cone-like objects

Higher-dimensional geometry (previously "Polyshapes").

cone-like objects

Postby PWrong » Fri Jun 09, 2006 11:29 am

I know this has been looked at before, but I don't think we've ever tried enumerating the possible cones, or combining cone-like objects with torii.

The main operation with cones is tapering. Now although usually we taper from some object into a point, I think it's also possible to taper from one object to another object, as long as we have certain symmetry conditions (I have no idea what these are yet). For instance, the coninder (linear extension of a cone) is the same as a cylinder, tapered into a line.

Here's a few 4D cone-like objects I've found, some of which have been considered before.

1. coninder
Linear extension of cone = cylinder tapered to a line

2. cylindrone
cylinder tapered to point

3. spherone
Sphere tapered to a point

4. Cone tapered to a point
= circle tapered to a line

5. Torus tapered to a circle
We start with a torus with major radius R, minor radius r. Then we reduce r as we go up the w axis.
This is the same as a cone, translated, then rotated so that the circlular base forms a torus, while the tip forms a circle.
I think we can also describe it as circle#cone, even though the torus product isn't well defined for non-spheres.

6. Torus tapered to a point
Start with a torus, and reduce R and r simultaneously up the w axis, so that for some w, R=r=0.
Or, we can start with a cone (circle in xz, pointing to w), translate the base along x without moving the tip, then rotate the base in the xy plane.
In torus product notation, this is cone#sphere.

7. Torus tapered to sphere.
Here we start with a torus, and reduce R with the w axis, holding r constant. When R=0, we get a sphere. This might be a pretty ugly shape, and I'm happy to count it as invalid.
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Postby Keiji » Fri Jun 09, 2006 4:52 pm

In each of those cases, the object you are tapering something to is a special case of the object you are tapering, which is why it is possible. A point is a special case of all objects in all dimensions, which is why tapering is usually done to a point.

You could easily have a cube tapered to a square, which would be the linear extension of a triangular prism, or a cube tapered to a line, which would be the linear extension of a triangular pyramid.

As for the torus tapered to a sphere, it is just as valid as any other self-intersecting polytope. Personally, I dismiss self-intersecting polytopes as invalid, but it's up to you whether you want to count it as valid or not.
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Re: cone-like objects

Postby Marek14 » Sat Jun 10, 2006 6:38 am

Basically, these would be the "round" counterparts to Mr. Klitzing's "segmentochora" which are polytopes where all vertices lie in one of two parallel hyperplanes.

Let's look at our "zoo" in 3D so far:

Cube, cylinder, sphere, torus, and now we are adding cone. We should have square pyramid, too, and even tetrahedron. And triangular prism, as square -> line

In 2D we have square and circle, but now we have to add triangle too (line tapered to point).

So what 4D shapes do we get with "tapering", if we limit ourselves to the round shapes?

Cylinder -> circle: would be triangle X circle duoprism. Has no vertices, three circular "lines", three disks and three cylindrical mantles as hedrixes, and four chorixes - three full cylinders, and one triangular prism bent in circle until its bases meet.

Cylinder -> line, or Cone -> cone: Two vertices, two circles and one line joining the vertices, two disks, one cylindrical mantle and two cone mantles, one cylinder, two cones, and one 3D mantle of this figure which is a prism of cone mantle.

Cylinder -> point, or cone -> circle: one vertex, two circles, two disks, one cylindrical mantle, and two cone mantles, one cylinder, two cones, and one global mantle of the figure

Sphere -> point: One vertex, one sphere, one ball and one mantle

You could taper the sphere to circle, too, by simply taking spherinder and cutting away... (imagine circle->line in this way, the intermediate shapes are not ellipses, but circles "with ends cut")

Cone -> point: 2 vertices, 1 circle and 1 line, 1 disk and 2 cone mantles, 2 cones and 1 global mantle.

Now one you've missed:

Cone -> line: As you move in w direction, the radius of the cone decreases to zero, transforming it into line. Has:
Three vertices (one of cone, two of line)
One circle and two lines (the cone circle, the line, and cone vertex -> point)
One disk and two cone mantles (cone disk, cone mantle, and cone circle -> point)
Two cones and a global mantle.

I'm not sure, but this might be a quarter of circular duotegum.
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Postby wendy » Sat Jun 10, 2006 7:31 am

We have the somewhat more precise:

1 coinder = circle - pyramid - prism

2 cylidrome = circle - prism - pyramid

3 soherone = sphere pyramid

4 = circle - line - pyramid

All of these amount to what can be defined in terms of simple products.

In any case, the tegum product is not mentioned, eg


di-circular tegum = circle tapering to circle
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Postby PWrong » Sat Jun 10, 2006 11:22 am

Ok, I'll try to write all the 3D objects out using a combination of RNS notation, "->" for the taper operation, and '0' for a point.

