cone-like objects

Higher-dimensional geometry (previously "Polyshapes").

Postby PWrong » Wed Jun 14, 2006 10:46 am

Tapering in RNS notation only allows tapering to a point.

Most tapered objects can be constructed in more than one way, so it doesn't really matter. For instance, we know that cylinder->line and cylinder -> circle are valid objects because they're equivalent to the extruded cone and triangle x circle respectively.

Even we only allow tapering to a point, the formula 2*(p(k)+p(k-1)) won't work because it doesn't take into account objects formed by a product of tapertopes.

I imagine the triangular torus would be the cartesian product of a circle in the XZ plane and a triangle in the XY plane, but I haven't gotten used to cartesian products just yet

triangular torus would be a torus product circle # triangle. If it was circle x triangle, it would be a 4D shape.

And I intended to include the square torus, but:

We can't include the square torus. It's not related to tapering, so including it would change the nice formula for the number of toratopes.
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Postby moonlord » Wed Jun 14, 2006 11:33 am

I think I've figured out why the formula doesn't work. It is a sum over partitions of all integers 1<=i<=k, although I haven't yet managed to get the final formula...

The main difference between the torus and the cartesian product is the following: in the torus product, the two base bodies have a common dimension. In the cartesian, they can be in disjunct (correct word?) hyperspaces. While the torus product transforms in a certain way the cartesian equations of the base bodies, the cartesian just takes both into consideration.

Example: cartesian product of circle in XY and circle in ZW gives the diframe duocylinder.

first circle: x^2 + y^2 = r_1^2
second circle: z^2 + w^2 = r_2^2
diframe duocylinder: x^2 + y^2 = r_1^2 AND z^2 + w^2 = r_2^2

Hope this helps.

@PWrong: Why would you stick to the nice formula? Perhaps it just isn't correct.
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Postby PWrong » Wed Jun 14, 2006 12:09 pm

I think I've figured out why the formula doesn't work. It is a sum over partitions of all integers 1<=i<=k, although I haven't yet managed to get the final formula...

I posted the formula for the toratopes near the end of this thread. I suspect the formula tapertopes will be much more complicated, so we need a lot more data and some better notations.

The main difference between the torus and the cartesian product is the following: in the torus product, the two base bodies have a common dimension.

Not always. Sometimes a low-dimensional object sit in high-dimensional space. Generally, the second object "fills up" the remaining dimensions of the first, then adds new dimensions. For instance, the duocylinder margin is a 2D shape in 4D space. The circle is a 1D shape in 2D space, so duocylinder margin # circle is a 3D solid in 4D space.
Duocylinder # disk is a 4D solid in 4D space.
Duocylinder # sphere is a 4D solid in 5D space.

Let A be a kD object in nD space, and let B be a jD object in mD. Then A#B is a (k+j)D object in (k+m)D. I think that's right.

Example: cartesian product of circle in XY and circle in ZW gives the diframe duocylinder.

That's right.

@PWrong: Why would you stick to the nice formula? Perhaps it just isn't correct.

Well, it's correct provided you don't add extra shapes to the list of rotopes. I derived it in the thread I linked to, and I've confirmed it up to 12D with mathematica. Obviously if you add random shapes like the square torus (which is ugly and can't be written in RNS notation anyway), the formula has to change. Another reason for not adding the square torus is that there are several different kinds. If you rotate the original square 45 degrees, you get a kind of diamond torus.
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Postby Keiji » Wed Jun 14, 2006 2:56 pm

PWrong wrote:
@PWrong: Why would you stick to the nice formula? Perhaps it just isn't correct.

Well, it's correct provided you don't add extra shapes to the list of rotopes. I derived it in the thread I linked to, and I've confirmed it up to 12D with mathematica. Obviously if you add random shapes like the square torus (which is ugly and can't be written in RNS notation anyway), the formula has to change. Another reason for not adding the square torus is that there are several different kinds. If you rotate the original square 45 degrees, you get a kind of diamond torus.


