Holes of Toratopes (wendy's version)

Discussion of shapes with curves and holes in various dimensions.

Holes of Toratopes (wendy's version)

Postby wendy » Fri Nov 20, 2009 7:46 am

Some figures, like the duocylinder [()()] have holes, although you can't stick anything through them!

Defining holes is tricky, but the method i use, is that there is a space that one can place a closed spherous fabric that can not be made to vanish. Such can be made impossible by placing a complementary connection, that spans the hole.

A duocylinder then has two kinds of hole, formed by the surface of circles that run around the two torii. This can be closed by adding spaning bases across the ends. The corresponding toric condition is hh (a 3d hole) [being 2d circle prevented by a 2d disk.

An example of a 31 hole would be a pea in a whistle. The pea could be surrounded by a sphere, but were there a connection between the pea and the whistle (like a sprig or line), this would prevent the sphere from forming. Circles evidently fall off. We see here that the surface is a 3-sphere, and the disk is a 1-sphere. The 2d dimensional thing is disjoint objects, which form a 21 hole.

A 13 hole is formed by an object with an internal hole, such as a closed box. The 1-sphere is formed by disjoint points, one inside the box and one outside the box. To prevent such dyads forming, ye ned a 3d disk to fill either the box or inside or outside the box.

In four dimensions, ye have 14, 23, 32, and 41 holes as solid, and any lesser set (eg 13, 22, 31, 12, 21, 11), as surface holes. The effect of such holes is that the surtope-sum (or euler characteristic), will be increased by one by an odd number and decreased by 1 for an even number. 4D holes, like 23 and 32, do not disturb the Euler characteristic.

Figures like the "tiger" have a supprisingly interesting set of non-obvious holes.
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Re: Holes of Toratopes

Postby Keiji » Fri Nov 20, 2009 9:56 am

So in n dimensions you have n types of "solid hole" (which I previously called "pockets") and n(n-1)/2 types of "surface hole".

So in 3D you have 31, 22 and 13 for solid holes and 21, 12 and 11 for surface holes.

I imagine from your examples that 22 could be a hollow box with a pole joining one face to the opposite face.

What would the three types of surface holes be? What type does the torus have?
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Re: Holes of Toratopes

Postby wendy » Sat Nov 21, 2009 8:29 am

The holes that involve 1, ie 1m, m1, refer to what is in that space a disjoint object. For example, an annulus is a 12 hole (since it supports a point-pair (1-sphere) stopped by a 2-disk). However in 3d, a polyhedron can have an annulus-face.

Although one is less likely to consider 21 type holes (disjoint objects), they are considered for completeness.

A 'solid hole' is a hole that exists in solid space. In Nd, the solid holes are numbers that add to N dimensions, so

13 = object with cavity inside
22 = single hole like a torus In 3d, genus corresponds to the number of 2's in the hole strings.
31 = disjoint objects (missing-link).

Holes involving 1 are disjoint surfaces, eg either disjoint objects or objects with a hidden interior. These are not usually considered, so we have "rigid" holes that can participate in chains etc. In 3d, the rigid chain is a string of loops (22).

In four dimensions, the rigid holes are 23, 32, which form different kinds of solid, but the surface is topologically identical. That is, a surface like a circle-sphere torus, will divide space into two spaces, one with a 23 hole, and one with a 32 hole. Central inversion, or "turning the surface inside out" can swap the hole-sequence, ie make a ab hole into a ba hole. (In the 1x case, the point of inversion inside the box, but not inside the cavity, will produce the box as the outside, and the two disjoint spaces are the interior and exterior of the box, ie an x1 hole).

Note that a space never looses its hole-ness, even should it loose its capacity to bound etc. A bi-circular torus is still a 22 hole, even in 15 dimensions. It's different to the knots, which in some large dimension unravels (eg a 22 knot, such as a piece of string forms, becomes a simple unconnected loop in 4d. You can still use 22 knots, as long as they are extended in prisms (ie 22P1), which is why you can tie hedra (planes) together in 4d.