2D:
2, 11, 1->0

3D: 9 objects
111, 21, 3, (21)
square pyramid = 11->0
tetrahedron = (1->0)->0
wedge = (1->0)->1 (this is a cross-section of cone -> line)
triangular prism = 11->1 = (1->0)1
cone = 2->0

Note that tapering is not associative:
(1->0)->0 = tetrahedron but 1->(0->0) = 1->1 = square

You could taper the sphere to circle, too, by simply taking spherinder and cutting away... (imagine circle->line in this way, the intermediate shapes are not ellipses, but circles "with ends cut")

I don't like the idea of that at all. There are several ways to turn a circle into a line, so I don't think we should pick one arbitrarily.

It looks like we've got some problems with the taper operation.
1. It's not associative.
2. It's obviously not commutative.
3. It's not defined for some pairs of objects.
4. We get four uninvited guests in 3D, and one in 2D.

We can't solve these problems by only tapering to a point. Maybe we need a different operation that will produce a cone. For instance, take any line through the origin in the xy plane, and rotate it around the y axis. This produces the double cone, which in some ways is more natural than the cone.
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Postby moonlord » Sat Jun 10, 2006 3:35 pm

Umm, can anyone explain the tapering process in more detail? When none of the elements is a point, I don't understand it at all... :(
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Postby Keiji » Sat Jun 10, 2006 4:36 pm

PWrong wrote:
You could taper the sphere to circle, too, by simply taking spherinder and cutting away... (imagine circle->line in this way, the intermediate shapes are not ellipses, but circles "with ends cut")

I don't like the idea of that at all. There are several ways to turn a circle into a line, so I don't think we should pick one arbitrarily.


A line is NOT a special case of a circle. It is a special case of an ellipse. A circle is also a special case of an ellipse. Therefore it is possible to determine a single way to taper from a circle to a line.

It looks like we've got some problems with the taper operation.
1. It's not associative.
2. It's obviously not commutative.

How are these problems?
3. It's not defined for some pairs of objects.

Then again, exponentiation isn't defined when the LHS is negative and the RHS is a non-integer. We don't go around trying to improve exponentiation. :P
4. We get four uninvited guests in 3D, and one in 2D.

Define an "uninvited guest"...

We can't solve these problems by only tapering to a point. Maybe we need a different operation that will produce a cone. For instance, take any line through the origin in the xy plane, and rotate it around the y axis. This produces the double cone, which in some ways is more natural than the cone.


That operation is already in CSG notation anyway, it's called L. I think we need this taper operation. Also, there is a problem with the "double cone" as you call it. It is self-intersecting. So the circle at one end has positive area while the circle at the other has negative area. This means the object has zero volume. We can't have objects with zero volume.

moonlord wrote:Umm, can anyone explain the tapering process in more detail? When none of the elements is a point, I don't understand it at all... :(


It's actually very simple. You just blend one object into another by changing variables. For example, in square -> line, we reduce the width of the square until it reaches zero.
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Postby PWrong » Sun Jun 11, 2006 3:35 am

A line is NOT a special case of a circle. It is a special case of an ellipse. A circle is also a special case of an ellipse. Therefore it is possible to determine a single way to taper from a circle to a line.

But a line and a circle are also special cases of the "circle with ends cut" that Marek suggested. So there is no unique way to taper from circle to line. I think it's best that we don't define circle->line at all. Similarly, the line is not a special case of a square, it's a special case of a rectangle. Unfortunately, this doesn't mean we can exclude the triangular prism, because it's an extruded triangle.


It looks like we've got some problems with the taper operation.
1. It's not associative.
2. It's obviously not commutative.

How are these problems?

Well, I suppose it actually can be commutative, since point->circle is just an upside down cone. But not being associative is a problem because we have to write brackets every time we want to taper twice.

Then again, exponentiation isn't defined when the LHS is negative and the RHS is a non-integer. We don't go around trying to improve exponentiation.

Actually, that's essentially why the complex numbers were invented. But in this case, we want to enumerate the possible cones in nD. Normally we would use our notation to write out all the possible shapes. But if some shapes aren't valid, we have to investigate each one individually.

Define an "uninvited guest"...

When we were trying to count torii before, we had objects like circle#square and "a pair of concentric cylinders" cropping up. All we wanted was the torus, so these other objects were uninvited guests.
Now we're trying to include the cone in our list of shapes, and we're getting another long list of 3D objects that noone asked for. Personally, I don't want to include triangles, pyramids, prisms and wedges.

Also, there is a problem with the "double cone" as you call it. It is self-intersecting. So the circle at one end has positive area while the circle at the other has negative area. This means the object has zero volume. We can't have objects with zero volume.

