There are three ways (three differenent axis about which) I can flip my cell phone - over the front (like a frontflip), about the center (like a disc of pizza dough being spun by a baker), and over sideways (like a dog rolling over, or like a pizza roller rotating as it rolls over the dough). Of all these ways to rotate, I've found that the first way, the frontflip-like rotation, is almost always unstable. It doesn't only rotate about that axis, even when I try really hard to make it do so. About the other two axes, however, it flips/rotates about only one axis, just as I intend it to.    Why does this happen?
I've noticed something curious about the rotation of a rectangular prism. If I take a box with height $\neq$ width $\neq$ depth and flip it into the air around different axes of rotation, some motions seem more stable than others. The 3 axes which best illustrate what I mean are:  (1) Through the centre of mass, parallel to the longest box edge.  (2) Through the centre of mass, parallel to the shortest box edge.  (3) Through the centre of mass, parallel to the remaining box edge.  It's "easy" to get the box to rotate cleanly around (1) and (2), but flipping the box around (3) usually results in extra twisting besides the rotation around (3) that I'm trying to achieve (obviously a "perfect" flip on my part would avoid this twisting, which is why I call it an instability). If you're not quite sure what I'm talking about, grab a box or book with 3 different side lengths and try it out (but careful not to break anything!).  What's special about axis (3)?   Image taken from .
I'm a computer student, learning math just for fun. Today I was  that I found something strange! I noticed that that wired function ${x^{x^{\cdot^{\cdot^{x}}}}}$ in zero, seems to converge to 1 when there are even powers and to 0 when there are odd powers! Then I attempt to prove it but I failed.  Then I did a little research and I found the  article on Wikipedia. This article says that my guess was right but without any proof. So I'm here to ask you about it.     If we define ${x^{x^{\cdot^{\cdot^{x}}}}}$ as ${^{n}x} :=\begin{cases} 1 &amp;\text{if }n=0 \\ x^{\left(^{(n-1)}x\right)} &amp;\text{if }n&gt;0 \end{cases}$      then prove:      $$\lim_{x\rightarrow0} {}^{n}x = \begin{cases}   1, &amp; n \text{ even} \\  0, &amp; n \text{ odd} \end{cases}$$ Correct way of declaring pointer variables in C/C++ Can I open Microsoft Word and Excel files? Can I open Microsoft Word and Excel files with Ubuntu? Add an image directly from the clipboard I've tried to find other questions covering this, but (surprisingly) I did not find any.  When I want to add a screenshot to a post, as far I know the only way to do it is to save it as a file and then upload the file, using Stack Overflow's image hosting. OK, much better than finding a hosting site yourself, but sometimes I wish I could just paste the image from the clipboard.  It's not as straightforward as it may seem, I know. , and it needs a Java applet to get it done. If humans were able to catch all sun energy reaching the earth for their use, will the climate change? I guess that energy will be used up and, at the end, will contribute to heat the earth, so I see no big differences...  please explain your point of view. Limit with inverse trignometry We have to find the value of $$\lim_{n\to \infty}\sum_{k=2}^n\arccos\left(\frac{1+\sqrt{(k-1)k(k+1)(k+2)}}{k(k+1)}\right)$$  I tried it a lot but could not find any start.   Please help me in this . Evaluating the sum after reordering an infinite series $1-\frac {1} {2}+\frac {1} {3}-\frac {1} {4}+\ldots $ Why do liquids boil when their vapor pressure equals the ambient pressure? Given that the boiling point of a liquid is the temperature at which the vapor pressure is equal to the ambient (surrounding) pressure, what significance does a liquid's vapor pressure have in the formation of bubbles that happens at and above the boiling point?    The definition of boiling point seems to imply that the pressure inside of the bubbles must be at least as great as the liquid's vapor pressure in order to balance the outside pressure, but is there any particular reason why the pressure inside of the bubbles is related to the vapor pressure?    The vapor pressure seems to be a measurement describing the tendency of the molecules to escape from the surface of the liquid, but I don't see how that relates to bubble formation within the liquid.  This question has bothered me for a while, so any help would be much appreciated. How to send mail from the command line? How to send mail from the command line? Procedure for reducing the degrees of freedom of an arbitrary system Nonlinear function continuous but not bounded I would like an example of a map $f:H\rightarrow R$, where $H$ is a (infinite dimensional) Hilbert space, and $R$ is the real numbers, such that $f$ is continuous, but $f$ is not bounded on the close unit ball $\{ x\in H : \|x\| \leq 1\}$.  