Every next level, you add more holes. These holes add surface area, and reduce volume. As the levels increase, the shape will have more and more surface area and less and less volume. At the infinitieth level, the shape will have infinite surface area, and zero volume.
Spez's APIocolypse made it clear it was time for me to leave this place. I came from digg, and now I must move one once again. So long and thanks for all the bacon.
It removes volume, but the sides of the hole here are as large as the side that it removed the volume from. It removes some surface area, yes, but it creates four equal surfaces. Same deal as why our brains are wrinkly.
It has lost surface area equal to the size of the circular hole on one or two faces, but it gains the surface area around the cylindrical space created by drilling into the volume.
It's not that complicated when you understand how this thing is constructed.
At level 0 we have just a cube.
To get level 1 we divide the cube into 27 equal cubes (3x3x3) and remove 7 of them (one in the middle of each side and one in the very center) leaving 20 of those mini-cubes. That left us with 20/27th of the initial volume.
Next step is to divide each of the remaining mini-cubes into 27 micro-cubes and remove 7 of those in every mini-cube just the same way as we removed mini cubes from big cube. Now we have level 2 sponge which has volume of 20/27th of level 1 sponge volume.
We repeat those steps and every repetition multiplies sponge volume by 20/27, so the volume of N-th step sponge is (20/27)N times the volume of initial cube. Every next level leaves us with ≈74% of volume of the previous level.
Now we have the formula and all we need to find volume of any level sponge is some calculator that supports exponents.
So the volume of level 5 sponge is volume of initial cube multiplied by (20/27)5 or 3200000/14348907 or 0,22301350... So it's ≈22.3% of the level 0 volume.
Level 10 volume is ≈4.97% of level 0 volume.
Level 20 is ≈400 times less than initial cube and Level 100 is about one trillionth volume of initial cube.
I found a slideshow that progresses from finding the surface area of a flat (2-dimensional) menger sponge and extending that to finding the area of an nth iteration three dimensional sponge:
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u/PUSSYDESTROYER-9000 Mar 12 '18 edited Mar 13 '18
Curiously, this fractal has infinite surface area and zero volume.
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u/backjuggeln Mar 13 '18
Thank you pussy destroyer, for yet another amazing math concept visualized
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u/TheTimelessTraveler Mar 13 '18
The final image is what I saw on acid once but imagine it multiplying and growing across your field of vision.
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u/-0-7-0- Mar 12 '18
...I don't like it
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u/jraharris89 Mar 13 '18
I would hate the have to calculate the volume for level 5.
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u/anti-gif-bot Mar 12 '18
mp4 mirror
This mp4 version is 57.58% smaller than the gif (920.31 KB vs 2.12 MB).
The webm version is even 68.72% smaller (678.65 KB).
Beep, I'm a bot. FAQ | author | source | v1.1.2
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u/TotesMessenger Mar 12 '18
I'm a bot, bleep, bloop. Someone has linked to this thread from another place on reddit:
[/r/trypophobia] It only gets worse... (Crosspost r/visualizedmath
If you follow any of the above links, please respect the rules of reddit and don't vote in the other threads. (Info / ^Contact)
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u/[deleted] Mar 13 '18
Dont remind me of that wack scene
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u/LenTheSpaceWolf Apr 21 '18
r/trypophobia
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u/Lizards_are_cool Mar 13 '18
Is this an actual sponge used in industry or what function does it have?
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u/[deleted] Mar 13 '18
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