And you can do that with a triangle as well, but that doesn't stop the triangular torus from being valid, as it is expressable in RNS notation. That doesn't apply to the square. So you can't exactly say you won't include it because of the orientation problems. ;)
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Postby PWrong » Wed Jun 14, 2006 3:52 pm

And you can do that with a triangle as well, but that doesn't stop the triangular torus from being valid, as it is expressable in RNS notation. That doesn't apply to the square. So you can't exactly say you won't include it because of the orientation problems.

I'm just listing reasons why the triangular torus seems unnatural. But if you insist on keeping it, we should at least make a list of objects excluding these things as well as including them. The list with the nicest formula will probably be the most natural set of objects.

I might write up a list of objects along with the number of parameters we can vary. I'm pretty sure that's the only way we're likely to find a formula (apart from counting up to 7D and using sloane's encyclopedia.)
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Postby PWrong » Tue Jun 20, 2006 1:53 pm

Now that I think about it, there's two ways we can count parameters. We could count an n-cube as having n parameters, or just one (size). We'll call the first number p, and the second q. We'll include both in the list. The table should also have the number of tapertopes we can derive from each toratope, that is T<sub>p</sub> and T<sub>q</sub>. The number of 5D tapertopes should be equal to the sum of the T<sub>p</sub>s in 4D, plus the number of toratopes in 5D.

Here's the table. I'm not sure if this is right
Code: Select all
2D:
shape   p   q     T_p   T_q
11      2   1     2     1
2       1   1     1     1

3D:
111     3   1     3     1
21      2   2     3     3
3       1   1     1     1
(21)    2   2     3     3

4D:
1111    4   1     4     1
211     3   2     5     2
22      2   1     2     1
31      2   2     3     3
4       1   1     1     1
(211)   2   2     3     3
(22)    3   2     5     2
(31)    2   2     3     3

EDIT:
(21)1   3   3     7     7
((21)1) 3   3     7     7
Last edited by PWrong on Wed Jun 21, 2006 11:44 am, edited 3 times in total.
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Postby moonlord » Wed Jun 21, 2006 10:01 am

I'm sure if I understood your post completely, but it seems to me that sometimes you get the same body from two different bodies, by using different transformations. Example: cube, rotated and cylinder, extruded both give the 211, the cubinder.

Also, I don't see two 4D bodies in your list :?, the (21)1 and ((21)1) - toric prism and tritorus.
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Postby PWrong » Wed Jun 21, 2006 12:15 pm

but it seems to me that sometimes you get the same body from two different bodies, by using different transformations. Example: cube, rotated and cylinder, extruded both give the 211, the cubinder.

Yes, that's true, but cylinder only has one representation in RNS. Perhaps a more relevent example is square -> line = extruded triangle. This is the reason for having two ways of counting parameters.

Also, I don't see two 4D bodies in your list , the (21)1 and ((21)1) - toric prism and tritorus.
You're right, I forgot those.

I've just realised that (1<sup>1</sup>1) shouldn't represent the circle#triangle at all. Putting brackets around an object isn't the same as the torus product, although the two operations are related. The "bracket" operations tends to curl things up. For instance, it turns a line into a circle, a square into a sphere, and a cylinder into a torus.

Let me demonstrate how the method I'm using works. Instead of using <sup>1</sup> all the time, I'll just use a dash ('). I'll still use <sup>2</sup> for 2 taperings. Suppose we count 11 as having two parameters (that seems to be the best way). Because 1'1=11', there are two ways to taper it:
1'1 = triangular prism, and 1'1' or [11]' = square pyramid.

Let's look at (21). The torus has two parameters, (the major and minor radii). We can taper these in three ways (although there may be a fourth way which I'll start a new thread on). We have:
(2'1) = torus -> circle
(21)' = torus -> sphere
(2'1)' = torus -> point

So far, I haven't considered objects formed by tapering, and then using some other operation. Hopefully all such objects can be produced by using tapering last. For instance, extruded triangle seems to be missing, until we realise it's covered by square -> line. I also haven't considered object being tapered twice. To do this, we'll have to look at the parameters of tapertopes. We've already decided that a triangle only has one valid parameter: size (otherwise we get wedges). Hopefully this won't be too much of a problem.
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Postby Keiji » Wed Jun 21, 2006 4:48 pm

PWrong wrote:I've just realised that (1<sup>1</sup>1) shouldn't represent the circle#triangle at all. Putting brackets around an object isn't the same as the torus product, although the two operations are related. The "bracket" operations tends to curl things up. For instance, it turns a line into a circle, a square into a sphere, and a cylinder into a torus.