Even if the other circle had a negative radius, the area is pi r^2 which is always positive. Anyway, the volume of a cone doesn't become negative just because you turned it upside down.
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Postby PWrong » Sun Jun 11, 2006 4:21 am

I've just noticed something interesting. Extrusion is a special case of tapering i.e. A -> A. But it's also a special case of the cartesian product where one of the objects is a line, i.e. A x 1.
Maybe there's some operation that includes both tapering and the cartesian product.
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Postby Marek14 » Sun Jun 11, 2006 6:49 am

The "circle with ends cut" thing is definitely not arbitrary :) If you recall the old discussion about rotatopes and my "graphotopes", circle with ends cut is a shape that occurs there frequently.

For example, the dome, or crind, with equation

max(x^2,y^2)+z^2 = 1

can be glued together from two "circle -> line" shapes, providing they are defined as I did. You see, tapering shapes can be generally tought of as "cuts", what you get when you split some other shape, defined without the tapering operation by itself.
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Postby PWrong » Sun Jun 11, 2006 7:59 am

The "circle with ends cut" thing is definitely not arbitrary

It's arbitrary in the sense that it's not the only object of which circle and line are a special case. It's not enough to simply "occur frequently".

Given two objects A and B, there may be infinitely many ways to continuously transform A into B.
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Postby wendy » Sun Jun 11, 2006 8:17 am

You can "taper" anything into anything, as long as they are in orthogonal spaces, eg xy vs. z, and there is a tapering axis, eg w.

At any point on the tapering axis from (w=0%) to (w=100%), there is a prism (or cartesian) section, that is x% X and y% Y. as one moves from w=0 to w=100, one makes the prism go from 100 % X (ie no Y height) to 100 % Y (ie no X height).

One can then generalise this to N dimensions, by noting, eg that there is a plane w+x+y+z = 1. For all positive values, this means that all lie between 0 and 1, at any point w,x,y,z, there is a prism of shape wW, xX, yY, zZ.

One can make W, X, Y, Z any possible shape, (inculding trees, circles, etc), as long as they are in orthogonal dimensions (ie the intersection of W and X is a point). The space containing w,x,y,z space (ie the thing where we're drawing percentages from), is orthogonal to W, X, Y, Z, and is called the "altitude" of the product, A.

The total sum is then A+W+X+Y+Z.

Fot the simplest case, we have A=1d, W=point, X=polygon giving a prism, and A=1d, W=point, X=circle gives a cone.

You can have in 4d, either

A=1d, W=point, X=3d solid => point pyramid.

A=1d, W=line, X=2d => polygon - line - pyramid

In higher dimensions one can have A, etc even higher. One can have, in 8d, for example,

A=2d, W=circle, X=circle, Y=circle, giving a tri-circular pyramid.

The shape i describe the rotations in 4d as, is a pyramid, as

A=1d, W=sphere surface (3d), X=sphere surface (3d), which gives 7d, but because it's surfaces, it gives a 1+2+2 = 5 dimensional manifold.

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Postby Marek14 » Sun Jun 11, 2006 10:39 am

PWrong wrote:
The "circle with ends cut" thing is definitely not arbitrary

It's arbitrary in the sense that it's not the only object of which circle and line are a special case. It's not enough to simply "occur frequently".

Given two objects A and B, there may be infinitely many ways to continuously transform A into B.


That's true. However, a dome is, in a way, "special case" of transforming circle into a line and back, in the sense that it's the only shape in its class with this particular transformation.

Of course, if you'd take square -> point from dome, which is possible too, you wouldn't get the true square pyramid, but rather one with arcs for lateral edges...

As Wendy said, the most "natural" circle -> line would be circle -> perpendicular line, which is identical to cone -> point.

It all has to do with notions of products that Wendy uses. If you have a tegum anything x line, then you can cut it in the middle, to get the anything -> point taper. Cutting prism in the half gives you the same prism (since we can change the lengths freely).

So the torus -> point taper is a half of torus x line tegum.

The other way to explore is the Klitzing's segmentochoron idea - after broadening the scope a bit, it tells us how to deal with shapes that have two "generators" in two parallel hyperplanes and which "morph" continuously one into another as you move between those two hyperplanes.
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Postby Keiji » Sun Jun 11, 2006 11:27 am

PWrong wrote:
Rob wrote:A line is NOT a special case of a circle. It is a special case of an ellipse. A circle is also a special case of an ellipse. Therefore it is possible to determine a single way to taper from a circle to a line.

But a line and a circle are also special cases of the "circle with ends cut" that Marek suggested. So there is no unique way to taper from circle to line. I think it's best that we don't define circle->line at all. Similarly, the line is not a special case of a square, it's a special case of a rectangle. Unfortunately, this doesn't mean we can exclude the triangular prism, because it's an extruded triangle.


