Say we're walking on the unit square and wish to make our way from (0, 0) to (1, 1). Obviously, we can go one unit up and one unit right, or, for the same distance, one unit right and one unit up. Similarly, we can go half a unit up and then half a unit right and then repeat these steps, or the mirrored alternative. And so on...  Each of the above options, as long as we allow ourselves to travel only finite distances and only in one axis direction at a time, will result in the same final distance, namely 2. Yet in the limit, these steps will have to collapse to $\sqrt{2}$, as we will by this point be traveling just on the diagonal.  First, are my above intuitions correct? And second, if so, how can we demonstrate the above to be the case?
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What is wrong with this proof?    Is $\pi=4?$
7009
I'm starting to read Steven Skiena's Algorithm Design Manual, and one of the examples discusses the nearest-neighbor solution to planing a route that visits each in a set of points and ends up back where it started.  The algorithm starts at some point, travels to the nearest neighboring point, then continues, traveling at each step to the nearest unvisited point, and finally returns to the initial point.  It's not the optimal solution (far from it), but the book mentions that      ... it looks at each pair of points (pi, pj) at most twice: once when adding pi to the tour, the other when adding pj   I can't figure out the scenario in which a particular (pi, pj) pair is considered more than once, since each step only considers points that haven't been visited yet.  When you're at pi, if you haven't yet visited pj you'll consider (pi, pj). But once you've left pi, when would you ever consider another pair that includes it?  Is the author referring to the case where pi is the first point visited and pj is the last before returning to complete the circuit? If so, does that really qualify as a case where the pair of points needs any consideration, since we have no choice but to return to the starting position since all the points in the set have been visited?
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Rephrasing     (one year later)    (original question is below)   Apparently the original question wasn't clear, or nobody knows an answer (or both).  So I will try to rephrase it.   Look at your favorite .  This question asks for a labeling system for the points.  Of course we could draw the grid on the Poincare disk, and use real numbers (the (x,y) coordinates of each point), but in practice that would raise all sorts of precision issues that could surely be avoided if one had a proper symbolic coordinate system.  You shouldn't need real numbers if you're staying on grid points!  Just like (2,3)+(0,1) is trivial to do by hand (or in your head) for the square grid, we'd like some form of symbolic coordinates (i.e. consisting of finite sequences of symbols) for a hyperbolic grid, in which simple operations on small grid vectors are similarly easy.  If you take some sequence of left and right turns in a hyperbolic grid city, what should you keep track of in your head so you can take the optimal route back to your starting point?  In a Euclidean square grid, it is {(int)X, (int)Y, (enum)direction you are facing}, and we all know how to use this.  Is there any hyperbolic grid where this question has a nice answer?    Original Question  If we consider the Euclidean plane, then a square grid works well with using two integers (the coordinates) as a representation for vectors that lie on the grid.  Does anybody know of a similarly nice representation for working in the hyperbolic plane?  Desirable features of any such system:  $-$ It should be easy to add two vectors.  $-$ It should be easy to see if two vectors are the same.  $-$ It should be easy to see if two vectors are close.  $-$ It should be easy to rotate a vector by an angle that is a symmetry of the grid.  $-$ Every point in the plane should be near some vector. (The grid shouldn't have large empty regions.)   Note that "addition" using continuous coordinates would most naturally be defined in terms of parallel transport, and it is neither commutative nor associative.  On a grid, however, the issue of orientation needs to be addressed differently, since parallel transport along arbitrary grid vectors will introduce rotations that are not a symmetry of the grid, so to remain on the grid, addition will need to be defined using something other than parallel transport.  Since hyperbolic grids have an exponential number of points (as a function of distance from the origin), our symbolic coordinates will need to have a length proportional to the length of the vector.  Don Hatch has a web page which gives some .  ("Easy" means an efficient algorithm that is doable by hand, like how the algorithms for addition and subtraction using the standard digit-sequence representation of integers can be used for the Euclidean square grid. Using hyperbolic functions of real numbers, always keeping track of enough digits to identify the nearest grid point, does not count as easy!)    Addendum: Many readers seem to be disturbed by a little voice saying "But vectors don't make sense in the hyperbolic plane!": What that voice is really saying is that the hyperbolic plane doesn't give you all the nice properties you are used to in Euclidean space or in vector spaces.  So if you define vectors as objects having specific properties you are used to (such as having a direction that is unaffected by parallel transport), then that definition may not correspond to anything in the hyperbolic world.  You could use the term isometry, but that is typically a fixed mapping from the space to itself, and in the hyperbolic plane the isometry corresponding to a vector would depend on the starting point. (However, isometries will be central to any solution, I assume.) And a geodesic is a fixed path within the space, no different from a hyperbolic line or segment.  The paper  offers one solution.  offer another.  I deliberately leave them undefined in this question, as I am looking for any workable system.