Let me demonstrate how the method I'm using works. Instead of using <sup>1</sup> all the time, I'll just use a dash ('). I'll still use <sup>2</sup> for 2 taperings. Suppose we count 11 as having two parameters (that seems to be the best way). Because 1'1=11', there are two ways to taper it:
1'1 = triangular prism, and 1'1' or [11]' = square pyramid.


We already established that the taper operation would apply to everything before it, due to the fact that the only "extra" shapes we get from not applying this rule are dupes, and that not applying this rule introduces the need for cumbersome square brackets. I'd rather keep this rule.

Let's look at (21). The torus has two parameters, (the major and minor radii). We can taper these in three ways (although there may be a fourth way which I'll start a new thread on). We have:
(2'1) = torus -> circle
(21)' = torus -> sphere
(2'1)' = torus -> point


Wrong:

(2'1) = Circle tapered extruded rounded = Conic prism rounded = torus -> circle
(21)' = Circle extruded rounded tapered = Torus tapered = torus -> point
(2'1)' = Circle tapered extruded rounded tapered = Conic prism rounded tapered = (torus -> circle) -> point (5D object)

There is still no notation for the torus -> sphere.

So far, I haven't considered objects formed by tapering, and then using some other operation. Hopefully all such objects can be produced by using tapering last. For instance, extruded triangle seems to be missing, until we realise it's covered by square -> line. I also haven't considered object being tapered twice. To do this, we'll have to look at the parameters of tapertopes. We've already decided that a triangle only has one valid parameter: size (otherwise we get wedges). Hopefully this won't be too much of a problem.


Why not just stick to what we already have?

The current RNS notation is fine. By adding in "parameters" and rules like "tapering last" you are confusing things. Tapering does not take parameters. It is done by copying the objects and scaling down until you reach a point.
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Postby moonlord » Wed Jun 21, 2006 6:19 pm

(21') = circle extruded tapered rounded = cylindric-cone rounded.

Can't see the result of the last operation, though. It's some kind of 4D torus, I think. Call it shape X.

(2'1') = circle tapered extruded tapered rounded = conic-prism tapered rounded.

Seems 5D to me. Shape Y.

(21')' = circle extruded tapered rounded tapered = shape X tapered.

5D, obviously. And the bomb on the funeral wheat porridge:

(2'1')' = circle tapered extruded tapered rounded tapered = shape Y tapered.

Sounds 6D, doesn't it? Yet, from all this we have one unaccounted 4D body, the shape X. I'd like to see your oppinions.
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Postby Keiji » Wed Jun 21, 2006 7:40 pm

None of the four shapes you proposed exist. :P To "round" anything, it needs to have at least one pair of opposite, flat hypercells - and obviousness will show that if the last operation applied to something was tapering, this condition won't be met.

Edit: That got me thinking. The duocylinder has no flat cells. So does the tiger really exist? :?
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Postby moonlord » Thu Jun 22, 2006 11:34 am

I'm pretty sure the (21') has two opposite 2D flat faces.

Rob wrote:[...] and obviousness will show that if the last operation applied to something was tapering, this condition won't be met.


I disagree. Tapering doesn't destroy any existing cell, no matter it's dimensionality. Or do you require the two opposite faces to have a certain dimensionality?

EDIT: And yet, I love proposing impossible bodies :D.

EDIT 2: I've just had a revelation (?). (X > point) > point, aka X'' is the same as X > segment. Or isn't it? I've got to ponder on this...
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Postby PWrong » Thu Jun 22, 2006 1:37 pm

I just finished my last exam today, so I might not be completely sober yet.

We already established that the taper operation would apply to everything before it, due to the fact that the only "extra" shapes we get from not applying this rule are dupes, and that not applying this rule introduces the need for cumbersome square brackets. I'd rather keep this rule.
You established that, not me. That rule doesn't allow all the torus-derived tapertopes.

There is still no notation for the torus -> sphere.
That's exactly why your notation doesn't work.

