Wikipedia:Reference desk/Archives/Mathematics/2009 April 23

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April 23

In geometry, is a square with rounded corners still considered a square? If not, what is it called, mathematically?

Normally I can answer any question my daughter asked, but this one stumped me! —Preceding unsigned comment added by 216.61.187.254 (talk) 21:30, 23 April 2009 (UTC)

Might be Squircle. Zain Ebrahim (talk) 21:35, 23 April 2009 (UTC)

In practical use it's more probably a composition of line segments and circular arcs, though, which is simply called a "rounded square" in the same article. But the squircle is obviously more mathematically interesting. —JAO • T • C 21:41, 23 April 2009 (UTC)

Yes, right. Squircle is not what OP is looking for but OP maybe pleasantly surprised to find a closed match of rounded square defined actually by an algebraic equation. Squircle nowhere has straight edges, which OP's question seems to insist on. Though as a side note even square may not have straight edges in Non-Euclidean geometry. But, I hope we are dealing with Euclidean geometry here. - DSachan (talk) 21:52, 23 April 2009 (UTC)

Also rounded square, or smoothed square, or similar. The fact is that not every object in mathematics has a standard aknowledged name; just the most common and used ones (for instance: the trigonometric functions sin(x), sin(x)/cos(x) and 1/sin(x) all have special names, but sin(cos(x)) has none). If in a given context (a book, a paper, a theorem) one needs to use frequently something which is not otherwise so common to deserve a special name, one may just introduce a name to be used there. --pma (talk) 21:53, 23 April 2009 (UTC)

Quotient spaces, and cell complexes

I'm currently reading Hajime Sato's Algebraic Topology: An Intuitive Approach, but I'm having a bit of trouble understanding how to understand what particular quotient spaces look like (despite the book's name). I've taken an introductory course in Algebraic Topology, but was hoping for a bit more rigour regarding quotient spaces, but the book hasn't really helped. It's easy enough to visualise simple examples (such as a closed two-dimensional ball which has its boundary identified to a single point being homeomorphic to S²), but for more complicated examples, I'm just not comfortable stretching these shapes around in my head - I'd like some sort of rigorous method to determine what the quotient space looks like, but haven't been able to find one.

Another (somewhat related) issue I'm having with Sato is the idea of cell-complexes. I've come across these before too, but after describing how to construct general spaces using them, he bounds into examples without really explaining what to do. He, for example, say that the torus, T² can be written as $T^{2}=({\bar {e}}^{0}\cup _{h_{1}}({\bar {e}}_{1}^{1}\cup {\bar {e}}_{2}^{1}))\cup _{h_{2}}{\bar {e}}^{2}$ where the ${\bar {e}}^{i}$ are closed i-balls, and the $h_{i}$ are 'attaching maps', which take the boundary of one part of the complex onto a lower-dimensional part, and which he does not actually give the details of. (For instance, I'm interpreting h₁ to be $h_{1}:\delta ({\bar {e}}_{1}^{1}\cup {\bar {e}}_{2}^{1})\to {\bar {e}}^{0}$). I've covered similar ground to this before, having studied n-simplices, which are each obviously homeomorphic to ${\bar {e}}^{n}$ (and with boundary operators in place of the 'attaching maps', however the boundary operators were never used to glue), but I can't seem to get the two ideas to gel.

In my mind, the above example starts with two disjoint closed intervals, ${\bar {e}}_{1}^{1}$ and ${\bar {e}}_{2}^{1}$ whose boundaries (end points) are attached/glued by h₁ to a single point, ${\bar {e}}^{0}$, giving a sort of figure of eight shape. This is then glued by h₂ to the boundary of ${\bar {e}}^{2}$, S² (edit: obviously I meant S¹) ...which I again have no idea how to visualise, or see how this could possibly give a torus.

