Rob wrote:PWrong wrote:Like I said before: since (x1) is every point of a circle replaced with x
Where'd you get that idea? (31) is sphere#circle, ((21)1) is torus#circle, (211) is circle#sphere.
(31) is every point in a circle replaced by a sphere.
((21)1) is every point in a circle replaced by a torus.
(211) is every point in a sphere replaced by a circle.
That's how I see it anyway. I would understand if you switched (31) and (211), but there is no possible way you can replace every point in a torus with a circle and expect to get the tetratorus.
I am sorry, but that's exactly how it is. Maybe you should stop being so categorical
Work with me here:
Tetratorus (A,B,C) is a set of all points in 4-space with a set distance C from the surface of torus (A,B). (said torus is, similarly, set of all points in a 3-space with a set distance B from a circle with radius A)
Now, what would be the same set defined only in 3-space? In 3-space, set of points with a given distance from torus are two more torii - one outside, one inside of the original one. In other words, we are replacing each point with a 1-D circle, i.e. pair of points, and this 1D circle is lying on a normal to surface at each given point.
Now, in 4D, the normal to 2D surface is also a 2D surface, i.e. a plane. Points of tetratorus lie in these normal planes - and in each of these planes, they are points of the same distance from origin (where the plane intersects the mother torus), i.e. circles.
Note: maybe more intuitive would be to read (x1) as "surface of x, BLOWN UP in D(x)+1-dimensional space". Thus torus is what you get when you blow up a circle in 3D, (31) is when you blow up a surface of sphere in 4D, and tetratorus is what you get when you blow up a surface of torus in 4D. Blowing up in n additional dimension is simply replacing each point with (n+1)-dimensional sphere, although I agree it might look a bit counterintuitive.