In 3d, a closed line-loop can be tied in knots, because it being a surface of a 2-sphere, supports 22 linkages, which happens as far as N=3. Likewise, the surface of a closed sphere 3, supports 33 linkages, which permits "knotted loops" in 5d, even though the surface itself has no holes. Of course, you could tie a hedrix that is itself has one or more 2 holes (the criss-cross cap has a single 2-hole), into knots as well.
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Re: Holes of Toratopes (wendy's version)

Postby Keiji » Sat Nov 21, 2009 3:00 pm

I split this since otherwise it would get confusing.

A "12" hole in 2D could be a (hollow) circle. In 3D, a "13" hole is a (hollow) sphere. So what is a 12 hole in 3D? If you take a hollow cube and stick a 12 hole in one face, you get something equivalent to a solid cube. If you stick a 12 hole in two faces, you get something equivalent to a solid torus. Assuming that 22 refers to a torus, would that be a solid torus or a hollow torus?
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Re: Holes of Toratopes (wendy's version)

Postby wendy » Sun Nov 22, 2009 7:43 am

A 12 hole is a hollow circle in a hedrix or 2-space.

Since a 2-space is solid in 2d, the hole is also solid. But in higher dimensions, one can still have 2-spaces, and of course figures in 2-space that have holes.

An example of a figure with a 12 hole, might be something with an annulus face, say, a cylinder where there is a second circle at one end. You can have 12 holes where the figure has a solid 13 hole, such as a cylinder with a hole bored through it, or a ring of eight cubes, in a 3*3*1 arrangement, with the centre cube removed.

A 12 hole, for example, has a different effect on the surtope-count, then say, a 13 hole. A triangle inside in a hexagon has the same surtope count - 1n, 9v 9e, 1h as an enneagon, but a 13 hole in 3d (say a cube inside a cube), gives something that can not be replicated by a normal polytope, ie 1n 16v, 24e, 12h, 1c, giving an euler-excess of 2, which no convex polytope can hold.
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Re: Holes of Toratopes (wendy's version)

Postby PWrong » Sun Nov 22, 2009 10:05 am

Holes involving 1 are disjoint surfaces, eg either disjoint objects or objects with a hidden interior. These are not usually considered, so we have "rigid" holes that can participate in chains etc. In 3d, the rigid chain is a string of loops (22).

So these holes are related to the H0 group and the Hn group where n is the dimension of the surface. H0 always counts the number of path components, and Hn counts pockets.

Central inversion, or "turning the surface inside out"

Does this turn (31) into (211) or keep the shape the same? Is it a bit like this?
http://en.wikipedia.org/wiki/File:Inside-out_torus_%28animated,_small%29.gif
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Re: Holes of Toratopes (wendy's version)

Postby wendy » Mon Nov 23, 2009 8:10 am

I have not figured out what H_0 and H_n might mean.

If you took a 31, you could turn it inside out, to give a 13. A 31 hole allows spheres to surround an object, this being broken by a line-disk to form between them. To turn this surface inside out in the manner of 22, do this.

A 31 hole, is formed by two spheres, separated in space. The 3 refers to an imaginary sphere drawn around one of the objects, which can not disappear by sizing. The 1 refers to a (missing) solid line that connects the two spheres.

Put a hole in one of the spheres (in the manner of the 22), and turn the sphere inside out (so to include the second sphere), the outside is now inside etc, and the genus of the sphere has been reversed (from 31 = two spheres to 13 = sphere + cavity).

Holes with 1 in them are rather hard to follow, because they typically involve disjoint objects. However, various slices of the 22 torus gives 21 (disjoint circles), and 21 (annulus). It's useful to study this relation, because various slices of 23 and 32 yield 22. (along with 13 and 31).

