Definition of the "taper" transformation

[houserichichi]

Care to explain the (or point me in the direction of your) definition of "tapering"? Is it setting up a homeomorphism?

Sorry for the interlude

moonlord: Split from "the fourth tapered torus".

[moonlord]

The taper operation, as I see it, means performing the following:

1. Take two bodies, A and B. Each of them is embed in the minimal dimensional hyperspace (the bounding space of each - alpha and beta). Place the two bodies in such a manner that these two hyperspaces do not intersect.
2. Find a way to continuously transform A into B by changing it's parameters, like length, height, trenght, radius and so on. If it doesn't exist, tapering is not possible (not sure here).
3. Transform A into B (cont.) and in the same time translate the current body towards B so that when the transformation is complete (A has become B), the result is in the position of B.
4. Consider the whole "trail" one body, C.

Therefore, A -> B = C.

I'll give an example.
1.Take A a disk - alpha is a plane. Take B a segment - beta is a line. Place the line parallel to the plane, but not in the plane.
2. A way to transform a disk to a segment is to consider them both ellipses, with radii (r1,r2), respectively (l,0) - r1=r2 is the radius of the disk, l is the length of the segment.
3. While translating A towards B, linearily transform r1 into l, by modifing it's length, and shrink r2 to zero.
4. The reunion of the disk, the segment and all the middle ellipses is the disk -> segment result.

Problems with my definition of the taper operation:
1. There is more than one way to taper A into B, depending on their position. In my example, the line could also be positioned in another 3D realm than that of the circle, perpendicular to its plane. The result is then the same with cone -> point. See [url="http://tetraspace.alkaline.org/forum/viewtopic.php?t=588&postdays=0&postorder=asc&start=30"]this.[/url]. Scroll towards the end of the page...
2. The taper product only considers the "full" bodies, those with net space equal with the bounding space.

[PWrong]

Care to explain the (or point me in the direction of your) definition of "tapering"? Is it setting up a homeomorphism?

I'm not quite sure what a homeomophism is. As far as I can tell, it's the topological equality, so a coffee cup is homeomorphic to a doughnut.

I don't know how it applies to tapering. I think almost all tapertopes are homeomorphic to each other, except the torii.

[bo198214]

ahm, novice question: isnt the tapering simply the union of all lines between A and B?
I dont think that it has to do with homeomorphism if I taper a point to a circle, they are not homeomorphic.

[houserichichi]

That's the example I had in my head that forced me to ask the question. I see now what they're doing. It seems to me that tapering is two things at once, but please correct me if I'm wrong.

I visualize a circle of radius r being tapered to a line of length l (where l < r). So along some axis we continuously shrink r down to l. On the perpendicular axis we shrink r down to zero. If we also imposed that the embedding space of the circle is parallel (in some sense of the word) to that of the embedding space of the line, then the tapering isn't the actual deformation, but it's the union of lines between embedding space of the circle and the embedding space of the line, assuming they're both embedded in some higher-dimensional space.

If the deformation has an inverse and is a bijection between the two spaces then it's clearly a homeomorphism. However, as bo198214 said, a circle and point are clearly not homeomorphic.

Is tapering the entire process of shrinking circle down, stretching it out long, then "dragging" it through higher-dimensional space into the other embedding space? I suspect my reasoning is off because it seems quite intuitive to others...I think my process is right, I just don't know what's considered "tapering" and what's not.

Sorry to hijack this thread. Mods are welcome to split this off - I just want to get an explanation of what tapering is.

[PWrong]

then the tapering isn't the actual deformation, but it's the union of lines between embedding space of the circle and the embedding space of the line

That's right, it's a union of lines. Except we can't actually taper from circle to line, except via a cylinder, and that gives a 4D shape. Also, usually when we taper from A->B, the embedding space of A is parallel to the embedding space of B, but I think circle->line is a special case.

