carltg_

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

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.