One can replicate a hole of type pq by means of the comb product, which is a repetition of surfaces (for each point on the surface of p, there is a copy of the surface of b, and vice versa. To the product of 21 gives two circles

A 23 hole, formed by a spheric prism wrapped into a circle, would turn inside out to a 32 hole, a sphere with a circle-section.

Consider for example, a pipe with cylinder section, and length. Were you to connect it end to end like a hose, the thing becomes a 23 hole. If you roll it down like a sock, so the top is rolled down the side of the cylinder to meet the bottom, then the thing is a 3-sphere covered by a 2-circle, giving a 32 hole.

So 'line ## circle' gives two circles, evidently there's a circle repeated for each vertex, and a dyad repeated for each point on a circle (eg 12-oclock on both cicles is a dyad).

A torus is circle ## circle. and so on.

The comb product is pondering (ie it reduces a dimension for each application, so polygon * polygon = polyhedron), so in four dimensions, one can have a torus of type 222 (eg 'tiger'). The hole model for this is not all that clear at this moment, but a cluster of 23 and 32 holes is expected. Holes inside holes indeed. [consider the effect of punching a thin choric sheet (3d sheet) in four dimensions, with various punches (so different shape holes go through). Imagine what a toric punch (22) would do.

22 would produce clearly a hole that contains at least one 23 hole (that a line running through the hole and back around the edge of the sheet would not close), and at least one 32 hole (which would be a sphere, half above and half below the plane of the sheet, so that the equator is entirely inside the torus.

You can close the former by placing a 3d dome over the punch-hole. This would prevent the string (ie circle), but not the sphere. Since the sphere appears as a circle in the hole, it needs to be prevented by a hedric disk (2-disk) spanning the base of the torus.

Topologically, turning this inside out, will give the same figure, because the 23 and 32 holes are reversed.
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Re: Holes of Toratopes (wendy's version)

Postby Keiji » Mon Nov 23, 2009 8:57 am

This does shed some light on the subject for me, though as usual, I'll have to think about it more later. For now, I'll satisfy myself listing types of holes to make sure I've got it right.

11 = Two disjoint points
21 = Two disjoint circles
31 = Two disjoint spheres
n1 = Two disjoint n-spheres

12 = Two concentric circles
13 = Two concentric spheres
1n = Two concentric n-spheres

22 = Torus

One thing I just want to clarify, though:

wendy wrote:A 23 hole, formed by a spheric prism wrapped into a circle


This implies a spherinder (spherical prism) joined end to end gives a toraspherinder and that a cubinder gives a toracubinder.
Therefore 23 is toraspherinder and 32 is toracubinder.

wendy wrote:Consider for example, a pipe with cylinder section, and length. Were you to connect it end to end like a hose, the thing becomes a 23 hole. If you roll it down like a sock, so the top is rolled down the side of the cylinder to meet the bottom, then the thing is a 3-sphere covered by a 2-circle, giving a 32 hole.


This implies that a cubinder (cylindrical prism) joined end to end gives a toraspherinder and that a spherinder gives a toracubinder.
Therefore 32 is toraspherinder and 23 is toracubinder.

Which way around is it? I've believed the first case as the naming and toratopic notations make sense that way, but I recall some argument that it's actually the opposite.
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Re: Holes of Toratopes (wendy's version)

Postby PWrong » Mon Nov 23, 2009 9:45 am

I have not figured out what H_0 and H_n might mean.

There's a certain symmetry here, I don't understand your notation either :P. I'm starting to get it though.

Like I said, H_0 counts path components, H_n counts pockets. Except instead of numbers they're copies of the group of integers Z.
H_1 is, very roughly, the group of circles that aren't the boundary of something. A torus has two unique kinds of circles that are not the boundary of any disk. So H_1 of the torus is Z+Z. This can be made much more precise. Anyway this thread is still about your notation, I'm just trying to understand it by comparing it to homology groups.