(2'1') = circle tapered extruded tapered rounded = conic-prism tapered rounded.
What's this "rounded" operation? Putting brackets around an object is complicated. It's not even strictly defined yet. For any given object A, the bracketed object (A) doesn't necessarily exist. When we taper something, we start with the object, then we taper it. That's why we need the parameters.

I'm pretty sure the (21') has two opposite 2D flat faces.
I'm pretty sure the (21') doesn't exist.
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Postby Keiji » Thu Jun 22, 2006 2:57 pm

moonlord wrote:I'm pretty sure the (21') has two opposite 2D flat faces.


You mean: you're sure the 21' has two opposite flat faces.

It does! But guess what? 21' is a 4D object, so it needs a pair of opposite flat CELLS to be able to be rounded - not faces.

I just finished my last exam today


Haha, me too! Coincidence, eh? :P

You established that, not me. That rule doesn't allow all the torus-derived tapertopes.


And the only ones that are excluded by that rule so far are the obscure shapes, such as wedges and self-intersecting torii.

There is still no notation for the torus -> sphere.

That's exactly why your notation doesn't work.


Nope, it's why the torus -> sphere should be excluded. Why couldn't I say there's no notation for a tree, so your notation doesn't work? :roll:

For any given object A, the bracketed object (A) doesn't necessarily exist.


Like I said, (A) exists if A has at least one pair of opposite flat hypercells.

When we taper something, we start with the object, then we taper it. That's why we need the parameters.


And like I said as well, why can't we do other operations after tapering? That removes the need for parameters, and removes the ugly shapes that certainly shouldn't be tapertopes.
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Postby PWrong » Thu Jun 22, 2006 3:16 pm

The duocylinder has no flat cells. So does the tiger really exist? Confused

The cells of a duocylinder are flat torii.

And the only ones that are excluded by that rule so far are the obscure shapes, such as wedges and self-intersecting torii.

Both our notations exclude the wedge. Also, we don't really need the square brackets in either notation. My square pyramid is 1'1', while yours is 11'. I don't think self intersection is a problem. The torus itself is self intersecting if the major radius is smaller than the minor radius.

And like I said as well, why can't we do other operations after tapering? That removes the need for parameters, and removes the ugly shapes that certainly shouldn't be tapertopes.

The parameters method is the most natural way of tapering. I think I'm already close to finding a formula with it.
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Postby moonlord » Thu Jun 22, 2006 4:14 pm

Now, it seems I start saying bullshit :(.

I've thought about the lack of definition for "rounding" but decided to give it a shot. Missed, obviously. So, to sum up, in order to be able to round something (nD) the object needs to have two opposite (n-1)D hypercells that are flat. Flat, as in zero curvature. But how can you deduce this from the body's equations (cartesian/parametric/whatever)?

PWrong wrote:The cells of a duocylinder are flat torii.


Now, what are those? Things like LLE? :?

Rob wrote:
PWrong wrote:I just finished my last exam today

Haha, me too! Coincidence, eh?:p


Funny thing, this also holds for me. I'm either tired or the exams made me stupid :roll: ...
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Postby Keiji » Thu Jun 22, 2006 9:51 pm

PWrong wrote:
Rob wrote:The duocylinder has no flat cells. So does the tiger really exist? :?

The cells of a duocylinder are flat torii.


Define a flat torus, then. Preferably in RNS or CSG notation.

PWrong wrote:
Rob wrote:And the only ones that are excluded by that rule so far are the obscure shapes, such as wedges and self-intersecting torii.

Both our notations exclude the wedge. Also, we don't really need the square brackets in either notation. My square pyramid is 1'1', while yours is 11'. I don't think self intersection is a problem. The torus itself is self intersecting if the major radius is smaller than the minor radius.


Self-intersecting torii are disregarded because a torus doesn't have to be self-intersecting. However, a torus -> sphere will always self-intersect.

PWrong wrote:
Rob wrote:And like I said as well, why can't we do other operations after tapering? That removes the need for parameters, and removes the ugly shapes that certainly shouldn't be tapertopes.

The parameters method is the most natural way of tapering. I think I'm already close to finding a formula with it.


