I hear this point parroted all of the time, but I think it is a misunderstanding and a poor visualization. Consider the same situation, but instead of focusing on the radius of the center sphere, focus on the distance between the spheres on the corners to the origin. For 1-dimension, these 'spheres' are unit intervals and so the distance is 1 (Central radius is 0). For 2-dimensions, these are circles at a distance of root(3) (Central radius is root(2)-1). 3-D: root(3) (Central radius is root(3)-1). Etc. So, it isn't the central circle getting more 'pointy' allowing the central radius to increase, but rather that the corner circles are getting further from the origin, allowing larger N-spheres (increasing proportional to the root of N). Thus, pointy is not the right way to conceptualize these spheres. For the more visual folk, I would recommend drawing this out and you can see this in action.
More clearly, if a sphere became 'spikey' then the distance on the surface of the spike should be further than a neighboring point, which is NOT the case.
Not trying to attack you, I just see this same point over and over and think that this warrants more thought
Yes that is true, but there are other ways to see spiky-like behavior.
First, the volume of spheres (or balls rather) in higher dimensions goes to zero as the dimension grows. Said another way, to keep unit volume on a ball you need to grow the radius more and more (which I interpret as spiky).
Second, the volume of spherical caps grows like ~exp(- d h^2 /2), in particular the caps lose volume fast in higher dimensions. To interpret this as "spikyness" I like to visualize it as two balls intersecting (which is just 2x the cap volume). If they are of the same radius, but their centers are just slightly off their intersection volume goes to zero quickly!
Not really, time doesn't make something difficult, just tedious. For the cat puzzle, smaller pieces doesn't really make a difference. Once you get a corner it is easy to line up and knock them down. So, just tedious
While 9kHz[3e-7/cm] (for an earth mode radio example) is lower than the plasma frequency of the ionosphere, 9MHz[3e-4/cm] (critical so that the ionosphere acts as a waveguide), would this be significantly lower, and hence have a high attenuation--a large imaginary component to the refractive index? If so, would this excite the ionosphere? Genuinely curious
The genus is insufficient to determine if an object is equivalent to the other. Orientability distinguishes the mobius strip and the torus, a torus is orientable whereas a mobius strip is not. Therefore, topologically speaking they are not equivalent.
you can make a mobius strip with paper. then get a pencil and try to orient it in the mobius strip. that is, make it normal to the paper then move it around. you will see that if you go though the strip and go back to the starting point the pencil will be in the other direction. thus, the orientation is not continuous so the surface is not orientable
