6 ms·

Or even just a convergent infinite series. Infinitely many positive numbers that sum to a finite number.

reagency 11y ago
Or in general, any infinite size n-dimensional structure can be embedded in some finite size n+1 dimensional structure.
wz1000 11y ago
I don't think that's paradoxical. We deal with convergent infinite series all the time in everyday life. 10/3(3.333...) is 3 + 3/10 + 3/100/ + 3/1000 + 3/10000... π(3.1415...) is 3 + 1/10 + 4/100 + 1/1000 + 5/10000... Its quite easy to see that both of these series will be finite, and many numbers, lets say 4 or 3 + 5/10 will be greater than either of them.
thaumasiotes 11y ago
People don't tend to find that unintuitive. The Greeks had a nice geometric example in a square of side length 1: ┌┬┬─┬───┬───────┐ ├┘│3│ │ │ ├─┘-│ │ │ │ 64│ │ │ ├───┘ │ │ │ │ │ │ 3/16 │ │ │ │ │ ├───────┘ │ │ │ │ │ │ │ │ 3/4 │ │ │ │ │ │ │ └───────────────┘ 3/4 + 3/16 + 3/64 + 3/256 + ... is easy to visualize as successively filling in three quarters of an ever-smaller residual square. Intuitively, no matter how finely you detail it, you're never going to stop fitting inside the original area-1 square. edit: better text art
- lmm 11y ago
  Now squeeze the top-left piece so it's half as wide but twice as tall (i.e. the same height as the full rectangle), and do this recursively. Same area, right? Then stack the L shapes on top of each other. Then you have one of these horns.
  - thaumasiotes 11y ago
    Huh? The horn is a three-dimensional object; the square exists in 2-space. You can't make the horn from pieces of the square even if you allow deformation.
raldi 11y ago
This sum's finiteness seems pretty intuitive: 9 + 0.9 + 0.09 + 0.09 + 0.009...
- sesquipedalian 11y ago
  Good example. Minor nitpick, you listed 0.09 twice.

Or in general, any infinite size n-dimensional structure can be embedded in some finite size n+1 dimensional structure.

I don't think that's paradoxical. We deal with convergent infinite series all the time in everyday life. 10/3(3.333...) is 3 + 3/10 + 3/100/ + 3/1000 + 3/10000... π(3.1415...) is 3 + 1/10 + 4/100 + 1/1000 + 5/10000... Its quite easy to see that both of these series will be finite, and many numbers, lets say 4 or 3 + 5/10 will be greater than either of them.

thaumasiotes 11y ago

People don't tend to find that unintuitive. The Greeks had a nice geometric example in a square of side length 1: ┌┬┬─┬───┬───────┐ ├┘│3│ │ │ ├─┘-│ │ │ │ 64│ │ │ ├───┘ │ │ │ │ │ │ 3/16 │ │ │ │ │ ├───────┘ │ │ │ │ │ │ │ │ 3/4 │ │ │ │ │ │ │ └───────────────┘ 3/4 + 3/16 + 3/64 + 3/256 + ... is easy to visualize as successively filling in three quarters of an ever-smaller residual square. Intuitively, no matter how finely you detail it, you're never going to stop fitting inside the original area-1 square. edit: better text art

lmm 11y ago
Now squeeze the top-left piece so it's half as wide but twice as tall (i.e. the same height as the full rectangle), and do this recursively. Same area, right? Then stack the L shapes on top of each other. Then you have one of these horns.
- thaumasiotes 11y ago
  Huh? The horn is a three-dimensional object; the square exists in 2-space. You can't make the horn from pieces of the square even if you allow deformation.

raldi 11y ago

This sum's finiteness seems pretty intuitive: 9 + 0.9 + 0.09 + 0.09 + 0.009...

sesquipedalian 11y ago
Good example. Minor nitpick, you listed 0.09 twice.