5 ms·
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
by 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 agoi 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 agoI should perhaps have said "a (topological) sphere can be a knot". Corrected, thanks!
- jjgreen 5y agoYou want a list? https://mathoverflow.net/questions/180846/ https://mathoverflow.net/questions/180846/
- Jeff_Brown 5y agoaw hells yes
- datameta 5y agoPotentially a typo in there - one of the cases must be odd?
- Jeff_Brown 5y agoYou 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 agoMeasure 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-...