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I thought that the concept of "gemoetric shape" is clear enough - squares, ellipses, triangles, you know. But then I found several papers, such as this one, which define "shape" as "an un-parameterized immersed sub-manifold". I have read about immersion and submanifolds, but I still don't understand, how is this definition related to the intuitive meaning of a "geometric shape"? And why is it formally defined in such a way?

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  • $\begingroup$ It is not entirely relevant, and yet it may be precisely what you are after: have you ever read the introduction to Yaglom's Geometric Transformations? It is basically a short essay on "What is Geometry?". The moral is, basically, that formalizing our intuitive notion of "shape" is actually more complicated than one might imagine. $\endgroup$
    – user1729
    Aug 28 '14 at 9:43
  • $\begingroup$ Thanks, I will take a look. $\endgroup$ Aug 28 '14 at 9:49
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First, note that in order to talk about a submanifold, we must already have in hand a background manifold in which the submanifolds live, i.e., in which they are subsets---your examples are all immersed $1$-manifolds in $\mathbb{R}^2$ (just the usual plane of high school plane geometry).

Now, the most important feature of manifolds is that they be "locally Euclidean", that is that around a point they look like Euclidean space, $\mathbb{R}^n$---the upshot of this condition is that we for many questions about Euclidean space we can ask the same or at least analogous questions about manifolds. One practical consequences of this definition is that we can always write down coordinates for a manifold---this is roughly idea of using latitude and longitude to specify places on a sphere, and what gives us a consistent way of producing maps of the world, despite that the world is round and maps are (usually) flat. This also means that in a sense, each point looks like every other point in a sense. (For $1$-dimensional manifolds. this more or less means we can trace out the shape by drawing a single curve, maybe closing up but definitely not crossing itself.)

One can write describe geometric objects that don't fit this description but they're always "irregular" in some way. For example, if we glue two circles at a point, then we've made a space that has a point (where the circles meet) that are different from all other points on the shape, a notion which we can make formal.

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  • $\begingroup$ Thanks, your explanation makes it more understandable. But what about polygons - aren't the vertices different than the other points in that they are not smooth? $\endgroup$ Aug 28 '14 at 10:32
  • $\begingroup$ Erel, that's a good question---the answer is that from the point of view of manifolds (without requiring any more structure, so these are "topological manifolds" in technical language), there's not difference between corners and other points. This is obviously not very satisfying for some purposes, because it means that from this point of view (i.e., the viewpoint of topology), all polygons are the same! We can rectify this situation by demanding that we /do/ keep track of this kind of information. There are various formal ways to do so, and they give us new notions of shape. $\endgroup$ Aug 28 '14 at 11:59
  • $\begingroup$ One notion isn't intrinsically better than another, and different notions lead to different kinds of questions we can ask about shapes. For example, we can ask whether the sphere and torus (doughnut shape) are the same from the p.o.v. of topology. It turns out that they aren't, and this sort of question lead to the development of ideas together called "algebraic topology" that tries to answer this sort of question. $\endgroup$ Aug 28 '14 at 12:03
  • $\begingroup$ If I understand correctly, a 1-manifold is only the line that surrounds the shape (i.e. the circle and not the disc). Is there a standard way to describe "the area that lies inside the 1-manifold"? $\endgroup$ Aug 28 '14 at 12:54
  • $\begingroup$ Yes, that's right, though let me point out that a line (or generally, curve that doesn't close back in on itself) is also a manifold, but these are essentially the only $1$-manifolds. Anyway, yes, the the area inside a "circle" (polygon, whatever) in the plane is an "open" $2$-manifold. If you want to include both the boundary and the inside together, you have a "$2$-manifold with boundary"---this is more or less what it sounds like, and note that you can distinguish boundary points and "interior" points of such shapes! $\endgroup$ Aug 28 '14 at 13:06

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