|Feb10-12, 01:56 AM||#1|
Assuming Complex Shape a Simple Shape. What is this called?
What topic in mathematics would something fall under if you assumed a complex shape to be a more simple shape? Such as assuming a recliner is actually a cube, or a tree it actually a cone or prism, or a tire is actually a cylinder? Would this be topology, or something else? Simple geometry?
|Feb10-12, 02:36 AM||#2|
Hey chiako and welcome to the forums.
This is essentially what topology does. Topology allows you to describe continuity in a way that deals with deformations of an object or representation.
So yes when you have a situation where you deform something but can't change the topology you can't say turn a donut into a ball because that would mean getting rid of the hole. This isn't really a rigorous explanation but I think you get the idea.
Again with something like contuinity, if you deform a continuous thing without change its topology it should stay continuous. There are many ways of describing this concept in a variety of contexts.
Also there are different kinds of topologies applied to a variety of fields which have their own ideas and nonclamenture.
|Feb10-12, 10:52 AM||#3|
I would call these "simplifying assumptions" rather than any kind of well-defined mathematical notion, since it seems like you are wanting the approximating shapes to have some geometric relationship to the things they approximate. There are lots of different mathematical notions related to "shape" (or, more typically, we think about different "spaces"), and they're characterized by the set of properties we choose to care about. For example, metric spaces care about distances, topological spaces care about continuity, Euclidean space cares about angles and distances (what non-math/physics folks have in mind when they think of "geometry"), projective space has its own set of invariant quantities that aren't so intuitive, and so on.
So in a topological space, we "don't care about distance", or more accurately, distance just doesn't exist there. The usual example is that you can turn a coffee mug into a donut, but it's more precise to say that they were already the same thing to begin with--the things that let us distinguish coffee mugs from donuts have to do with thickness, angles, concavity, and so forth, and these concepts just don't have any definition in a topological space. But if I take that donut and tear it into two pieces, then I can tell that I did something, because topological space does care about connectedness. (I can also tell that the pieces themselves are not donuts, using some other topological tools.)
So, back to your particular case: It seems like you are essentially "caring about" all the same things for your approximating objects (sizes, angles, overall shape) as you do for the original objects. So you aren't looking at them in a different context mathematically, but rather just "simplifying" them. Of course, doing this in an automatic, efficient, consistent way probably raises all sorts of interesting mathematical questions, so I'm not saying math isn't involved. Probably not topology, though.
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