31 = Two disjoint spheres

A pair of disjoint spheres has two path components, but also two pockets. Are the holes of a single sphere represented by 3?

What would you call the holes of a pair of disjoint torii? 221?

13 = Two concentric spheres

I like the idea that concentric spheres have a different kind of hole than disjoint spheres, but this means holes are not invariant under continuous deformations. You can easily turn two concentric spheres into two disjoint spheres. Isn't this a problem?
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Re: Holes of Toratopes (wendy's version)

Postby Keiji » Mon Nov 23, 2009 10:58 am

31 can be transformed into 13 by turning it inside out as wendy said. There is no continuous transformation between a pair of disjoint spheres and a pair of concentric spheres, because the orientation matters - if you take a pair of disjoint spheres and move one inside the other (assuming you're in 4D already, which to me already makes the idea invalid), the space between them is invalid as it is both inside and outside. Correcting the orientation is not a continuous transformation.
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Re: Holes of Toratopes (wendy's version)

Postby PWrong » Mon Nov 23, 2009 12:37 pm

31 can be transformed into 13 by turning it inside out as wendy said. There is no continuous transformation between a pair of disjoint spheres and a pair of concentric spheres, because the orientation matters

But turning a shape inside out is a continuous transformation :\

Correcting the orientation is not a continuous transformation.

Sure it is, except that orientation isn't really a part of a manifold so much as an extra thing you add to the manifold.
If it was, you can take a circle with an orientation defined by a normal vector pointing outwards at each point. Simply rotate each vector around in the r-z plane (r being the vector). In other words, turn the circle "inside out". This is clearly a continuous rotation, I could define it as a continuous action of the reals on R^3. The circle is the same but the orientation is now inside instead of outside.
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Re: Holes of Toratopes (wendy's version)

Postby Keiji » Mon Nov 23, 2009 3:03 pm

Turning an n-bounding-space object inside out inside the n-dimensional space is not a continuous transformation.

You may be able to continuously turn a circle inside out in three dimensions, but you can't do it in two.
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Re: Holes of Toratopes (wendy's version)

Postby PWrong » Mon Nov 23, 2009 3:56 pm

Hmm, so much for this then.

Note that a space never looses its hole-ness, even should it loose its capacity to bound etc. A bi-circular torus is still a 22 hole, even in 15 dimensions. It's different to the knots, which in some large dimension unravels (eg a 22 knot, such as a piece of string forms, becomes a simple unconnected loop in 4d. You can still use 22 knots, as long as they are extended in prisms (ie 22P1), which is why you can tie hedra (planes) together in 4d.
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Re: Holes of Toratopes (wendy's version)

Postby Keiji » Mon Nov 23, 2009 5:38 pm

I never really understood that either; I can't see any plausible way for holes to be retained in higher dimensions so I've just assumed otherwise.

If a hole is extended in a prism, it increments the first number (as far as I know).
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Re: Holes of Toratopes (wendy's version)

Postby Keiji » Tue Nov 24, 2009 1:31 pm

Wait a minute.

I just realised while working on the list of toratopes that "toratopic duals" (turning a toratope inside out) are homologous, and their hole-sequences are equal.

There is not a direct relationship between wendy's idea of holes and yours - hers cares about orientation, yours doesn't.

So yes, there is a continuous transformation between the toraspherinder and the toracubinder (that is, 23 and 32, or the other way round), but only if you ignore orientation.
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Re: Holes of Toratopes (wendy's version)

Postby PWrong » Wed Nov 25, 2009 2:52 am

Is that only in 4D? That is, could you correct the orientation by rotating it in 5D? I'd like to see an explicit homeomorphism between the two shapes, and also for Tiger ~ Ditorus. I came up with one but I don't think it's invertible.
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Re: Holes of Toratopes (wendy's version)

Postby Keiji » Wed Nov 25, 2009 7:26 am

Yes, you could correct it from 5D, exactly the same way as you'd correct your concentric spheres from 4D.
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Re: Holes of Toratopes (wendy's version)

Postby wendy » Wed Nov 25, 2009 8:00 am

Holes form a rather difficult topic, some of it not intuitive.