Sorry for this massive outpouring, I just really can't get my head around it. Thanks a lot! Icthyos (talk) 22:27, 23 April 2009 (UTC)

Don't apologize - we are here to help you. Topology is a subject on which there are too few a number of textbooks; at least half of which try to simplify the subject down. I have attempted to understand why this is the case (when I first saw the definition of a topology (a long time ago), I did not see the purpose of having such a complex set-theoretic definition, but after a week I got used to it and saw the definition more intuitively than ever - the definition of a topology is really ingenious and I respect Hausdorff for this). Quotient spaces are rigorously defined as follows: Let X be a topological space and ~ an equivalence relation on X. Let X^* be the set of all equivalence classes under this relation. Define U to be open in X^* iff the union of the equivalence classes that belong to U, is open as a subset of X. Then X^* is a topological space with this topology - it is called the quotient space of X by the equivalence relation ~ (verify that this is a topological space, if you have not done so already). Note that some authors prefer to define the quotient space using the quotient map (which is probably a less confusing way to define it), but I think this definition is alright too. Think about the following - it should improve your intuition of the concept:

a) Consider the natural projection $p:X\to X^{*}\,$. Is this map continuous? Is the inverse of this map continuous? Is this map surjective? Deduce some properties of X^*, given properties of X, using this map.

b) Suppose X is homogeneous (i.e the homeomorphism group acts transitively on X) (equivalently, if x is in X and y is in X, there exists a homeomorphism of X taking x to y). Under what conditions is X^* homogeneous?

c) Consider the circle with ~ defined by - x ~ y iff x is antipodal to y (i.e x = - y). Consider the (two-point) equivalence classes (actually viewed as a single entity)- one equivalence class ((1,0) and (-1,0)) essentially determines the x - axis. As you move clockwise around the circle, the equivalence classes are distinct until you have rotated 180 degrees. So essentially, to consider the quotient space, you have to examine the geometry of the equivalence classes - i.e points on the upper semicircle. Essentially, the equivalence classes "close" to (-1,0) are also close to (0,1) because both points are equivalent. What is the quotient space X^* homeomorphic to?

d) With a similar equivalence relation defined on the sphere, note that the resulting quotient space is a two-manifold. However, it is impossible to embed the resulting quotient space in R³. What is the smallest n, such that this quotient space can be embedded in Rⁿ? What is the smallest n for which the resulting quotient space can be embedded via a diffeomorphism in Rⁿ (define a smooth structure on the quotient space using the natural smooth structure on the sphere, and the projection (i.e quotient) map).

e) Define x ~ y on an arbitrary topological space iff there is a homeomorphism of that topological space carrying x to y. Find a space for which the resulting quotient space is the Sierpinski space. Is there a topological space for which this is homeomorphic to the countable discrete space?

f) Is the quotient space of a manifold, again a manifold? If not, under what conditions is it a manifold? Under what conditions does it have the same dimension (as a manifold) as the original manifold?

g) Let G act on a topological space X. Use this action to define a quotient space of X. Prove that the resulting quotient map to the quotient space, is a covering map, under certain conditions.

I think that these questions are useful to develop your intuition. As I don't know your exact level, I can't say whether these questions will be challenging for you or not. However, most should be relatively easy. If you have any other questions, or any difficulties, feel free to ask again. I will help you on your other question, if no-one else will do so, but I have got to go now. --PST 03:00, 24 April 2009 (UTC)

For the torus example in particular, I think you can consider this by looking at the fundamental polygon for the torus, which is realising the torus by the following construction . You can see that this has only one point ($e^{0}$), two 1-cells and one 2-cell. So you attach these together as the above diagram says, and you get the torus.