Then I would suggest we give a more verbose definition of the taper operation.

object>[end1;end2]

This will take object and taper it resulting in a shape one dimension higher where the ends are 1 unit and -1 unit in the next dimension. The variables for the object are specified in end1 and end2. For example:

ellipse>[r<sub>1</sub>=1,r<sub>2</sub>=1;r<sub>1</sub>=1,r<sub>2</sub>=0]

would give my definition of a circle tapered to a line.

PWrong wrote:
Rob wrote:
PWrong wrote:It looks like we've got some problems with the taper operation.
1. It's not associative.
2. It's obviously not commutative.

How are these problems?

Well, I suppose it actually can be commutative, since point->circle is just an upside down cone. But not being associative is a problem because we have to write brackets every time we want to taper twice.


Given the new definition proposed above, commutativity and associativity won't be a problem, because the LHS must always be an object and the RHS must always be a pair of lists of equations.

PWrong wrote:
Rob wrote:Then again, exponentiation isn't defined when the LHS is negative and the RHS is a non-integer. We don't go around trying to improve exponentiation.

Actually, that's essentially why the complex numbers were invented. But in this case, we want to enumerate the possible cones in nD. Normally we would use our notation to write out all the possible shapes. But if some shapes aren't valid, we have to investigate each one individually.


I see. Again, given the new definition, this isn't an issue as all results are defined.

PWrong wrote:
Rob wrote:Define an "uninvited guest"...

When we were trying to count torii before, we had objects like circle#square and "a pair of concentric cylinders" cropping up. All we wanted was the torus, so these other objects were uninvited guests.
Now we're trying to include the cone in our list of shapes, and we're getting another long list of 3D objects that noone asked for. Personally, I don't want to include triangles, pyramids, prisms and wedges.


I don't see the problem with this. The wedge is a rather interesting object to me anyway, and besides, I'd like to be able to visualize more objects in the 4th dimension (tesseracts are boring :P )

PWrong wrote:
Rob wrote:Also, there is a problem with the "double cone" as you call it. It is self-intersecting. So the circle at one end has positive area while the circle at the other has negative area. This means the object has zero volume. We can't have objects with zero volume.

Even if the other circle had a negative radius, the area is pi r^2 which is always positive. Anyway, the volume of a cone doesn't become negative just because you turned it upside down.


You are correct that the circle's area would become positive, and that would mean the cone's volume is also positive. But the radius of the circle is indeed negative (think about it: if you (once again) use the defintion I proposed above, there's no way to get a double cone without setting one of the circle's radii to a negative value). This could cause problems in 4D, for example a double sphere-cone would have an r^3 in it, so the volume of the sphere cells would be negative.
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Postby PWrong » Sun Jun 11, 2006 1:47 pm

I don't see the problem with this. The wedge is a rather interesting object to me anyway, and besides, I'd like to be able to visualize more objects in the 4th dimension (tesseracts are boring )

Ok, lets try counting the objects that can be formed using only tapering and extrusion i.e. no rotation and no spherating. I'm not sure of the names for the 4D objects.

2D: two objects
square and triangle

3D: five objects
cube = 111
square pyramid = 11->0
tetrahedron = (1->0)->0
wedge = (1->0)->1
triangular prism = 11->1 = (1->0)1

4D: too many objects
tesseract = 1111
cube pyramid = 111 -> 0 = (11->0)->11
simplex = ((1->0)->0)->0
square pyramid cylinder = (11->0)1 = 111->1
tetrahedron cylinder = ((1->0)->0)1
wedge cylinder = ((1->0)->1)1 = ((1->0)1)->11
triangular prism cylinder = (1->0)11 = 111->11
wedge -> point = ((1->0)->1)->0
wedge -> line = ((1->0)->1)->1
(there may be several different versions of this last shape, if any :()

I'm sure there are even more shapes than this.

Here's a conjecture:
Extrusion distributes over tapering, that is: (A->B)1 = A1->B1
For example, we know that:
(1->0)1 = 11->1
(11->0)1 = 111->1
(1->0)11 = 111->11
If this conjecture is true, then we have:
(11->0)1111 = (111->1)111 = (1111->11)11 = (11111->111)1 = 111111->1111

Maybe we could reduce the number of shapes if we require that all lines be equal, so that triangle becomes "equilateral triangle". We lose the wedge, because a line is not a special case of an equilateral triangle.
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Postby PWrong » Sun Jun 11, 2006 1:59 pm

Here's a couple I missed:
square pyramid -> line = (11->0)->1

wedge -> triangle = triangular prism to line
= ((1->0)->1) -> (1->0) = ((1->0)1)->1

This second one suggests another identity:
(A->B)->A = (A->A)->B = A1->B
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Postby Keiji » Sun Jun 11, 2006 3:00 pm

Care to tell me what you think of my proposed tapering notation?

