New problem

Discussion of shapes with curves and holes in various dimensions.

New problem

Postby PWrong » Thu Dec 31, 2009 6:06 am

Hi, hope everyone's having a good holiday.

I just realised that a toratope and it's expanded rotatope have the same holes in the minimal frame, but not in other frames. In fact they don't even have the same number of frames. So it's not sufficient to work out the homology groups just for rotatopes. We have a whole lot more stuff to work out

For example, a filled in (31) is basically a sphere, but a filled in (211) is a circle and a 4-frame 32 is a pentasphere. I'll figure some more out later. I'll be pretty busy for most of the next two weeks though.
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Re: New problem

Postby wendy » Thu Dec 31, 2009 7:19 am

Holes are something to do with bound regions, whether filled in or not. Any unmarked closed surface, is necessarily a hole (since it leaves no room to vanish).

Simply doing things like changing the surface markings and placing additional walls changes the topology.

There is room for 'solid holes', which admit spaces bounded in "solid space" (eg 4d in 4d), which, in connection with solids, gives what most people perceive a hole to be.
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Re: New problem

Postby PWrong » Thu Dec 31, 2009 10:18 am

Holes are something to do with bound regions, whether filled in or not.

Well no, if you fill in a hole it's no longer a hole :\.

We assumed that every toratope is homeomorphic to (can be deformed into) its expanded rotatope, and that therefore they have the same hole structure. However if you fill in a toratope and its expanded rotatope, they're no longer homeomorphic so we have to calculate the holes separately.

Example: Torus and Duocylinder.

2-frame torus has [1,2,1]. That means it comes in 1 piece, has 2 circular holes and 1 pocket.
3-frame torus has the inside of the tube filled in, so it's only got one (circular) hole left. It's still in one piece, so it's [1,1]

2-frame duocylinder is also [1,2,1].
3-frame duocylinder is homeomorphic to a glome, so it's [1,0,0,1]
4-frame duocylinder is completely filled in like a ball, so it has no holes at all and is [1].

So the min-frames are the same but the other frames are different. Do you follow?
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Re: New problem

Postby wendy » Fri Jan 01, 2010 7:28 am

It pretty much depends on what you're looking at.

A torus, in three dimensions, is a surface that divides space into two. If you don't create a choron (3d surtope ), then the two sides of the surface belong to the same 3-space, and one can create a point-pair on either side of this surface, that is unable to vanish. Filling in a region makes the regions belong to different things, and the point-pair has no business being there.

You can regard the torus as a kind of projection of the duocylinder. It's one of those stretch out Engels diagrams. When you close it up, the outside becomes a new 3d region that belongs to it, and a 4d region is created as the interior.

So, holes of the torus (and the duo-cylinder), become

1d. None
2d. two circles + point.
3d. two disks
4d. none.

The 3d disks serve to break the two toric faces into cylinders, the top and bottom join. Inserting extra disks just creates more faces: a new face per disk. It also creates new circles etc, points etc.
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Re: New problem

Postby PWrong » Sat Jan 02, 2010 4:53 am

Sorry, I'm not following you. Did you understand my post?
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Re: New problem

Postby PWrong » Mon Jan 04, 2010 12:50 pm

In particular, I don't get this at all:
So, holes of the torus (and the duo-cylinder), become

1d. None
2d. two circles + point.
3d. two disks
4d. none.


We know all about the holes of the torus, it's quite simple and nothing like this. :\
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Re: New problem

Postby wendy » Tue Jan 05, 2010 9:19 am

The maths is hard by me, but i use models and intuition.

Genus, and holes, are a property of a particular piece of fabric, regardless of how it becomes mounted. So something like a rectangle, connected in the manner of a torus, permits the existance of two closed loops that cross at a single point. This is regardless of whether it is unmounted (a 2d closed fabric), or the surface of a torus, or the margin of a duocylinder.

A solid divides is an assembly of spaces of varying dimensions, of which the highest dimension is the content, and all other spaces are incident on the content. A sphere, a cone, a cube, a square, are all solids.

Each space, including the exterior, are evaluated for "non-vanishing surfaces". Where there is no limit on the surface, a point is inserted to match (bound) the nulloid of the thing.

A duocylinder is a collection of spaces, the 4d interior, the 4d exterior, two torii in 3d, a torii in two dimensions, and no lesser space. The two 4d figures do not support non-vanishing spheres: they need on addition.

The two three-dimensional spaces have the topology of a cylinder, connected end to end. There is a circle forming inside this, which can be broken by placing a disk inside this. However, this disk must be mounted, which means that circles are added to the 2d surface.

The 2d surface by itself, supports a pair of circles, intersecting at a single point. Putting in one such circle does not prevent parallel circles forming. You need the two crossing circles to prevent this happening. These are indeed the mountings of the two disks above.

The two circles each need a point to break them to prevent the space itself from being reckoned as nonvanishing. The point of intersection does.

In the table, i give the dimensions of the spaces, and the relative mountings that must be added to prevent spheres forming

1d - No space of 1d exists
2d - The sole hedron requires two circles (lines), and a point of intersection to prevent the non-vanishing spaces forming
3d - There are two 3d spaces, which each require a disk (2d space) to prevent non-vanishing 1-loops from forming.
4d - The two 4-spaces do not allow holes to form.
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Re: New problem

Postby PWrong » Tue Jan 05, 2010 10:57 am

1d - No space of 1d exists
2d - The sole hedron requires two circles (lines), and a point of intersection to prevent the non-vanishing spaces forming
3d - There are two 3d spaces, which each require a disk (2d space) to prevent non-vanishing 1-loops from forming.
4d - The two 4-spaces do not allow holes to form.


Are the two circles just the boundaries of the two disks? If I have this picture right, then the two disks also have a point of intersection. As I said in the other thread, your vanishing sphere test is overcomplicated. It's equivalent to adding a point at infinity, so you're not really adding anything useful, you're just changing the embedding space.

Anyway, this thread is supposed to be about the fact that some shapes, that we thought were equivalent, actually have completely different hole structures in non-minimal frames. The frame concept has been explained in other threads and is pretty simple.
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