I don't think there really can be a general method for determining what the quotient space looks like, you need to consider what the attaching maps involved actually are and see if you can relate that to anything you know; you can realise many spaces by a similar construction as above, for example I remember a nice example of taking a cube and identifying opposite faces with a quarter-twist, you can put a cell structure on this quite easily by counting points, edges, faces and the whole volume, and you end up with some space with fundamental group $Q_{8}$, the quaternion group, which I thought was quite nifty. - Xedi^Talk 02:56, 24 April 2009 (UTC)

Thanks a lot for your responses, they've been a great help. Xedi: I see now how those attaching maps on those cells give the torus, I'm just a bit miffed that I couldn't build it from the ground up, without using that diagram. Thanks!

PST: I've not come across a lot of those definitions before. Ordinarily I'd dive right in, but it's exam crunch time at the moment, and I feel guilty when I wander away from curricular-based topics! It's definitely helped consolidate the ideas I have about quotient topologies, though. What you said about understanding the set theoretic definition of a topology struck a chord with me, though. As I assume is the usual way, I was first taught the subject from the point of view of metric spaces, and once comfortable with that, the notion of distance was stripped away, and we defined a topology to be a certain collection of subsets of a space that maintained the 'nice' properties of the open sets in a metric space. What I don't understand is, why do we define a topology to have those specific properties? (unions and finite intersections of open being open etc.) Just because they allow us to prove certain theorems? If the subject were taught without the lead in from metric spaces, how would one justify the definition of a topology? I suppose I just don't see how a 'structure' is defined on the space in such abstract terms - without some notion like distance, it's harder to visualise. Thanks again, Icthyos (talk) 20:28, 25 April 2009 (UTC)

If you feel confortable with metric spaces, and if you find them reasonable objects, you will sooner or later accept topological spaces as well. There are several reasons to abstract and consider topological spaces, even if one is only interested in metric spaces. One is that, even in a metric space, certain properties are stated and studied better in terms of open sets rather than distances (compactness, connectedness, continuity of functions... every topological property, in one word). Another reason is that, especially if you like more metric spaces, you would be happy to put a suitable distance in a set and make it a metric space (example: is a certain notion of convergence in functional analysis a convergence w.r.to a suitable distance?). As a matter of fact, the problem of metrizability was initially the big motivation to study general topological spaces --in the meanwhile people became acquainted with the generalized notion, and learnt to do without distances. A third reason that comes to my mind, is that certain natural and useful constructions that one may perform on metric spaces, to build other ones, in some case give rise naturally to a topological space, but maybe not a metric space (uncountable products, quotients, spaces of mappings, weak topologies in functional analysis...). So you are quite naturally led to go out of the class of metric spaces, in much the same way you are led to consider rational numbers when making operations with natural numbers. --pma (talk) 15:49, 26 April 2009 (UTC)

Let me note that a topological space is essentially a metric space - although not in the way that most people expect. For example, I can derive a metric out of any topological space using other mathematical disciplines, although this "metric" will not necessarily satisfy the axioms. But who cares? :) Mathematics is a general subject and although working with objects that satisfy so many similar properties to Euclidean space is easy, it is interesting to study objects which don't. After all, people like Euclidean space just because that is our universe. But what if we were living in another Banach space? Actually, throw away that idea, because Banach spaces are again too similar to Euclidean space. The point I am trying to make is that, in my view, a lot of mathematics is still to come - in particular, I strongly feel a new (crucially important) field of mathematics will be invented. After all, it was only in the last century that many famous areas of mathematics were fully axiomatized. However, it does not seem that this will occur in our lifetime. --PST 02:12, 27 April 2009 (UTC)

On a different note, I feel that exams are not a good thing. Rather than letting students pick up interest themselves, they are rather forced to "do it for good marks". This is not to say that all exams are bad. Nowadays there are take-home exams which allow one to actually think rather than do it in a few hours. However, mathematics is not about the time it takes to prove something, but rather the quality of that which you prove, no matter how long it takes. --PST 02:17, 27 April 2009 (UTC)

there are take-home exams which allow one to actually think...and to put a post here 82.84.117.68 (talk) 18:51, 28 April 2009 (UTC)