Holes

Discussion of shapes with curves and holes in various dimensions.

Holes

Postby wendy » Thu Dec 03, 2009 8:19 am

I wrangled some of the numbers on the theory that a hole = missing patch, where the complete figure is spherous (topologically convex).

The first point is that any unmarked surface over content is necessarily has hole (since the surface is closed, and can not vanish). A point prevents such from happening.

For example, the common sphere-circle comb (the base form of both of 4d's two kinds of solid hole, enumerates as having a hole configuration of:

1c.2h.1e.1v ie 5 different kinds of hole!

Of course, when one writes it as 1c.1h + 1h.1e + 1v, the three holes thus formed are the two solid holes, 1c, 1h, with their intersections on the surface, and the 1v required because the surface is boundless. (it's where the hole surfaces 1h+1e intersect.) The solid hole content is 1c, 1h.

Because unmarked surfaces have more holes than some marked surfaces, we might want to reconsider the definition of hole as a missing patch spanning a solid region.

Four dimensions has only one kind of hole, but it can be external or internal, a given figure might have several holes, both external and internal.

Because a surface divides space into interior + exterior, the internal holes correspond to the complement of the external ones: that is, on turning a thing inside out, "tunnels" become "bridges" and vice versa.
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Re: Holes

Postby PWrong » Thu Dec 03, 2009 9:28 am

The first point is that any unmarked surface over content is necessarily has hole (since the surface is closed, and can not vanish).

This is what we've been calling a "pocket", correct? You can cover the shape with a copy of itself. This cover has no boundary and isn't the boundary of anything. So there is one hole of the same dimension as the shape. I don't know what you mean by "a point prevents such from happening", or by marked and unmarked surfaces.

For example, the common sphere-circle comb (the base form of both of 4d's two kinds of solid hole, enumerates as having a hole configuration of:

1c.2h.1e.1v ie 5 different kinds of hole!

Do you mean sphere#circle? In terms of homology that has four holes, one of each type. Its homology sequence is [1,1,1,1]. In your notation I thought it would be 1c.1h.1e.1v, where does the extra h come from?

As for the missing patch theory, I think I can recover it. Our problem was with things like adding a disk to the inside of a sphere. The difference between that and adding a disk to the inside of a torus is the following. You can take the disk and deform it, without moving the boundary, until it lies entirely on the sphere. You can't do that with the patches we were talking about on the torus. Thus, I claim it doesn't count. With this restriction, I think the patch method will work as a way to count homology groups, at least for toratopes.
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Re: Holes

Postby wendy » Fri Dec 04, 2009 8:24 am

1. non-vanishing surface.

A non-vanishing sphere is a test for holes. Essentially, vanishing means going to zero or infinity, without crossing any part outside of the space it falls in.

For example, in a plane, you can have two points. A circle that includes both can go to infinity, ie vanish. A circle that includes one, can neither vanish to zero or to infinity, so indicates a hole exists. In order to prevent such a loop forming, one has to patch the hole with a fabric that is orthogonal to the hole and spans the space. We create a line between the two points, and so all circles that now form must include either zero or none.

The surface of a sphere of n dimensions, does include an n-dimensional sphere, which is strictly immobile. To prevent this from happening, we need to include a point, which means that the surface no longer can contain an n-sphere.



2. The calculation for the circle-sphere torus is as follows:

We start in five dimensions with a prism, being the circle-sphere prism, each with an extra point.

1h.1e.1v.1n *# 1c.1h.0e.1v.1n The prism product ignores n, but includes the first term, so

we find 1 001 001 * 1 001 000 001 = 1 002 002 002 001 001 = 1p 2t 2c 2h 1e 1v 1n

The next step is to fold the thing flat from 5d to 4d, while preserving all the connections. This removes 1p 1t, since this t becomes the outside. Think schegel diagrams.

We get then 1t 2c 2h 1e 1v 1n

This contains the torus 1t 1c 1n, subtracting this we get 1c 2h 1e 1v as the hole consist.

The missing patch theory suggests that a patch will prevent the forming of non-vanishing spheres in any space, so let's see

1c, 1h are required to prevent non-vanishing loops on the outside/inside. Circles are prevented by 1c, the spheres are by 1h.

The remaining three holes 1h, 2e. 1v form inside the 3d surface. This space is a circle-sphere comb (the product of surfaces).

You need a full sphere 1h to stop circles running around the height. Adding this allows one to open the figure into a sphere-prism, the curved side. All circles on this will vanish, including those that go around the figure but complete spheres do not. You can see that a circle on a unmarked sphere, or a sphere with a point on it, can still vanish in the space of the sphere. To prevent these spheres forming, we need a line from top to bottom, which is another line.

