Awesome video clip by the way. Straightforward idea and an extremely very well presented. Ns really preferred it and I don't think the of every manim video clip that comes out. In reality don't think it an extremely much.
You are watching: Shape with infinite perimeter and finite area
Also, you asked a concern that was exciting to try by myself and let a minute to ponder in the video, however you spoiled the answer in the thumbnail!
Good question around not subtracting the overlap. Since the perimeter converges to infinity, but the subtracted portion converges to infinity slow (since it’s just a 4th the size), that won’t rotate out to matter.
If you desire to read an ext about the unlimited perimeter, finite area phenomenon, the quintessential instance is Koch’s snowflake. For three dimensional geometry, with volume instead of area, the most typical example would certainly be Gabriel’s horn.
Good question. Reasoning of the perimeter as just the border (which doesn’t include where the squares overlap), you have actually to add the two sides the the square top top the outside and also the tiny bit of the bottom next that’s exposed.
Technically that would be 2(1/n) + (1/n - 1/(n+1)), however that is still less than 3(1/n). There’s additionally the height of the an initial square which ns didn’t mention. The would have been good to incorporate this, so many thanks for bringing it up.
And you’re right, i did coincidentally spoil that in the thumbnail. Yet as others stated there space many possible shapes, so nothing let that avoid you!
I evaluate the feedback world gave me top top my very first video utilizing manim:
I tried the end a voiceover this time because that a easier topic at a slow pace, interested in listening what I have the right to work top top next!
doesn't it strategy a specific value? and also since it's an limitless sum, lot like exactly how we deserve to say 1 + 1/2 + 1/4 + 1/8 +.... Ideologies 2 and so it equals 2, we deserve to say this ideologies a certain value?
The perimeters room what’s dubbed an harmonic series. 1 + 1/2 + 1/3 + (...). The harmonic series diverges - it doesn’t approach a certain value
Why carry out multiple every sidelength through 3? ns don't yes, really see how this works and also in mine mind the makes an ext sense to simply do 4x1 and also then include up the harmonic collection multiplied by 2.
If girlfriend squish all the other squares versus the bottom the the top 1 you deserve to see the by including squares you're only adding side length however the top and bottom perimeter stay equal.
It doesn't readjust much however it simply makes more sense to me.
seems like we're in some sense cheating in some way. I typical "cheating" in a an extremely specific method here:
if we're including an infinite summation of confident "numbers", but only with a total sum that "2", then we're putting a totality lot (an boundless amount) of "numbers" into the sum that execute not run like the rest of the number we typically think about. That's the only method that could work. We're slicing points so thin toward the finish that whatever those late elements are, we deserve to no much longer think about them the same way we think around a "number" in the feeling of integers or reals, ie day-to-day numbers, due to the fact that they no longer meaningfully add to the total. It's not simply that they're small, they're all meaninglessly tiny after whatever suggest you watch close to the total.
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it seems prefer a cheat to usage the same idea that a number for those things, they don't have the very same ontologic commitment the we intend from everyday numbers.
personally, ns think we're bad kidding ourselves come treat separation together zero-cost (free or axiomatic in part sense) because that both maths and physics,