3 ms·

It is disturbing to find one's intuition failing in higher dimensions.

malux85 5y ago
exciting*
Jeff_Brown 5y ago
Yes! Let's list more! I've got a couple, maybe (depends on your intuition i guess): In 4d a (topological) sphere can be knotted. Hyugens's principle: When a wave is created in a field in N-dimensional space, if N is even, it will disturb an ever-expanding region forever (think a pebble hitting a pond's surface) whereas if N is odd the wavefront will propagate forever but leave no disturbances in its wake (think of a flashbulb, or of someone shouting in an infinite space full of air but no solids to echo off of).
- smegsicle 5y ago
  i can see why you can't tie a circle in a knot, but why can't you tie a regular sphere into a knot?
  - Jeff_Brown 5y ago
    I should perhaps have said "a (topological) sphere can be a knot". Corrected, thanks!
- jjgreen 5y ago
  You want a list? https://mathoverflow.net/questions/180846/ https://mathoverflow.net/questions/180846/
  - Jeff_Brown 5y ago
    aw hells yes
- datameta 5y ago
  Potentially a typo in there - one of the cases must be odd?
  - Jeff_Brown 5y ago
    You were right, it's fixed now, thanks!
- contravariant 5y ago
  > In 4d you can tie a sphere in a knot. Arguably you can do that in 3D, if you accept the horned sphere as a knot. Though I suppose that does raise the question of what you are willing to call a sphere. Regardless it's an embedding of a sphere that cannot be deformed into a unit sphere so I think the analogy holds.
- feoren 5y ago
  Measure a group of humans on N traits and take the individual average of each trait. For surprisingly small N (think 10-ish, but obviously depending on your group size), it's highly likely that no human in your group (or even in existence) falls within 10% of the average in every trait. This is roughly equivalent to the statement that less and less of the volume of an N-sphere is near the center as N increases. Sometimes called "the flaw of averages". Of course I learned about this from another HN post recently: https://www.thestar.com/news/insight/2016/01/16/when-us-air-force-discovered-the-flaw-of-averages.html https://www.thestar.com/news/insight/2016/01/16/when-us-air-...
nabla9 5y ago
It fails in multiple ways. For example the volume (hypervolume) gets concentrated close to the surface of the sphere when dimension grows. For example, if you have symmetric multidimensional probability distributions around the zero it becomes weird.
- kgwgk 5y ago
  What does become weird?
  - nabla9 5y ago
    Consider normal distribution. n-dimensional vectors with independent N(0,1) with expectation 0. When n increases the mass of the probabilities are around a sphere of radius n^0.5, nowhere near the origin.
    - kgwgk 5y ago
      True, a huge volume with low density will contain more mass than a small volume with large density. But we can say that it gets “concentrated” far from the origin only in the same sense that Japan’s population is “concentrated” outside of Tokio…

exciting*

Yes! Let's list more! I've got a couple, maybe (depends on your intuition i guess): In 4d a (topological) sphere can be knotted. Hyugens's principle: When a wave is created in a field in N-dimensional space, if N is even, it will disturb an ever-expanding region forever (think a pebble hitting a pond's surface) whereas if N is odd the wavefront will propagate forever but leave no disturbances in its wake (think of a flashbulb, or of someone shouting in an infinite space full of air but no solids to echo off of).

smegsicle 5y ago
i can see why you can't tie a circle in a knot, but why can't you tie a regular sphere into a knot?
- Jeff_Brown 5y ago
  I should perhaps have said "a (topological) sphere can be a knot". Corrected, thanks!
jjgreen 5y ago
You want a list? https://mathoverflow.net/questions/180846/ https://mathoverflow.net/questions/180846/
- Jeff_Brown 5y ago
  aw hells yes
datameta 5y ago
Potentially a typo in there - one of the cases must be odd?
- Jeff_Brown 5y ago
  You were right, it's fixed now, thanks!
contravariant 5y ago
> In 4d you can tie a sphere in a knot. Arguably you can do that in 3D, if you accept the horned sphere as a knot. Though I suppose that does raise the question of what you are willing to call a sphere. Regardless it's an embedding of a sphere that cannot be deformed into a unit sphere so I think the analogy holds.
feoren 5y ago
Measure a group of humans on N traits and take the individual average of each trait. For surprisingly small N (think 10-ish, but obviously depending on your group size), it's highly likely that no human in your group (or even in existence) falls within 10% of the average in every trait. This is roughly equivalent to the statement that less and less of the volume of an N-sphere is near the center as N increases. Sometimes called "the flaw of averages". Of course I learned about this from another HN post recently: https://www.thestar.com/news/insight/2016/01/16/when-us-air-force-discovered-the-flaw-of-averages.html https://www.thestar.com/news/insight/2016/01/16/when-us-air-...

nabla9 5y ago

It fails in multiple ways. For example the volume (hypervolume) gets concentrated close to the surface of the sphere when dimension grows. For example, if you have symmetric multidimensional probability distributions around the zero it becomes weird.

kgwgk 5y ago
What does become weird?
- nabla9 5y ago
  Consider normal distribution. n-dimensional vectors with independent N(0,1) with expectation 0. When n increases the mass of the probabilities are around a sphere of radius n^0.5, nowhere near the origin.
  - kgwgk 5y ago
    True, a huge volume with low density will contain more mass than a small volume with large density. But we can say that it gets “concentrated” far from the origin only in the same sense that Japan’s population is “concentrated” outside of Tokio…