We have now created two new surfaces 1h 1e, both of which admit holes. None the same they cross, and this crossing provides a point for these figures to prevent the surface forming as a non-vanishing sphere.

We could also demonstrate the piecewise construction of this, by

v.n create a point.
h.e create the circle, with its interior. The circle contains v. The dimensions are wx
c.h create an entirely different sphere, so that it lies in wyz space, in opposite w.
t.c The surface created by rotating the sphere around the centre of the circle, and the interior ditto.
depending on closure the t could contain the leading c or the h from the previous spaces.

The bits that are extraneous are the first three elements, the beginning of the second and third are in some 4-space
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Re: Holes

Postby PWrong » Fri Dec 04, 2009 10:40 am

Your explanation of non-vanishing surfaces is very similar to the definition of homotopy, if mixed up a bit. However there are many nonvanishing circles you missed. If you wrap a circle around a point twice, it can't vanish either way, nor can you deform it into a circle that wraps around once. Also, you can surround two points but wrap one of them backwards, like a figure 8.

I don't follow your derivation for 32. Using the missing patch theory, you can fill in the circular hole with a disk, and the spherical hole with a ball. This leaves a pocket which you can fill in with a 4-ball. Hence [1,1,1,1].
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Re: Holes

Postby wendy » Sat Dec 05, 2009 8:16 am

The test for holes is a non-vanishing sphere. A surface that crosses itself is not a sphere, since you can use the crossing to create simpler loops that do the same function. In any case, a non-vanishing sphere is intersected by the missing patch at a point, both the sphere and the patch to fall in the space of the patch. Of course, if the loop crosses itself, then it has a kind of hole in it already, and reduces to a sum of simpler loops that do not cross itself.

In terms of the 32 derivation, there is no room for a type 4 hole (ie a teric patch). Point-pairs on both the interior and the exterior can vanish without crossing the surface, so there is no room for a solid region to become filled.

In terms of the other two patches, the patch must be mounted: that is, its surface must be made of elements that exist on the figure it is to be mounted on. Since the intersection between the elements are the surfaces of the holes, the choric hole needs a hedric surface to become attached, and the hedric hole needs a latric surface. These intersect in a point. In this way, all of the possible holes in the surface of the torus are closed too.
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Re: Holes

Postby PWrong » Sat Dec 05, 2009 10:41 am

Your non-vanishing sphere test is equivalent to calculating the homotopy group of the embedding space, with the shape taken out and an extra point at infinity, so the sphere can vanish either by shrinking to a point or by expanding to infinity. Since R^n with a point at infinity is homeomorphic to an n-sphere, you're really talking about the holes of the complement of a shape embedded on a sphere.

A surface that crosses itself is not a sphere, since you can use the crossing to create simpler loops that do the same function

Well you can define a sphere like that if you want, but you'll find it's not as useful as the standard definition. A sphere in topology is simply the image of some continuous map from the geometric sphere onto the space. It can cross itself in the space, because in the whole space (preimage x image) it doesn't cross because the parameters are different. You'll find it very difficult to talk about holes of the projective plane and klein bottle.

The non-vanishing sphere test is a reworked version of homotopy and patches are equivalent to homology if you do it right, but homotopy and homology are irreconcilably different. On the other hand, the non-vanishing sphere test is done in the embedding space instead of in the shape, so maybe it won't have these problems. So possibly the non-vanishing sphere test is equivalent to the method of patches. I wouldn't bet on it though. I really like the missing patch theory, but I don't think we need to mess around with nonstandard homotopy theories.

In terms of the other two patches, the patch must be mounted: that is, its surface must be made of elements that exist on the figure it is to be mounted on. Since the intersection between the elements are the surfaces of the holes, the choric hole needs a hedric surface to become attached, and the hedric hole needs a latric surface. These intersect in a point. In this way, all of the possible holes in the surface of the torus are closed too.

Ok, I get the idea of mounting a patch onto its boundary. But you have still pockets left behind. Let's go back to a standard torus. You fill in the middle hole with a disk mounted on a circle. Then you put another disk inside the tube. But there's still space inside. You have to fill it with a ball. You can't completely fill in the torus with two disks. Nor can you forget the second disk, because a ball won't fill up the whole tube or else it's no longer a ball. So the torus requires two disk patches and a ball patch. Same goes for the duocylinder 22 since they're homeomorphic. Do we agree on this?

Similarly, you can't fill in 32 without one disk, one ball and a gongyl (solid 4D sphere).
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