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I've very recently started to try to understand special relativity. I'm want to get a decent understanding of the twin paradox. I'll post what I've done so far and highlight what's gone wrong for me.  The situation is that Alice and Bob both begin at point $x_1$ in Alice's coordinate system $(x,t)$ (we have orientated the axis so that y and z will not matter). Bob then instantaneously moves off with velocity $v$ in the positive $x$ direction. Bob's coordinate system is now $(x',t')=(\gamma (x-vt), \gamma (t-v\frac{x}{c^2}))$. In Alice's coordinates, Bob reaches point $x_2$ after having moved distance $d$, then instantly turns around and travels back to Alice at $-v$. I want to show that Alice's space time interval $\Delta s_A^2$ is greater than Bob's $\Delta s_B^2$, as each path's space time interval is proportional to the proper time passed along that path.    According to Alice: $\Delta s_A^2=-c^2\Delta t^2 + \Delta x_A^2$ where $\Delta t=\dfrac{2d}{v}$ and $\Delta x_A=0$ as she didn't move, but the time elapsed is the time it took bob to travel the distance and then come back. Thus $\Delta s_A^2=-\dfrac{4c^2d^2}{v^2}$  Still according to Alice, Bob's $\Delta s_B^2=-c^2\Delta t^2+\Delta x_B^2$, where the change in time is the same, but now bob has moved distance $d$ twice, thus $\Delta x_B=2d$ ($\Delta x$ refers to total distance traveled rather than displacement which is 0 in this case). Now we have $\Delta s_B^2=-\dfrac{4c^2d^2}{v^2}+4d^2$.     The |size| of Bob's spacetime interval now definitely smaller than Alice's, and this would be all OK, except that when I do the calculations in Bob's frame, they don't agree. This is contradictory to the fact that $\Delta s^2$ is conserved under Lorentz transformations.    According to Bob: $\Delta {s'}_A^2=-c^2\Delta t'^2 + \Delta {x'}_A^2$ where $\Delta t'=\dfrac{2d'}{v}$. I'm not sure if I'm right in saying that $v$, the relative velocity of the two frames, is the only velocity upon which both Alice and Bob will agree on, other than the speed of light. Anyway, $d=\gamma d'$ as lengths contract by $\gamma$ i.e. $d'$ is smaller than $d$ by factor $\gamma$. Also, $\Delta {x'}_A=2d'$, so we have $\Delta {s'}_A^2=-\dfrac{4c^2d'^2}{v^2}+4d'^2=-\dfrac{4c^2d^2}{\gamma ^2 v^2}+\dfrac{4d^2}{\gamma ^2}=4d^2\dfrac{v^2-c^2}{\gamma^2 v^2}\neq \Delta s_A^2$  Although I haven't included them here, my calculations for $\Delta {s'}_B^2$ and $\Delta s_B^2$ agree. Sorry for the ultra-long post, but any help would we well appreciated!
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I'm a little confused by this question I've come across in do carmo while studying for my final. (Sect 3.2 #7) Show that if the mean curvature is zero at a nonplanar point, then this point has two orthogonal asymptotic directions. Can someone show me how to solve this? So obviously, $k_1+k_2=0$ but I don't know where to go from there. I'm also confused by the concept of asymptotic directions.
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Consider any integer, suppose a number 3. We can break any such integer down into a sequence of 1's. So, we can write: $$ 3\equiv1+1+1 $$ 3 represents what we get when we add 1 to 1 to another 1. Conceptually, what I am getting confused with, is the fact that notionally, at least according to this definition, there is one unit at the beginning, to which we add another, and yet a third. There seems to be a sequence here, which would imply there being an order. Am I missing something obvious?
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How many ways are there to stack coins on top of the other (2D stack) without any coin falling down ? Here's an example for $n=3$:  Now this is most likely just like the monotonic path of Catalan number with up and down instead of up and right but I just don't get why is that ? Well, any advice would be appreciated.
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Let $p &lt; q &lt; r &lt; s$ be real numbers. Let $l$ be the geodesic with endpoints at $p$ and $q$ and let $m$ be the geodesic with endpoints at $r$ and $s$. (a) Prove that there is a unique geodesic segment from $l$ to $m$ that is perpendicular to both. (We sometimes say $l$ and $m$ are ultraparallel since they are not only parallel but they do not share endpoints also.)  (b) Now suppose $p &lt; q &lt; r$ are real numbers and $l$ is a geodesic whose endpoints are $p$ and $q$ and $m$ is a geodesic whose endpoints are $q$ and $r$. Then $l$ and $m$ are parallel, but prove that there is not geodesic segment from $m$ to $l$ that is perpendicular to both. That is, unlike the ultraparallel case, there are no points on $l$ and $m$ that realize the shortest distance from $l$ to $m$.
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Possible Duplicate:          I was thinking on the following line of thoughts: $1 =  \sqrt{1} = \sqrt{-1 \cdot -1} = \sqrt{-1} \cdot \sqrt{-1} = i^2 = -1$  Of course this is not true, but I was wondering which step in this 'line of thoughts' is forbidden to make?   Thanks for the explanation.
