Holes in torii

Discussion of shapes with curves and holes in various dimensions.

Holes in torii

Postby moonlord » Thu Aug 10, 2006 8:42 am

For the sole purpose of this post, I will invent some terminology.

A hole-x is a hole in a body that, no matter how the body is transformed, is topologically equivalent to a hypersphere-x. Therefore, a pocket is a hole-0.

EDIT: The above sucks. Please read on to understand what I'm refering to.

In 0D holes aren't possible, because there is no place for a body and a hole. That takes, in the trivial case, two points of space. In 1D holes would split objects. Let's say that the circle-0 (hollow) has a hole. Anyway, these are defined more than determined. In 2D there can only exist holes-0 in shapes. A hole-1 would split it.

In 3D, on the other hand, there are two possible kinds of holes in a body. There are the "pockets" (0D), as the one in the sphere (hollow), and there are "torus holes" (1D), as the one in the full torus. The hollow torus has one of each kind. In fact, every hollow (2D) version of a 3D body has a pocket.

My conjecture is that in 4D, there are three kinds of holes. Pockets in all hollow (3D) versions, planar as in the extruded torus (21)1, and linear holes. I suspect that in the tritorus there is a linear hole. Can anybody help with this?
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Re: Holes in torii

Postby Marek14 » Thu Aug 10, 2006 12:47 pm

moonlord wrote:For the sole purpose of this post, I will invent some terminology.

A hole-x is a hole in a body that, no matter how the body is transformed, is topologically equivalent to a hypersphere-x. Therefore, a pocket is a hole-0.

EDIT: The above sucks. Please read on to understand what I'm refering to.

In 0D holes aren't possible, because there is no place for a body and a hole. That takes, in the trivial case, two points of space. In 1D holes would split objects. Let's say that the circle-0 (hollow) has a hole. Anyway, these are defined more than determined. In 2D there can only exist holes-0 in shapes. A hole-1 would split it.

In 3D, on the other hand, there are two possible kinds of holes in a body. There are the "pockets" (0D), as the one in the sphere (hollow), and there are "torus holes" (1D), as the one in the full torus. The hollow torus has one of each kind. In fact, every hollow (2D) version of a 3D body has a pocket.

My conjecture is that in 4D, there are three kinds of holes. Pockets in all hollow (3D) versions, planar as in the extruded torus (21)1, and linear holes. I suspect that in the tritorus there is a linear hole. Can anybody help with this?


Well, yes - there are three kinds of holes although these are basic kinds and I suspect there might be more complex classification. Imagine, for example, a hollow torus - the hollow is a 0-hole, but maybe it would deserve a subcategory of its own? We can see the holes easily through slicings.

(31): we can use the slicing of two concentric spheres. This starts as a spheric shell which splits in two spheres - outer and inner, with the body of figure being in the shell between them. At each point, there is a 0-hole inside of the figure, so they all line in 4th dimension to form a 1-hole.

((21)1) - tetratorus: If we slice tetratorus in a similar way, we will start with a single toric shell that splits on outer and inner torus. There is a PLANAR hole, since the linear hole through torus persists throughout the whole process (in the tetratorus sections in my "slicing toratopes with hyperplanes" thread, you can actually see it better, as in some pictures, you can pass a plane through the center without intersecting the figure).
However, this planar hole also connects with secondary hole (formed inside of the toric shell), giving it an interesting inner structure I haven't fully grasped yet.

(22) - tiger: Tiger has TWO planar holes in absolutely perpendicular planes. If you slice it in any cardinal direction, you will get a series or toroidally symmetrical figures whose linear holes join up to form first planar hole. However, slices in the middle will look as two separated toroidal bodies, and the space between them is the second planar hole.

(211): A good slice to use here is a circle inflating to become a torus, then deflating back. The linear holes in the figures join up to become a planar hole.
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Postby wendy » Fri Aug 11, 2006 7:16 am

In four dimensions, there are two kinds of hole: tunnel and bridge.

In four dimensions, one has a closed latrix linking with a closed hedrix. A tunnel has the latrix outside. One can make one of these by taking a linear pin, and sticking it through a body.

A bridge has the latrix inside, and the hedrix outside. This is as if a line from inside the body, made a jump to a different part of the surface.

When one deals with the surtope consist, there is a third kind of hole, that is only a surface effect. The duo-cylinder, or bicurcular prism, has two of these.

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