Actually, $H$ could be replaced by any Banach space (but not just a normed space-- that's too easy).  My motivation is that if $f$ is linear, this is impossible; but I have next to no intuition about non-linear functions.  Edit: Here's an example for $c_0$ which is even differentiable (disclaimer: I found it here: ).  Define $f:c_0\rightarrow F$ (where F is your field, real or complex) by $$ f(x) = \sum_{n=1}^\infty x_n^n \qquad (x=(x_n)). $$  You can estimate the sum by a geometric progression, so it does converge.  A bit of checking shows that f is Frechet differentible (so certainly continuous).  But $f(1,1,\cdots,1,0,\cdots)=n$ (if there are $n$ ones) so $f$ is not bounded on the closed unit ball.  What I don't immediately see is how to adapt this to $\ell^2$, say. Within 300 km of Tokyo: Where for flora, landscape, nature? I plan to visit Japan for 10 days and apportion 5 days to Tokyo. For the remainder, I want to escape the megalopolis and enjoy picturesque backwoods or countryside. Please recommend? Criteria:   No dangerous animals or untamed boondocks. No  or snakes. I won't drive, and must rely on public transport and walking less than 5 km in total.  No shrines, temples or any religious edifice. OPTIONAL: I fancy  , the more  the better.   This instantiates, doesn't duplicate, . Bias of maximum likelihood estimators for logistic regression How to clear bash history completely? Problem with camera orientation in motion tracking sorry for my English, I write from Italy. I'll try to explain my problem. I'm tracking a footage with a solver error 0.2973. When I try to set the floor, the orientation of the camera is wrong (see screenshot).  So I tried to manually orient the camera, but even if well oriented respect to the xyz axes, geometry is incorrect as you can see from the screenshots below (the selected marker appears to be on the same plane but it is lower).  The video was recorded by a roof of a house so much higher than the road. It seems that Blender is not able to recognize this height and set the floor in a wrong point of view. Can someone help me solve this problem? Thank you In either case, one coin flip resulted in a head and the other resulted in a tail. Why is {H,T} a different outcome than {T,H}? Is this simply how we've defined an "outcome" in probability?   My main problem with {H,T} being a different outcome than {T,H} is that we apply binomial coefficients (i.e. we count subsets of sets) in some common probability problems. But if we take {H,T} and {T,H} to be different outcomes, then our "sets" are ordered, but sets are by definition unordered...   I feel as though the fact that I'm confused about something so basic means that I am missing something fundamental. Any help or insight whatsoever is greatly appreciated! Question Staging SE Been a few blogs on , and I get the point. However, would a staging SE help solve some of these issues like community overlap. I understand , but how can someone not yet a part of a community know to do so. Some of the point of SO is that you often don't know what to ask exactly, and we help clear that up. So, I wonder, rather than &quot;playing hot potato&quot; would it be useful to have a site that's simply for temporarily holding questions and such.  It would be the alternative to migrating to another site where you dump all responsibility on that site. This area, all communities are responsible for. It would be a good place to ask a question if you don't know where to ask. The blog I linked talks about phrasing to a specific audience. For someone that doesn't know, the audience is better at selecting which questions are directed at them. As SE grows, the amount of communities will mean more overlap. Right now it may be reasonable to expect people to be able to predict the best community, but will it always be that way?  Some restrictions:  It would be accessible by all communities. The question could be edited, but not answered. The question could be exposed to more easily tracked/understood/etc. automated janitorial work.  Just a helpful thought. How can I extract text from images?  I am not talking about scanned files, but garden variety images, such as when you take a high-def picture of a blackboard at class, and it is nicely handwritten; or when you photograph a page from a recipe book and want the recipe in text format.  Any free and open software for that?  I tried tesseract, and the results were awful. Recursively rename files and directories Let $K=F(x)$ be the rational function field over a field $F$ of characteristic $0$, let $L_1=F(x^2)$, and $L_2=F(x^2+x)$. How to show that $L_1\cap L_2 = F$?