Generally, you can't taper from A to B unless B is a special case of A, or A and B are special cases of a single object. A point is a circle with radius 0, so circle -> point is allowed. Line is a cylinder with radius 0, and circle is a cylinder with length 0, so line->circle via cylinder is allowed. The in-between objects have to be tapertopes though, you can't use an ellipse or a circle with bits cut off.

It would be nice to have a more strict definition. We could use your help, since you know about homeomorphisms and things.

[houserichichi]

Wait, isn't a line a special case of a circle (one where we pick perpendicular diameters and shrink one down to zero?

Now, I'm thinking this: what if we consider your tapering as a "world line" of sorts. Say you have a plane A that is underneath and parallel to some plane B. In between A and B there are infinitely many planes. In plane A we draw a circle. In plane B, directly above the circle, we draw a line of length equal to its diameter. Now, start the clock. Every instant of time that passes we shrink the diameter that is perpendicular to the line (were we to superimpose the circle on the line) closer to zero by equal increments. However, we do this on the "closest" plane directly above the one we were just on.

In words it's terrible to describe so just picture a circle in plane A and on top of that an ellipse and on top of that a skinnier ellipse, and on top of that an even skinnier ellipse, etc etc etc until ultimately the ellipse is so skinny that it's the line (and we're actually all the way up to plane B). Since there are infinitely many planes stacked up we have a higher-dimensional (3-space) object, sort of an upside down cone except at the top is a line rather than a point. Get what I'm trying to explain?

Is that what tapering is or am I still an idiot or do I need to draw pictures (because jebus knows I sure love Microsoft Paint).

[PWrong]

Wait, isn't a line a special case of a circle (one where we pick perpendicular diameters and shrink one down to zero?

Both line and circle are special cases of an ellipse, but a line isn't a special case of circle. If A is a special case of B, then every A is a B, but not every B is an A.

Your description of tapering is correct, but if you do it with an ellipse it doesn't count as a tapertope. The resulting shape is too ugly, and anyway there are infinitely many ways to transform a circle into a line.

[bo198214]

Oh I thought he meant a circle with radius infinity and center at (some) infinity...

[moonlord]

PWrong: So we're only going to consider A -> B via C when all of them are tapertopes. That rules out the circle -> line via ellipse and via cut circle.

EDIT: I think I've found out a way to uniquely define tapering. We're only going to consider A -> B via AxB, where A and B are tapertopes, and so is AxB (cartesian product of A and B). What do you think?

[PWrong]

We're only going to consider A -> B via AxB, where A and B are tapertopes, and so is AxB (cartesian product of A and B). What do you think?

Well, that works for circle->line (since circle x line = cylinder) and the cone (circle x point = circle), but it doesn't work for circle -> sphere via torus (circle x sphere = the rotatope 32) or the triangular prism, which is square -> line (because square x line = rectangular prism, not rectangle)

[houserichichi]

I'm real sorry, I still think I'm missing something here so I'll use moonlord's taper algorithm for the circle to line argument and someone tell me which step I've got wrong.

1. Take two bodies, A and B. Each of them is embed in the minimal dimensional hyperspace (the bounding space of each - alpha and beta). Place the two bodies in such a manner that these two hyperspaces do not intersect.

Body A: Circle
Body B: Line

2. Find a way to continuously transform A into B by changing it's parameters, like length, height, trenght, radius and so on. If it doesn't exist, tapering is not possible (not sure here).

If we draw 2 perpendicular axes on a 2 dimensional sheet of paper (embedding space) and draw a circle of radius 1, for instance, then we can shrink the "height" down to zero. What I literally mean is connect everything in the upper-half plane to a single corresponding point in the lower half plane with vertical lines parallel to the y-axis like in the diagram below



(clearly I got lazy with drawing lines so imagine a blue line connecting every point above the x-axis with a corresponding point below it) Now shrink each of these lines down to length zero (excluding the endpoints because they're already at "height" zero). Now we have a line, no?

3. Transform A into B (cont.) and in the same time translate the current body towards B so that when the transformation is complete (A has become B), the result is in the position of B.