At the moment, i have three branches for it: linkages, holes, and fabric (manifold) topology.

Linkages deal with making a connection out of separate solid bodies. It deals also with knots and self-knotting fabrics.

You can tie a knot between fabrics, if the spaces add at least to the solid-space surface, and knotting itself if the fabric is half of the surface. What this means, eg is that ye can knot latrices (1-fabric) and hedrices (2-fabrics) things in 4d, or knot 2 hedrices in 4d, but not two latrices (as one can in three dimensions). Evidently, a surface (like a hedrix in 3d or a chorix in 4d), does not allow anything to go around, and can not be knotted.

Self-knotting of fabrics like knot theory in 3d, seems to to be of fabrics of m dimensions, in a space of 2m+1. Here we see that the joys of things like the knots in 3d, do not generally happen in higher dimensions, because the crossings can pass each other.

Weaving is a kind of knotting, but every time i try to think it through, the only way to stop the various threads from passing each other, is that the thing must be a weave of marginices (ie N-2 fabrics, such as lines in 3d).

Linkages deal with pairs of open spheres, a latrix becomes a circle, and a hedrix a sphere. Once space is extended it is possible to lift the individual parts through hyperspace.

Fabric Topology suggests that some kinds of knotting is intrinsic to the space. That is, the 22 torus can exist in finite 2-space, to the extent that you can pass from a point to itself in two different ways, and not meet except at the start and finish. This means that hedrices continue to have "genus", even though they are no longer "holes" (or rather the holes are outside of space).

Coxeter also talks of 'j-circuits that do not bound', such as the triangle faces of the great dodecahedron. These are kinds of hole, although the nature is a missing hedron (2-patch), there is nothing missing in the surface.
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Re: Holes of Toratopes (wendy's version)

Postby Keiji » Wed Nov 25, 2009 5:05 pm

Wendy, could you answer this question please?

Keiji wrote:One thing I just want to clarify, though:

wendy wrote:A 23 hole, formed by a spheric prism wrapped into a circle


This implies a spherinder (spherical prism) joined end to end gives a toraspherinder and that a cubinder gives a toracubinder.
Therefore 23 is toraspherinder and 32 is toracubinder.

wendy wrote:Consider for example, a pipe with cylinder section, and length. Were you to connect it end to end like a hose, the thing becomes a 23 hole. If you roll it down like a sock, so the top is rolled down the side of the cylinder to meet the bottom, then the thing is a 3-sphere covered by a 2-circle, giving a 32 hole.


This implies that a cubinder (cylindrical prism) joined end to end gives a toraspherinder and that a spherinder gives a toracubinder.
Therefore 32 is toraspherinder and 23 is toracubinder.

Which way around is it? I've believed the first case as the naming and toratopic notations make sense that way, but I recall some argument that it's actually the opposite.
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Re: Holes of Toratopes (wendy's version)

Postby wendy » Thu Nov 26, 2009 9:40 am

When mn is used as a toric or comb product, the great feature (eg "the thing looks like a hollow circle") is represented by the first number. So something that looks like a circle (or hollow 2d thing), and has a spheric (or 3d section) gives a 23 hole.

Both the figures, despite the cylinder section, give a 23 hole, since the primary shape has the form of a circle or polygon, which is hollow, the lot wrapped in a 3d section.

The second quote ought give a spherical section, not a cylinder section. Rolling the top downwards outside the tower, would lead to a thing at the bottom, that has a section the same as the original tower, and the bit around it would now be a circle. So it's a 32 hole.

mn holes are not particular kinds of shape but topological entities: the teacup into a doughnut sort of thing.
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Re: Holes of Toratopes (wendy's version)

Postby Keiji » Thu Nov 26, 2009 9:55 am

So, wait.