Not really. The most natural way is simply to scale to a point, which is a continuous transformation of the shape along the axis from base to apex. With the parameters method, you aren't transforming the shape, you are completely re-creating it.

moonlord wrote:So, to sum up, in order to be able to round something (nD) the object needs to have two opposite (n-1)D hypercells that are flat. Flat, as in zero curvature. But how can you deduce this from the body's equations (cartesian/parametric/whatever)?


No idea. But I can say that if an object in RNS notation has at least one 1 not in brackets and after any taper operations, it definately has at least one pair of opposite flat hypercells. This was how I arrived at the conclusion that it's impossible to round a duocylinder.

moonlord wrote:
PWrong wrote:The cells of a duocylinder are flat torii.


Now, what are those? Things like LLE? :?


LLE would be the curved face of the cylinder.

moonlord wrote:
Rob wrote:
PWrong wrote:I just finished my last exam today

Haha, me too! Coincidence, eh?:p


Funny thing, this also holds for me. I'm either tired or the exams made me stupid :roll: ...


Lol! Well you shouldn't have to worry about that any more. Maybe we'll have some rather more intelligent discussions once people get their brains back :mrgreen:
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Postby PWrong » Fri Jun 23, 2006 10:03 am

I'm either tired or the exams made me stupid ...


In my case, what made me stupid last night was a combination of tiredness, exams and about six beers. :lol:

PWrong wrote:
The cells of a duocylinder are flat torii.


Now, what are those? Things like LLE?


It's topologically the same as a torus, but it's flat and it lives in 4D.

Not really. The most natural way is simply to scale to a point, which is a continuous transformation of the shape along the axis from base to apex. With the parameters method, you aren't transforming the shape, you are completely re-creating it.


What do you mean? There's nothing discontinuous about the parameters method. You simply start with the original object, take one of the parameters, and scale it down to zero as you move up the axis. The great thing about it is once you've tapered something, you don't need to do anything else to it. If you extrude it or "round" it or anything, you just get another tapertope that's already been made.

Here's a list of tapertopes, alongside the objects that can be created by tapering them.
I can't use the sup tags in code, so I'll just use ('') instead of <sup>2</sup>.

Code: Select all
2D:
11   | 1'1, 1'1'
2    | 2'
1'   | 1''

3D:
11   | 1'11,  1'1'1, 1'1'1'
21   | 2'1,   21',   2'1'
3    | 3'
(21) | (2'1), (21)', (2'1)'
2'   | 2''
1''  | 1'''



Before I finish this list, I'll look at the parameters of the remaining objects.

1'1 = triangular prism
This has 2 parameters: the size of the triangle, and the length.
Tapering the triangle size gives a tetrahedron prism. [1'1]->1 = 1<sup>2</sup>1
Tapering the length gives (triangular prism -> triangle) = 1'1->1'
This shape is tricky. I can't think of a simple notation for it.

1'1' = square pyramid
Now I'm not sure about the parameters. If we say the two side lengths are parameters, then we allow square pyramid -> triangle. From drawing this shape, it looks like one of its cross sections is the wedge. That's no good. So I think it might be wise to give 1'1' only one parameter (size). That way our only shape is 1<sup>2</sup>1<sup>2</sup>. Now the problem is that we've left out 1<sup>2</sup>1'. I'm not sure how to interpet this one.

So we have one object without a notation and one notation without an object. I'll try to visualise these again later.

moonlord: Remind me not to post in the morning anymore. I hit edit instead of quote and realised this after submiting the post. Hopefully, I still had the old page in the history and recreated PWrong's post... :|
Last edited by PWrong on Sat Jun 24, 2006 3:42 pm, edited 1 time in total.
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Postby Keiji » Fri Jun 23, 2006 2:12 pm

PWrong wrote:
PWrong wrote:
The cells of a duocylinder are flat torii.

Now, what are those? Things like LLE?

It's topologically the same as a torus, but it's flat and it lives in 4D.


That doesn't define what it is.

Not really. The most natural way is simply to scale to a point, which is a continuous transformation of the shape along the axis from base to apex. With the parameters method, you aren't transforming the shape, you are completely re-creating it.

