"A shape whose impossibility might have been an elegant theorem, but whose existence may be much more elegant." Chandler Davis, Editor-in-Chief, The Mathematical Intelligencer http://www.gomboc.eu/gomboc_english.html
an impossible shape is one sided. unless a line is counted as a shape, hey i just had a thought. you see a straight line, no matter how thin it is, its still a type of rectangle. because if you zoom in close enough you would be able to see all 4 sides. like this - <--- that is still a rectangle even though it looks like a line. peace.
In math a line has dimension 1, so no matter how much you zoom in, it willl never be a rectangle which has dimension 2. So that could be an impossible shape. Or do we define shapes as something with dimensions higher than 1?
It's a bit like a mexican jumping bean. I was watching the film waiting for it to do something dramatic, so was a little disappointed. It does move as though it couldn't decide what to do next, which is the point I suppose.
whats the difference between a rectangle and a straight line? if you colour in the rectangle its just a big fat line. peace.
A line (mathematically speaking) has only length no width. So it has dimension 1. A rectangle has length and width and has dimension 2. So a fat line is a rectangle.
" The "Gömböc" is the first homogeneous, convex object having just one stable and one unstable point of equilibrium."
I can't tell why the editor made that comment... he seems to imply that there is an elegant theorem that was supposed to prove that there are no homogenous convex mono-monostatic bodies, but I can't find any such proof :shrug: Describing impossible shapes is easy. Trivially, a square circle is impossible. Almost as trivially, a regular convex 3D polyhedron with exactly ten faces is impossible. Or, a convex 3D polyhedron with more faces than edges is impossible. Here's a less trivial impossible shape, by the wonderful M.C. Escher: Please Register or Log in to view the hidden image!
but can a line be so tiny that it has no width dimension? no matter how small you can draw a line, i can always zoom in with a microscope and describe how wide it is in micro measurements. peace.
A line is a mathematical construct...no one can really "draw" a line physically. In that technical sense, this . is not a "point" because it has a definite (though small) height and width.
Yeah, but what we draw is just an approximation of line not mathematical line. Just the same as a drawn tiger is not a real tiger. Edit: too late.
Hi EFoC, A true line is physically impossible. Any line that you draw is not a true line, but a 2D shape. The question in this thread isn't so much about physical possibility, but theoretical possibility - possibility in the land of mathematics.
I wonder if it counts if the viewer of said object is from a different dimension? If the analog for a cube in the 4th dimension is a hypercube, then perhaps there is an analog in another-space that renders three sided/circular squares that aren't considered true triangles/circles somehow. Maybe there is a way to redefine the "z-axis", or use modified spatial dimensions (IE - height, width, and something non-perpendicular to the others, or completely different in it's own function) Of course, I'm just tossing a foggy wild idea in there. BTW - not trying to derail the original topic (technically, the discussion is about the Gömböc).
Definitely. Topologists will always specify the number of dimensions they're working in, because different theorems apply in different dimensions. There's a joke about this: A topologist and an applied mathemetician attend a public lecture on geometry in thirteen-dimensional space. "How did you like it?" the topologist asks afterward. "My head's spinning", says the applied mathemetician. "How can you possibly visualize thirteen-dimensional space?" "Easy!" says the topologist. "Just visualize an n-dimensional space, then set n = 13."