I'll list all those objects here (note: the simplexes are NOT regular):

1D:
line = point>[;]

2D:
square = line>[l=2;l=2]
triangle = line>[l=2;l=0]

3D:
cube = square>[l=2;l=2]
square pyramid = square>[l=2;l=0]
triangular prism = rectangle>[w=2,h=2;w=0,h=2]
tetrahedron = triangle>[b=2,h=2;b=0;h=0]
plough = triangle>[b=2,h=2;b=0;h=2]
wedge = triangle>[b=2,h=2;b=2;h=0]

4D:
tesseract = cube>[l=2;l=2]
cubic pyramid = cube>[l=2;l=0]
square dipyramid = cuboid>[w=2,h=2,d=2;w=0,h=0,d=2]
triangular diprism = cuboid>[w=2,h=2,d=2;w=0,h=2,d=2]
square pyramid wedge 1 = square pyramid>[l=2,h=2;l=0,h=2]
square pyramid wedge 2 = square pyramid>[l=2,h=2;l=2,h=0]
triangular prism pyramid = triangular prism>[b=2,h=2;b=0,h=0]
triangular prism wedge 1 = triangular prism>[b=2,h=2;b=0,h=2]
triangular prism wedge 2 = triangular prism>[b=2,h=2;b=2,h=0]
pentachoron = tetrahedron>[b=2,d=2,h=2;b=0,d=0,h=0]
tetrahedral wedge 1 = tetrahedron>[b=2,d=2,h=2;b=2,d=0,h=0]
tetrahedral wedge 2 = tetrahedron>[b=2,d=2,h=2;b=0,d=2,h=0]
tetrahedral wedge 3 = tetrahedron>[b=2,d=2,h=2;b=0,d=0,h=2]
tetrahedral diwedge 1 = tetrahedron>[b=2,d=2,h=2;b=2,d=2,h=0]
tetrahedral diwedge 2 = tetrahedron>[b=2,d=2,h=2;b=0,d=2,h=2]
tetrahedral diwedge 3 = tetrahedron>[b=2,d=2,h=2;b=2,d=0,h=2]
plough wedge 1 = plough>[b=2,d=2,h=2;b=2,d=0,h=0]
plough wedge 2 = plough>[b=2,d=2,h=2;b=0,d=2,h=0]
plough wedge 3 = plough>[b=2,d=2,h=2;b=0,d=0,h=2]
plough diwedge 1 = plough>[b=2,d=2,h=2;b=2,d=2,h=0]
plough diwedge 2 = plough>[b=2,d=2,h=2;b=0,d=2,h=2]
plough diwedge 3 = plough>[b=2,d=2,h=2;b=2,d=0,h=2]
diwedge 1 = wedge>[b=2,d=2,h=2;b=2,d=0,h=0]
diwedge 2 = wedge>[b=2,d=2,h=2;b=0,d=2,h=0]
diwedge 3 = wedge>[b=2,d=2,h=2;b=0,d=0,h=2]
triwedge 1 = wedge>[b=2,d=2,h=2;b=2,d=2,h=0]
triwedge 2 = wedge>[b=2,d=2,h=2;b=0,d=2,h=2]
triwedge 3 = wedge>[b=2,d=2,h=2;b=2,d=0,h=2]

There are indeed a lot of wedge-based objects...

Wedge vs plough: In the plough, you shrink the base of the triangle. In the wedge, you shrink the height instead - this gives you two curved faces...
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Postby PWrong » Sun Jun 11, 2006 3:50 pm

Wedge vs plough: In the plough, you shrink the base of the triangle. In the wedge, you shrink the height instead - this gives you two curved faces...

I said the wedge is a cross-section of cone -> line, which means we shrink the base. So my wedge is the same as your plough. I don't mind which name we use, but I think we should reject the one with curved faces.

Care to tell me what you think of my proposed tapering notation?

It seems a bit too long. It's good for precisely defining an object, but we usually want to describe the object in general, ignoring the parameters.

What about this?
Use ordinary RNS notation, but to taper an object, put an asterisk over the parameter you want to reduce to zero. If it's a radius, put the asterisk next to the brackets.

For instance, cone is (xy)*, because we want to reduce the radius.

triangle = x*
cone = (xy)*
square pyramid = [xy]*
triangular prism = x*z
tetrahedron = (x*)*
wedge = unsure, possibly x(**)

cylinder -> line = coninder = (xy)*z
cylinder -> point = (xy)*z*
cylinder -> circle = circle x triangle = (xy)z*

torus -> circle = ((xy)*z)
torus -> sphere = ((xy)z)*
torus -> point = ((xy)*z)*
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Postby Keiji » Sun Jun 11, 2006 3:56 pm

PWrong wrote:
Wedge vs plough: In the plough, you shrink the base of the triangle. In the wedge, you shrink the height instead - this gives you two curved faces...