What do you mean? There's nothing discontinuous about the parameters method. You simply start with the original object, take one of the parameters, and scale it down to zero as you move up the axis. The great thing about it is once you've tapered something, you don't need to do anything else to it. If you extrude it or "round" it or anything, you just get another tapertope that's already been made.


Sorry, but I really do find reordering operations a lot easier than messing around with parameters and getting objects that don't exist. :roll:

1'1' = square pyramid
Now I'm not sure about the parameters. If we say the two side lengths are parameters, then we allow square pyramid -> triangle. From drawing this shape, it looks like one of its cross sections is the wedge. That's no good. So I think it might be wise to give 1'1' only one parameter (size). That way our only shape is 1<sup>2</sup>1<sup>2</sup>. Now the problem is that we've left out 1<sup>2</sup>1'. I'm not sure how to interpet this one.


Simple, one parameter is height and the other is length of the sides of the square base.
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Postby moonlord » Sat Jun 24, 2006 6:47 am

PWrong wrote:
Code: Select all
2D:
11   | 1'1, [b]1'1[/b]
2    | 2'
1'   | 1''

3D:
11   | 1'11,  1'1'1, 1'1'1'
21   | 2'1,   21',   2'1'
3    | 3'
(21) | (2'1), (21)', (2'1)'
2'   | 2''
1''  | 1'''



Shouldn't the "bold" object be 1'1'? I mean, the triangular prism appear twice and there is no square pyramid... Also, it seems you miss them from the list of 3D tapertopes, too.

As a side note, I've been thinking about circle/disk -> segment, but it seems you can't give it a proper notation. It's quite... ugly, anyway. Should we include it? More general, circle/disk -> segment should be treated as ellipse -> ellipse, and (I think) will be 4D if one of the ellipses isn't degenerate.

PWrong wrote:1'1 = triangular prism
This has 2 parameters: the size of the triangle, and the length.
Tapering the triangle size gives a tetrahedron prism. [1'1]->1 = 1<sup>2</sup>1
Tapering the length gives (triangular prism -> triangle) = 1'1->1'
This shape is tricky. I can't think of a simple notation for it.


Firstly, please explain what are the size and the length of a triangle are.

EDIT: I understand now: the size of the triangle is it's side, the length is reffering to the prism and it's the height. Oh well.

It seems you consider triangles equilateral. What if we consider a triangle to have two parameters, the side and the height?

EDIT 2: Crap, I am tired. It has been discussed before and it gives birth to wedges and other artefacts :|.

Is 1''1 the same with 11''? If this is the case, then we might identify some bodies. In this case, the tetrahedral prism and the square bipyramid seem to be the same body...

How about 1'1 -> 1'? Isn't it 5D? Oh and, by the way, you can also taper a 1'1 to a point, giving the triangle prism pyramid, and I think it's notation should be 1''1'. Which seems to be the same with the tetrahedral x triangle. Can you visualise 1'' x 1', when reffering to the transformation? I personally cannot.

PWrong wrote:1'1' = square pyramid
Now I'm not sure about the parameters. If we say the two side lengths are parameters, then we allow square pyramid -> triangle. From drawing this shape, it looks like one of its cross sections is the wedge. That's no good. So I think it might be wise to give 1'1' only one parameter (size). That way our only shape is 1<sup>2</sup>1<sup>2</sup>. Now the problem is that we've left out 1<sup>2</sup>1'. I'm not sure how to interpet this one.


1''1'' is then the square bipyramid, and the 1''1' is the triangle prism pyramid, as I think I correctly found out above.

EDIT 3: One note on the notation: 1''1'' might give the impression to be 6D, when it's actually [11]''. I hit another thing, whether in general, X<sup>n</sup>Y<sup>m</sup> is the same with [XY<sup>m-n</sup>]<sup>n</sup>, considering n<m.

Questions, quoestions...
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Postby PWrong » Sat Jun 24, 2006 3:59 pm

Shouldn't the "bold" object be 1'1'? I mean, the triangular prism appear twice and there is no square pyramid... Also, it seems you miss them from the list of 3D tapertopes, too.

Yep, my bad. I've just edited it.

As a side note, I've been thinking about circle/disk -> segment, but it seems you can't give it a proper notation. It's quite... ugly, anyway. Should we include it?

