By Watheq Al-Bnosha
For $8\times 8$ I got two shapes
the first shape I got was by dividing $8$ into two circles and multiply each circle by $8$ as it is shown in Figure 1.
Fig. 1
we can imagine it like four torus on top of each other and the figure $8$ is in every single point of those torus, as we see in Figure 2.
Fig. 2
The second shape I got was by moving the figure $8$ around every point in the other $8$, as we see in Figure 3.
Fig. 3
For Figure 1 and 3, I think the Cartesian product is non-connected. I think Figure 2 seems correct to me as two torus are connected with other two torus in shape $8$.
ReplyDeleteI agree with you, but if you focus on Figure 3 you would see how 8 is moving around every point on the other 8.
ReplyDeleteI got the point you trying to say, but we discussed last week that $8\times S^1$ is not equal to double torus. Instead, it was two torus on top of each other. I am not sure if Figure 3 is right but I think that Figure 2 is right.
ReplyDeleteI agree that figure 3 should not be a double torus (actually it is not at all, since it is not a tube all around but that kind of double tube, but still, I get your point that should not be "like a double torus"). But also I agree with Watheq idea of moving the 8 around the other 8. We just have to look closer what happens with this movement when you get near the intersection, to make compatible these two visions. And maybe we need to improve a little bit the drawing to reflect this.
ReplyDeleteDo this: color the horizontal 8 say one O red and the other O blue. Then follow the movement of the 8 around the first red circle. Then around the second blue circle. Paying attention the the points that are common.
Hint: from outside, the whole thing looks like a double torus, but there is a lot more going on "inside" the surface.
Chintan, the Cartesian product of connected things, is always connected.
ReplyDeleteIn figure 1, the division you did in two circles is still connected at a point, I am not sure if I understand the drawing of four torus, one of top of another like a pile?
ReplyDeleteI think Figure 1 cannot be the solution, because then four torus are not connected to each other. I got the point that Cartesian product need to be connected, but I think Figure 1 doesn't fulfills that.
ReplyDeleteFigure 1 needs clarification. In the meanwhile, think of a way to improve Figure 3, what is exactly happening in the middle?
ReplyDelete