I said the wedge is a cross-section of cone -> line, which means we shrink the base. So my wedge is the same as your plough. I don't mind which name we use, but I think we should reject the one with curved faces.


I think we shouldn't, because it is a valid object, and certainly isn't self-intersecting.

Care to tell me what you think of my proposed tapering notation?

It seems a bit too long. It's good for precisely defining an object, but we usually want to describe the object in general, ignoring the parameters.

What about this?
Use ordinary RNS notation, but to taper an object, put an asterisk over the parameter you want to reduce to zero. If it's a radius, put the asterisk next to the brackets.


In your notation, it is impossible to express the wedge or the plough, and most likely other objects. I'll stick to mine until a better replacement is found :P
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Postby PWrong » Mon Jun 12, 2006 3:38 am

I think we shouldn't, because it is a valid object, and certainly isn't self-intersecting.

Actually, I just realised that both the wedge and the plough have curved surfaces. The reason is you're trying to fit a surface to four points, and you only need three points to define a plane. Here's the parametric equations of the two curved surfaces.
x = s(t-1), y = 1-s, z = t
x = s(1-t), y = 1-s, z = t
where s and t range from 0 to 1

I think that any object that we include in our list should consist purely of surfaces with constant curvature. But the wedge is the only object we've found that has a curved, non-spherical surface. This, if nothing else, should tell us that it doesn't belong in our list. The fact that my notation can't express it also suggests that it shouldn't be there in the first place.

I agree that it's an interesting shape, and we should look at it in more detail, but it doesn't really fit in. Fortunately, this reduces our list of objects from 28 to only six, all of which we can visualise easily.

This also means we have to reject cone -> line and cone -> circle.

Here's all the remaining 4D shapes in my notation.
tesseract = xyzw
cubic pyramid = [xyz]*
square pyramid cylinder = [xy]**
triangular diprism = x*zw
triangular prism pyramid = [x*z]*
pentachoron = x***

spherone = (xyz)*
cylinder -> line = coninder = (xy)*z
cylinder -> circle = circle x triangle = (xy)z*
cylinder -> point = (xy)*z*
cone -> point = (xy)**

torus -> circle = ((xy)*z)
torus -> sphere = ((xy)z)*
torus -> point = ((xy)*z)*
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Postby Keiji » Mon Jun 12, 2006 7:13 am

Interesting...

Then again, the taper operation can make a trapesium, and you could have made even more curved object fun with that. 3 variables in a 2d shape is hmm :lol:

Anyway, I agree with you now, given that the ones your notation doesn't support are the exact same ones as those that have curved surfaces, then I suppose it would help to ignore them. Besides, who would want to write about a lot of wedges and diwedges and triwedges that I don't even know the shape of (most of the 4d ones that is)? :P

You had some mistakes in your list. The "square pyramid cylinder" is actually a square dipyramid, as noted below. There were various incorrect expressions (look through them if you want to know which ones). Finally, the cylinder->circle and the torus->sphere have no expression and are invalid.

I've also given names to your x->y shapes, and have also included the extra torii that you forgot about (e.g. the triangular torus).

Now, can I suggest an improvement to your notation. Firstly, the * is used to mean cartesian product in CSG notation and this may become an operator in RNS notation too. Secondly, the stars get a little overwhelming, especially in the simplexes.

Here it is: One star is <sup>1</sup>, two stars is <sup>2</sup>, etc. As in superscript. I've included all other rotopes and <4D shapes in a sensible order, and the "original" RNS expressions (the ones that use numbers instead of letters) for completeness. So, they become:

1D: 1 object
line = x = 1

2D: 3 objects
square = xy = 11
circle = (xy) = 2
triangle = x<sup>1</sup> = 1<sup>1</sup>

3D: 9 objects
cube = xyz = 111
square pyramid = [xy]<sup>1</sup> = [11]<sup>1</sup>
cylinder = (xy)z = 21
torus = ((xy)z) = (21)
sphere = (xyz) = 3
cone = (xy)<sup>1</sup> = 2<sup>1</sup>
triangular prism = x<sup>1</sup>z = 1<sup>1</sup>1
triangular torus = (x<sup>1</sup>z) = (1<sup>1</sup>1)
tetrahedron = x<sup>2</sup> = 1<sup>2</sup>