I agree, it's too ugly, and there's no notation for it. Circle only has one parameter, and the ellipse isn't one of our 2D shapes.

It seems you consider triangles equilateral.

They don't have to be equilateral. You can have any angles you want, but you can't change them.

Is 1''1 the same with 11''?

In my notation they're the same. I think in Rob's notation, the dash applies to everything before it, so 11'' is the same as 1''1''.

How about 1'1 -> 1'? Isn't it 5D?

Nope. 1'1 is a 3D shape. Tapering it just adds a dimension. Generally, dim( A->B ) = dim(A) + 1 or dim(B) + 1, whichever is bigger.

Oh and, by the way, you can also taper a 1'1 to a point, giving the triangle prism pyramid, and I think it's notation should be 1''1'.

That seems right.

Can you visualise 1'' x 1', when reffering to the transformation?

Nope. Usually, if I can visualise A and B, then I can visualise A->B and A#B, but not AxB.

I hit another thing, whether in general, XnYm is the same with [XYm-n]n, considering n<m.

I'm hoping it is, but I'm not sure. If that's not the case, then we might need a better notation.
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Postby moonlord » Sat Jun 24, 2006 5:41 pm

This is weird. Think about 1'1 -> 1'.

It is a rectangle. You taper one side so that you get a triangle prism lying on a side. Now you taper again, the side of the rectangle that hasn't been used. If you haven't used it before, you can place it before the first tapering, so that it seems you can get 1'1 -> 1' in the following manner: point extrude (line now) taper (triangle) extrude (prism) taper (the new extrusion segments). It should be something like 1'1'.

So it seems we might need to expand our notation. [X]' means extruding all parameters of X at once, in the same new dimension, and X'Y' means extruding parameters X and Y in two new dimensions. So, by this notation, we have the tapertopes, considering the variant with the highest net dimensionality:

EDIT: removed list. Will post a image with a graph soon.
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Postby Marek14 » Sun Jun 25, 2006 7:11 am

PWrong wrote:
How about 1'1 -> 1'? Isn't it 5D?

Nope. 1'1 is a 3D shape. Tapering it just adds a dimension. Generally, dim( A->B ) = dim(A) + 1 or dim(B) + 1, whichever is bigger.

Actually, it doesn't have to be. I think it's dim(A U B) +1 which can be different (try tapering circle to a line that is perpendicular to its plane)
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Postby PWrong » Sun Jun 25, 2006 7:24 am

How can you do that? :?
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Postby Marek14 » Mon Jun 26, 2006 5:57 am

PWrong wrote:How can you do that? :?


Treat the circle as being a cylinder of diameter 1 and height 0, and as you go "upwards" in fourth dimension, continually decrease the diameter and increase the height, to end with a line of diameter 0 and height 1.

It's the same thing as cone tapered to a point, just seen with more symmetry
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Postby PWrong » Mon Jun 26, 2006 12:55 pm

Ok, I get it. It's constructed in a similar way to the (2<sup>1</sup>1)<sup>-1</sup>.
I guess it could be called 2<sup>1</sup>1<sup>-1</sup>
So we have an interesting identity: 2<sup>1</sup>1<sup>-1</sup> = 2<sup>2</sup>
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Postby moonlord » Tue Jun 27, 2006 7:34 am

It seems there is more than one way to taper disk -> segment, so how are we going to deal with that?

Besides, what happens if you don't increase the height? Don't you get the same thing?

Why not stick only to parameter-to-a-point tapers?
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Postby PWrong » Wed Jun 28, 2006 4:54 am

Besides, what happens if you don't increase the height? Don't you get the same thing?

Then you just get a cone.

Why not stick only to parameter-to-a-point tapers?

2<sup>1</sup>1<sup>-1</sup> is a parameter-to-point taper. Radius decreases to 0 in one direction, length decreases to 0 in the other direction. That said, I'd rather find a formula for the ordinary tapertopes before we include these.

It seems there is more than one way to taper disk -> segment, so how are we going to deal with that?

The other ways include "disk->segment via ellipse" and "disk->segment via cut circle". But ellipses and cut circles aren't real tapertopes, so they don't count.
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