Consider a sheet of paper. You can make it into first a cylinder, and then a torus. In four dimensions, this torus can be extended into a kind of torus-cylinder, which can then be bent into a torus.
Somehow, something is missing. You can fold a torus from a sheet of paper in the manner described. There is no origion on the torus, because the join is perfectly relative.
Finding A#B requires that we look at the tangent space of A, and put a B in there.
I fail to see how you get 5d from folding a sheet, laid out in the xyz space, firstly into a circle in wx, then in a circle in wy, and then in a circle in wz. It gives something that is scattered over wxyz, which us 4d when i last checked.
Also, it should be remembered, that the torus, like any comb product, or indeed, any cartesian product, has a grain, and therefore a basis for setting up standard coordinates.
A cylinder is a prism-product of a line and a circle. We see that there is a grain, in that you can determine height from circumference on the surface.
When you bend a cylinder in 3d, to make a torus, it does not become 4d, but the axis of bending includes one of the axis of the base. This is the axis of pondering (or dimension-loss).
There is also some nonsense that this is different to the spheration of the margin of the bi-circular prism. This is in fact also a torus, since one multiplies three circles, in prism product, and then fold the result downwards. The comb product of a tri-circle is a chorix (3fabric) in six dimensions, and this folds down onto four dimensions in an unknown number of ways.
But i can not see how two different figures can arise from the tri-circular torus. They are all topologically 3hc+3ch, so they ought be deformable into each other.
Note that a simple torus product, eg the dodecagon-dodecahedral torus, is topologically hc, while it gives a dodecahedron-dodecagon torus of genus ch. These are indeed different!
Deformable doesn't mean the same. I don't know what your h,c notation means. Could you define it in terms of one of our notations, or using equations?
Note that a simple torus product, eg the dodecagon-dodecahedral torus, is topologically hc, while it gives a dodecahedron-dodecagon torus of genus ch. These are indeed different!
They're not simple, and they're not different. They're both equally non-existent. I've said plenty of times that the torus product is only defined for a sphere on the right hand side.
Users browsing this forum: No registered users and 4 guests