4D: 27 objects
tesseract = xyzw = 1111
cubic pyramid = [xyz]<sup>1</sup> = [111]<sup>1</sup>
square dipyramid = [xy]<sup>2</sup> = [11]<sup>2</sup>
cubinder = (xy)zw = 211
toracubinder = ((xy)zw) = (211)
duocylinder = (xy)(zw) = 22
tiger = ((xy)(zw)) = (22)
cylindrical pyramid = [(xy)z]<sup>1</sup> = [21]<sup>1</sup>
torinder = ((xy)z)w = (21)1
tetratorus = (((xy)z)w) = ((21)1)
toroidal pyramid = ((xy)z)<sup>1</sup> = (21)<sup>1</sup>
spherinder = (xyz)w = 31
toraspherinder = ((xyz)w) = (31)
glome = (xyzw) = 4
spherone = (xyz)<sup>1</sup> = 3<sup>1</sup>
coninder = (xy)<sup>1</sup>w = 2<sup>1</sup>1
conindral torus = ((xy)<sup>1</sup>w) = (2<sup>1</sup>1)
circular dipyramid = (xy)<sup>2</sup> = 2<sup>2</sup>
triangular diprism = x<sup>1</sup>zw = 1<sup>1</sup>11
triangular diprismidal torus = (x<sup>1</sup>zw) = (1<sup>1</sup>11)
triangular prismidal pyramid = [x<sup>1</sup>z]<sup>1</sup> = [1<sup>1</sup>1]<sup>1</sup>
triangular toroidal prism = (x<sup>1</sup>z)w = (1<sup>1</sup>1)1
triangular ditorus = ((x<sup>1</sup>z)w) = ((1<sup>1</sup>1)1)
triangular toroidal pyramid = (x<sup>1</sup>z)<sup>1</sup> = (1<sup>1</sup>1)<sup>1</sup>
tetrahedral prism = x<sup>2</sup>w = 1<sup>2</sup>1
tetrahedral torus = (x<sup>2</sup>w) = (1<sup>2</sup>1)
pentachoron = x<sup>3</sup> = 1<sup>3</sup>

I'm using "toroidal" to mean "torus-based" (as an adjective), and the others you should be able to work out.

Note that the sum of all the numbers still equals the dimensionality of the shape. Also, note how the number of tapertopes in the nth dimension is 3<sup>n-1</sup>.

Now I have two suggestions. One, can we get rid of the square brackets? If they are present they always start at the beginning of the expression, and are unnecessary if we simply evaluate the expression from left to right.

Two, these "tapertopes" as I have named them are a superset of the rotopes. Can we therefore extend RNS notation to include these taperings, and replace rotopes on the Wiki with tapertopes?
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Postby Marek14 » Mon Jun 12, 2006 7:44 am

For the triangle -> line: Klitzing has it in his list of segmentochora. But if all lines are equal, its just the square pyramid (lying on its triangle face) :)
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Postby PWrong » Mon Jun 12, 2006 8:15 am

Here it is: One star is 1, two stars is 2, etc. As in superscript.
Ok, we'll use that. I was thinking of changing the star to something else anyway. I have a further suggestion though. When we use the xyz notation instead of numbers, maybe we should put letters in the superscripts. So the cone is (xy)<sup>z</sup> and tetrahedral prism is x<sup>yz</sup>w.

I've also given names to your x->y shapes, and have also included the extra torii that you forgot about (e.g. the triangular torus).

Would that be circle # triangle? If we include that one, we should probably include the square torus (circle#square) as well. I don't think that's a good idea.

Finally, the cylinder->circle and the torus->sphere have no expression and are invalid.

What's wrong with them? cylinder->circle is the same as triangle x circle, and it's expression is 21<sup>1</sup>. Torus->sphere may self-intersecting, but otherwise it's ok, and you've already included it in your list, under the name "toroidal pyramid".

Note that the sum of all the numbers still equals the dimensionality of the shape. Also, note how the number of tapertopes in the nth dimension is 3n-1.

That's a surprisingly simple formula, if it's correct. I think we'll have to check the 5D tapertopes

Now I have two suggestions. One, can we get rid of the square brackets? If they are present they always start at the beginning of the expression, and are unnecessary if we simply evaluate the expression from left to right.

So cube pyramid would be 111<sup>1</sup>, while triangular diprism is 1<sup>1</sup>11. What would 11<sup>1</sup>1 be?

Two, these "tapertopes" as I have named them are a superset of the rotopes. Can we therefore extend RNS notation to include these taperings, and replace rotopes on the Wiki with tapertopes?

I was thinking we might redefine "rotopes" to include the tapertopes, and restrict tapertopes to objects that use the taper operation at least once. So the cone is a tapertope, but a cylinder isn't. That way tapertope has a similar definition to toratope.
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Postby Keiji » Mon Jun 12, 2006 11:40 am

PWrong wrote:
Here it is: One star is 1, two stars is 2, etc. As in superscript.
Ok, we'll use that. I was thinking of changing the star to something else anyway. I have a further suggestion though. When we use the xyz notation instead of numbers, maybe we should put letters in the superscripts. So the cone is (xy)<sup>z</sup> and tetrahedral prism is x<sup>yz</sup>w.


That's a good idea.

