alkaline wrote:Today I read an article about Fermat's Last Theorem and the Fourth Dimension.
alkaline wrote: They describe the construction of the shape as taking a square, folding two sides together to make a cylinder, then folding the other two sides into the fourth dimension to make the four-dimensional shape that they call a "circle of circles". If you perform this procedure on a deformable square, you can create the torus in the third dimension. Does this mean that the torus and the duocircle are topologically equivalent? Mathworld has a page of Non-Orientable Surfaces. The square they show for the torus (which is an orientable shape) is basically the same that i would imagine would create a duocircle. Is any path on a duocircle a circle?
Polyhedron Dude wrote:The duocircle has only one face which separates the two cells - this face is the result of taking a square and curving it on both axises (I usually call this face a duoring - it is the product of two circles (hollow) - where the duocircle is the product of two disks). The duoring is topologically equivalent to the torus. Any path on the duoring part of the duocircle is a circle, but if we picked a path elsewhere on the duocircle, you could get squares!
Aale de Winkel wrote:I thought I caught up with you, but this confuses the hell out of me.
The duocircle looks to me indeed like a circle of circles :R [ cos(α) , sin(α) , cos(β) , sin(β) ], hence the name.
the duoring likewise a ring of rings or (R[sub]1[/sub],R[sub]2[/sub]) [ cos(α) , sin(α) , cos(β) , sin(β) ]
using the disc this way I get the duoball (r = 0..R) [ cos(α) , sin(α) , cos(β) , sin(β) ]
and the duodoball (r = R[sub]1[/sub] .. R[sub]2[/sub]) [ cos(α) , sin(α) , cos(β) , sin(β) ]
a really confused trionian :?
bobxp wrote:well it is easier and more practical to think of a ball as hollow and a sphere as solid.
Aale de Winkel wrote:bobxp wrote:well it is easier and more practical to think of a ball as hollow and a sphere as solid.
You get a lot of mathematicians against this point of view, everyone sees the sphere as hollow. To change t your way of thinking is far too IMpracticle.
You cought me just prior to editing out the mishap in my previous post, The tubes just airs the "sikaris episode" of voyager (mentioned elsewhere)
bobxp wrote:Then why did the mathematicians decide on these definitions then...
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