Does that mean that the toraspherinder and toracubinder both have both 23 and 32 holes, or does it mean the shapes that produce 32 holes are not toratopes?
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Re: Holes of Toratopes (wendy's version)

Postby wendy » Thu Nov 26, 2009 10:18 am

A 32 hole can be produced by taking a prism of a 3d figure with a cavity or internal hole in it. The general outline is that of a hollow sphere with thickness in four dimensions.

I don't know specifically know your torus figures completely, but torus-shaped figures produce holes of one kind only.

On the other hand, i believe figures like the tiger, and kindred 222 figures, produce a single 23 and a single 32 hole, but these are not apparent.
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Re: Holes of Toratopes (wendy's version)

Postby Keiji » Thu Nov 26, 2009 1:45 pm

I am myself completely confused about which way round the toraspherinder and toracubinder are. However, I can tell you that one of them is a sphere placed at every point in a circle, and the other is a circle placed at every point in a sphere. One is 3#2 and the other is 2#3, where # is the spheration operation and n is an n-sphere (where a "2-sphere" is a circle).

The toraspherinder is written ((III)I) in toratopic notation and its surface equation is sqrt((sqrt(x^2 + y^2 + z^2) - a)^2 + w^2) - b = 0.
The toracubinder is written ((II)II) in toratopic notation and its surface equation is sqrt((sqrt(x^2 + y^2) - a)^2 + z^2 + w^2) - b = 0.

Perhaps this is enough information to help you work out which is which? And if so, do tell, since I'd like to put my mind to rest on this matter :lol:
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Re: Holes of Toratopes (wendy's version)

Postby PWrong » Fri Nov 27, 2009 5:11 am

I'm posting from my new iPhone for the first time :D.

Toraspherinder is (211) = 2#3
Toracubinder is (31) = 3#2
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Re: Holes of Toratopes (wendy's version)

Postby Keiji » Fri Nov 27, 2009 6:31 am

Wait... the toraspherinder is (211)?

Bugger, that just about invalidates every toratope name on the entire wiki. Why didn't you tell me earlier? D:

It makes a lot more sense now, though.

So the toraspherinder is (211), 2#3, is a sphere at every point in a circle and has a 23 hole.
And the toracubinder is (31), 3#2, is a circle at every point in a sphere and has a 32 hole.

Starting with an uncapped spherinder 31 you can turn it in on itself to form a toracubinder (31) or you can bend it outside to form a toraspherinder (211).
The toracubinder and toraspherinder are toratopic duals (3d torus is a self-dual).

It's kind of annoying how the names swap over when you close an open toratope, though.
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Re: Holes of Toratopes (wendy's version)

Postby PWrong » Sat Nov 28, 2009 1:26 am

The names could be swapped around. It depends if you prefer toraspherinder to be a spherinder with ends attached or a spherinder with brackets on. But (31) = 3#2 follows from drawing a diagram and looking at the equations.
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Re: Holes of Toratopes (wendy's version)

Postby Keiji » Sat Nov 28, 2009 4:08 pm

Well, we could just say that closing an open toratope folds it through the inside rather than attaches the ends. Then the toraspherinder is both a closed spherinder and a spherinder folded through the inside. (And it means I don't have to go rewrite over 9000 wiki pages).
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Re: Holes of Toratopes (wendy's version)

Postby PWrong » Sun Nov 29, 2009 4:26 am

Speaking of the wiki, on the list of toratopes page there's a column called "hole sequence". It looks like homology groups but with a different notation. Has that always been there or is it new?
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Re: Holes of Toratopes (wendy's version)

Postby Keiji » Sun Nov 29, 2009 8:53 am

I calculated those here.
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Re: Holes of Toratopes (wendy's version)

Postby PWrong » Sun Nov 29, 2009 9:41 am

Ah ok. My conjecture 2 in that thread is wrong, so most of those will need updating. I'll do it when I get my computer working.
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