PWrong wrote:
I've also given names to your x->y shapes, and have also included the extra torii that you forgot about (e.g. the triangular torus).

Would that be circle # triangle? If we include that one, we should probably include the square torus (circle#square) as well. I don't think that's a good idea.


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 ;) And I intended to include the square torus, but:

Triangle is x<sup>y</sup>, triangular torus is (x<sup>y</sup>z)
Square is xy, so square torus should be (xyz)... but (xyz) is a sphere.

PWrong wrote:
Finally, the cylinder->circle and the torus->sphere have no expression and are invalid.

What's wrong with them? cylinder->circle is the same as triangle x circle, and it's expression is 21<sup>1</sup>. Torus->sphere may self-intersecting, but otherwise it's ok, and you've already included it in your list, under the name "toroidal pyramid".


21<sup>1</sup> = Cylinder tapered to a point - not a circle.

Similarly, "toroidal pyramid" is a torus tapered to a point - not a sphere.

PWrong wrote:
Note that the sum of all the numbers still equals the dimensionality of the shape. Also, note how the number of tapertopes in the nth dimension is 3n-1.

That's a surprisingly simple formula, if it's correct. I think we'll have to check the 5D tapertopes

Now I have two suggestions. One, can we get rid of the square brackets? If they are present they always start at the beginning of the expression, and are unnecessary if we simply evaluate the expression from left to right.

So cube pyramid would be 111<sup>1</sup>, while triangular diprism is 1<sup>1</sup>11. What would 11<sup>1</sup>1 be?


11<sup>1</sup>1 would be square pyramidal prism, or cube->line. And you are absolutely right that that is not included in my list. So much for the formula :roll: I actually got that formula from how you can do three things to an object: extrude, lathe or taper, but it doesn't always seem to work.

PWrong wrote:
Two, these "tapertopes" as I have named them are a superset of the rotopes. Can we therefore extend RNS notation to include these taperings, and replace rotopes on the Wiki with tapertopes?

I was thinking we might redefine "rotopes" to include the tapertopes, and restrict tapertopes to objects that use the taper operation at least once. So the cone is a tapertope, but a cylinder isn't. That way tapertope has a similar definition to toratope.


That's a better idea. But wouldn't it make the rotopes templates a bit big (especially for 4D and up)?
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Postby moonlord » Mon Jun 12, 2006 3:55 pm

As I haven't understood taperings to anything else than a point, my post will only refer to the old definition (T) of the taper operation. By the way, I suggest we expand the rotope definition to include tapered objects.

If you have n rotopes in k dimensions, let's see how many will there be in the k+1'th. Firstly, there are the base bodies, and their number is the number of partitions of k+1. Then, there are the spherated versions, as in (21123). As someone pointed out, their number is the number of the base bodies. Finally we have the tapered-to-a-point bodies; these count up to twice the number of partitions of k.

So in k dimensions, I believe there are 2*(p(k)+p(k-1)) rotopes.

I'd be very grateful if someone did explain me the tapering operation in detail...
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Postby Keiji » Mon Jun 12, 2006 4:07 pm

We decided to scrap my idea for tapering, and use PWrong's, which -is- tapering to a point. ;)
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Postby moonlord » Mon Jun 12, 2006 4:25 pm

Oh, nice. No more headaches :)... How about the numbers? Do you agree?
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Postby Keiji » Mon Jun 12, 2006 5:02 pm

Wouldn't it be p(k) + 2p(k+1)?
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Postby PWrong » Mon Jun 12, 2006 5:04 pm

Then, there are the spherated versions, as in (21123). As someone pointed out, their number is the number of the base bodies.

It's a bit more complicated than that. The actual formula for the number of toratopes is a complicated sum-product over the partitions of n. You'd be better off using T(n) to denote the number of toratopes (i.e. non-tapering rotatopes).

We decided to scrap my idea for tapering, and use PWrong's, which -is- tapering to a point.

Again, it's more complicated. Every object has at least one arbitrary parameter that can be tapered. Squares and triangles have only one. Cylinder has two: radius and height. A torus also has two: the major and minor radii. While there is only one way to taper a square, there are three ways to taper a cylinder: radius only, height only, and both together. In general, if an object has n parameters, there are 2<sup>n-1</sup> ways to taper it.

I think there's a formula for the number of parameters an object has, and we should be able to use that to work out a formula for the number of tapertopes.
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Postby Keiji » Tue Jun 13, 2006 7:55 pm

PWrong wrote:
We decided to scrap my idea for tapering, and use PWrong's, which -is- tapering to a point.

Again, it's more complicated. Every object has at least one arbitrary parameter that can be tapered.


PWrong, you're wrong ;)

Tapering in RNS notation only allows tapering to a point. If <i>x</i> is the base, <i>x</i><sup>1</sup> is the base tapered to a point. ;)
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