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I've drawn two random points on a two dimensional plane. If the distance between these two points is one unit, can a line of π units be drawn using a compass and straight edge?  Or, is there a proof that shows this is impossible?
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Just wanted to share a nice problem.I have given my method below.More answers are welcome.  Consider a rectangular integral grid of size $m*n$.A person has to travel from one end say $(0,0)$ to the diagonally opposite end $(m,n)$.He moves one step at a time towards the east or towards the north(that is,never moves towards the west or south at any time).How many distinct paths exist from the point A to the point C ?
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Hello,  in the picture there is an equation I can not understand. What I do not understand is how to get to the first step. I have been thinking and thinking but can not find an explanation. Could somebody please explain? Thanks.
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I don't know exactly how to describe it, but in a programmatic way I want to spiral outward from coordinates 0,0 to infinity (for practical purposes, though really I only need to go to about +/-100,000:+/-100,000)  So if this is my grid:  [-1, 1][ 0, 1][ 1, 1][ 2, 1] [-1, 0][ 0, 0][ 1, 0][ 2, 0] [-1,-1][ 0,-1][ 1,-1][ 2,-1]   I want to spiral in an order sort of like:  [7][8][9][10] [6][1][2][11] [5][4][3][12] etc...   Is there a formula or method of doing this?
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I came across an interesting puzzle:     You are climbing a stair case. It takes $n$ steps to reach to the top.   Each time you can either climb $1$ or $2$ steps. In how many distinct   ways can you climb to the top?   Is there a closed-form solution to the problem? One can compute it by creating a 'tree' of possibilities of each step. That is, I can either take 1 or 2 steps at each stage and terminate a branch once it sums to $n$. But this is would get really unwieldy very quickly since the maximum number of nodes in a binary tree is $2^{n+1}-1$, i.e., exponential. Is there an easier way to solving this puzzle?
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Here is the proof:     But I do not understand the following:  1-why the function needed to be bilinear to use the universal property?  2- what is he doing starting from the paragraph that starts with the statement "In the other direction ...." ?
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I read a little bit about , but I imagine the problem changes drastically when trying to fill a sphere with unit cubes. For example, how many unit cubes could one fit into a sphere of radius five? Clearly the upper limit would be $\left \lfloor \frac{4}{3}\pi \cdot 5^3 \right \rfloor$, but I imagine there will be plenty more empty space than just a small fraction of a cube.
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Just as background, I should say I am a mathematics grad student who is trying to learn some physics. I've been reading "The Theoretical Minimum" by Susskind and Hrabovsky and on page 134, they introduce  transformations. Here's the first example they use:  Consider a particle moving in the x,y plane under the influence of a potential, $V$, which depends only on the radius, with Lagrangian:   $L=\frac{m}{2}(\dot{x}^2+\dot{y}^2)-V(x^2+y^2)$  This is clearly invariant under rotations:  $x \rightarrow x\cos \theta + y\sin \theta$  $y \rightarrow -x\sin \theta + y\cos \theta$  All well and good. Now they say "consider what happens ... when the angle $\theta$ is replaced by an infinitesimal angle $\delta$." Already I could say "What the heck is $\delta$ really?", but I'm willing to play along with my intuition. Since $\delta$ is infinitesimal, we work to first order and say   $\cos \delta=1$ and $\sin \delta= \delta$.  Plugging this into our rotation formulas above, we obtain:  $x \rightarrow x+y\delta$  $y \rightarrow y-x\delta$  By differentiating, we see that:  $\dot{x} \rightarrow \dot{x}+\dot{y}\delta$  $\dot{y} \rightarrow \dot{y}-\dot{x}\delta$  Plugging these into the Lagrangian and ignoring terms higher than first order, we see that the Lagrangian is invariant under this transformation.  My main problem with all of this is that I don't understand what the physical nature of an infinitesimal transformation actually is. All I got from the above was that if you do this formal calculation by following rules like "only work to first order in $\delta$," then the Lagrangian is invariant. This is in contrast to the case where we have an actual transformation, like a rotation, where there is no question about what is going on physically.  I would also like to know how all of this relates to rigorous mathematics. In mathematics, I can't recall ever using infinitesimals in an argument or calculation, so it would be useful if there was some way of formulating the above in terms of limits/ derivatives/ differential forms (for example). I sense a connection to Lie Algebras, as the infinitesimal version of the rotation is $(I+A)$ where $I$ is the identity matrix and $A$ is an element of the Lie Algebra of $SO(2)$.   Here are some questions whose answers I believe may be useful to me (feel free to answer some or all):  -What is an infinitesimal quantity like $\delta$ to the physicist?   -Why do physicists argue using infinitesimals rather than "standard" calculus?  -What is the physical meaning of infinitesimal transformation? How does it relate to Lie Algebras?   -Is there a rigorous theoretical apparatus for justifying the computations shown above?  -What is meant by the Lagrangian being invariant under infinitesimal transformations?  If any of the questions seem too vague, please say so. Thanks in advance for your insights!