours ago | parent | prev | next [-]
[deleted]
▲almostgotcaught 2 hours ago | parent | prev [-]

I have no idea why you think the geometric series has anything to do with this - this is related to continuous but nowhere differentiable functions: https://en.wikipedia.org/wiki/Weierstrass_function

▲saithound 2 hours ago | parent [-]

> I have no idea why you think the geometric series has anything to do with this -

IgorPartola is perfectly right to mention geometric series, you can easily use a geometric progression to construct a shape with infinite perimeter and finite area, e.g. by gluing together rectangles with height one and width decreasing in geometric progression. With a bit more thought you can also construct a smooth shape having this property.

▲almostgotcaught 2 hours ago | parent [-]

> together rectangles with height one and width decreasing in geometric progression

The geometric series sums to 2 - your glued together rectangles will have perimeter 2*(1+2) and area 2*1.

▲saithound an hour ago | parent [-]

> your glued together rectangles will have perimeter 2*(1+2)

No. You should think through that perimeter calculation one more time, preferably while drawing a picture.

Here's a hint: the perimeter of a rectangle is no less than its height; you can glue so that the perimeter of each rectangle contributes at least 1 to the perimeter of the union.

▲genezeta 26 minutes ago | parent [-]

I think you're both right. But there are two ways to do what you said and you didn't specify which one.

First, a rectangle of height 1 and width 1/2. The perimeter is 1 * 2 + 1/2 * 2, two sides of height 1 and two sides of width 1/2.

You "glue" the second rectangle. As one may understand this, you glue them by putting them one beside the other standing up, i.e. you glue them along one of the heights. Sorry for the crude ascii art:

    ----   --     ------
    |  |   ||     |    |
    |  |   ||     |    |
    |  | + || ->  |    |
    |  |   ||     |    |
    |  |   ||     |    |
    ----   --     ------
Now you have a single rectangle, height 1, and width 1/2 + 1/4. The perimeter is 1 * 2 + (1/2+1/4) * 2. The "added perimeter" in this step is just 1/4 * 2 = 1/2.

Go on doing that and for a rectangle of width 1/n, you only add 2 * 1/n to the perimeter. In the end you get a single rectangle with height 1 and width 2. The perimeter is 2 * 1 + 2 * 2.

---

Now, maybe, you may want to specify that you glue the rectangles along their widths, not their heights.

That way, the resulting shape when you add the second rectangle is not a rectangle but an irregular shape with 6 sides. Sorry for the crude ascii art again:

         1
    ----------
    |        |n/2
    |        |      1
   n|         ----------
    |                  |n/2
    |                  |
    --------------------
             2
The added perimeter now is exactly 2 * 1 on each step. Now the final perimeter is infinite but the area is not.

But you didn't specify this option over the other one. And, honestly, if we talk about putting rectangles in a sequence, I think it's more common to think of the rectangles as standing up side by side with their heights together as in the first option. For the second option I would describe the rectangles as having a fixed width of 1 and decreasing heights.

▲nhinck2 5 hours ago | parent | prev [-]

Gabriels Horn for another example.

Doesnt seem that uncommon.

▲JadeNB 4 hours ago | parent [-]

Gabriel's horn is the same phenomenon one dimension up: finite surface area but infinite volume.

▲eru 4 hours ago | parent [-]

You mixed it up. The horn has infinite surface area but finite volume.