So like I said in my previous example, each time the circle "shrinks" a little bit it gets a little bit "closer" to the line. So it's similar to a cone except that it has a line for a tip rather than a point.

4. Consider the whole "trail" one body, C.

So my line-cone is the taper?



(from above: except that it should be a circle, not an ellipse. Oops)

*********

Now, since there are infinitely many ways to taper a circle to a line depending on orientation of the line with respect to the shrinking axis and/or the angle at which the circle approaches the line in the time translation, which one do we consider the canonical tapering? (You mentioned that the embedding spaces are parallel, but you did not mention whether A and B are "on top" of eachother in a strictly informal use of the word.

Also, if a circle cannot be tapered to a line could you please explain to me via the same steps I just took how a disc can? I just don't see where the difference is, especially if a circle can be tapered to a point. See, you say that two objects are taperable if they are both special cases of one thing. A circle is a special case of an ellipse (you said). A line is a special case of an ellipse too, no? Ellipse has two axes of different lengths. Shrink the length of one of these axes down to zero and you have a line. (I'm certain this is where I'm getting my definition mixed up.) Thus circle and line are taperable.[/img]

[PWrong]

You understand the taper operation fine. I think what you don't understand is the tapertope. Tapertopes are a set of objects produced by the taper operation (among others). There is a finite number of tapertopes in each dimension.

See, you say that two objects are taperable if they are both special cases of one thing.

The rule is that A -> B via C is a tapertope iff A, B and C are tapertopes and A and B are both special cases of C. Furthermore, all rotatopes and toratopes are tapertopes. So ellipse isn't a tapertope, and neither is circle -> line via ellipse. That doesn't mean circle -> line via ellipse doesn't exist, it's just not a valid tapertope.

[moonlord]

Besides, I didn't say circle -> segment is impossible, I just said we didn't think about them yet. The only problem we might foresee is that circle -> point (via circle) is a conical surface - that is, without the base disk. This, while it might seem reasonable that circle (1D) -> segment (1D) gives the full 2D cone - the surface of a cone.

The biggest problem of the taper operation now is that there are infinitely many C's to perform A -> B via C for given A and B. See, for example, circle/disk -> segment via ellipse, cut circle, cylinder, 211, which all give different bodies.

[PWrong]

Here's a new one. Start with the closed circle, break it at an arbitrary point so it becomes an open curve (i.e. it has two ends), and "uncurl" it as you go up.

[Keiji]

Here's a new one. Start with the closed circle, break it at an arbitrary point so it becomes an open curve (i.e. it has two ends), and "uncurl" it as you go up.

Now you're just getting silly.

That object is just a curved plane with a single point of convergance.

[pat]

For the whole circle-line tapering... is there a distinction being made between a circle and a disc? It seems to me that one can continuously deform a disc into a line... but it seems like there's a discontinuity when you go from a circle to a line. Points that are not neighbors suddenly become neighbors.

[moonlord]

Well we're (at least I am) considering the highest-dimensional version, that is the disk for this example. And what's wrong with that discontinuity? It also appears in the circle -> point, at the apex of the cone...

[pat]

There's nothing wrong with discontinuity, per se. But, if you take a hollow torus and taper it to point, then you've taken a 2d surface, tapered it, and ended up with an object that has at least one point where the object isn't locally homeomorphic to an open ball. I suppose, it's just a wacky edge condition.

[houserichichi]

After reading the wiki entry on rotopes (for no apparent reason) and scouring down through the chart, I finally understand what in the hell tapering is. I don't know why I couldn't grasp it before. Now I'll have to go back through all the old entries and see if I can't contribute something of merit.

I'm curious where the 'digit' notation arose from in the chart. I've seen it all over the geometry forum but haven't taken the time to learn it. Did we just arbitrarily define 1 = line, 2 = circle, etc? I'll hunt through and see what I can come up with.

Someone should change the name of this topic to "teach a dummy mathematician the basics".