A few comments, questions, etc. etc.

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

A few comments, questions, etc. etc.

Postby Kevin_Ahab » Thu Jul 27, 2006 10:55 pm

Okay - first of all, what do we call the transformation analogous to making something a prism from 2D to 3D. I mean, for instance, a rectangular prism is created by extruding a rectangle into the third dimension, so that the two rectangles do not lie on the same plane, are paralell, and all their perimenters connect. In 4D, I presume, a 3D figure would be extruded into 4D so that the two 3D objects do not lie in the same realm, are paralell, and all their surface areas connect. What is this called?

Also, the pyramid. Any object in 2D is made into a pyramid by placing a point not on the same plane and making all verticies on the 2D object go to it (or the whole of any round sides). A regular pyramid is one in which the point in the other plane is centered. What is the analogy of this in 4D.


Also, I take it there are 4 types of rectangular-prism-analogy in 4D. Consider. In 1D, the hyperrectangularprism (I call it), a line, has only one type-a line.

In 2D, there is 2-the rectangle, and the square.

In 3D, there is 3-the rectangle extruded into 3D in such a way that the new dimension, "height" is not equal to either side of the rectangle.
-2 - A square extruded into 3D in such a way that the new dimension is not equal to the side of the square (or a rectangle with height equal to one of its sides)
-3 - The cube (all sides equal)

In 4D, there would be 4!
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Re: A few comments, questions, etc. etc.

Postby Nick » Thu Jul 27, 2006 11:37 pm

Kevin_Ahab wrote:Okay - first of all, what do we call the transformation analogous to making something a prism from 2D to 3D. I mean, for instance, a rectangular prism is created by extruding a rectangle into the third dimension, so that the two rectangles do not lie on the same plane, are paralell, and all their perimenters connect. In 4D, I presume, a 3D figure would be extruded into 4D so that the two 3D objects do not lie in the same realm, are paralell, and all their surface areas connect. What is this called?


Perimeter is a measure of the lengths of the sides, and is not something that actually exists; its like saying pi is perpendicular to a plane. It just doesn't make sense (this applies to "surface areas"; instead, say "their corresponding vertices are perpendicular" as well).

To the point in question: We would need to create a definition for a 4d object with two parallel realms connected by their vertices. In 3d, we call such a thing a prism, but we would need a new name for a 4d one. I support the term "tetraprism". Therefore, I officially christen this shape a rectangular tetraprism!

Also, the pyramid. Any object in 2D is made into a pyramid by placing a point not on the same plane and making all verticies on the 2D object go to it (or the whole of any round sides). A regular pyramid is one in which the point in the other plane is centered. What is the analogy of this in 4D.


I believe you are thinking of the 16-cell (scroll down for a nice pic).


Also, I take it there are 4 types of rectangular-prism-analogy in 4D. Consider. In 1D, the hyperrectangularprism (I call it), a line, has only one type-a line.

In 2D, there is 2-the rectangle, and the square.

In 3D, there is 3-the rectangle extruded into 3D in such a way that the new dimension, "height" is not equal to either side of the rectangle.
-2 - A square extruded into 3D in such a way that the new dimension is not equal to the side of the square (or a rectangle with height equal to one of its sides)
-3 - The cube (all sides equal)

In 4D, there would be 4!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!


Actually, there are alot more in 3d:

Cube (All sides equal)
Rectangular Prism 1 (extended by length)
Rectangular Prism 2 (extended by width)
Rectangular Prism 3 (extended by height)
Rectangular Prism 4 (extended by length, and the line containing the centers of the bases are not perpendicular to the bases)
Rectangular Prism 5 (extended by width, and the line containing the centers of the bases are not perpendicular to the bases)
Rectangular Prism 6 (extended by height, and the line containing the centers of the bases are not perpendicular to the bases)

Unless your considering 1, 2, and 3 the same + 4, 5, and 6 the same; in which case there are three.

Assuming they aren't:

2d:
Possibilites: extended by length(1), width(2)
not perpendicular to base(3)
This creates 4 possibilities: 1, 2, 1+3, 2+3

3d:
Possibilities: extended by length (1), width(2), height(3)
not perpendicular to base(4)

This creates 6 possibilities: 1, 2, 3, 1+4, 2+4, 3+4

4d:
Possibilities: extended by length(1), width(2), height(3), trength(4)
not perpendicular to base(4)

This creates 8 possibilities: 1, 2, 3, 4, 1+5, 2+5, 3+5, 4+5

Conclusion: for ever n dimensional cubic(?) shape, the number of possibilities of being stretched is equal to the number of n-1 dimensional boundries it has (4 lines, 6 faces, 8 cells).

This is all under the assumption that we are only stretching these objects in one dimension. If these were all possible directions, then 3d would contain 3 "not perpendicular to base" possibilites, and 4d would contain (someone fill in this number please) "not perpendicular to base" possibilities.
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Re: A few comments, questions, etc. etc.

Postby Marek14 » Fri Jul 28, 2006 5:42 am

My take on the questions:

Kevin_Ahab wrote:Okay - first of all, what do we call the transformation analogous to making something a prism from 2D to 3D. I mean, for instance, a rectangular prism is created by extruding a rectangle into the third dimension, so that the two rectangles do not lie on the same plane, are paralell, and all their perimenters connect. In 4D, I presume, a 3D figure would be extruded into 4D so that the two 3D objects do not lie in the same realm, are paralell, and all their surface areas connect. What is this called?

It's usually simply called "prism". Any object that is a cartesian product of figures in absolutely perpendicular spaces can be called prism.

Also, the pyramid. Any object in 2D is made into a pyramid by placing a point not on the same plane and making all verticies on the 2D object go to it (or the whole of any round sides). A regular pyramid is one in which the point in the other plane is centered. What is the analogy of this in 4D.

16-cell is not a pyramid. Half of it would be an octahedral pyramid.

Also, I take it there are 4 types of rectangular-prism-analogy in 4D. Consider. In 1D, the hyperrectangularprism (I call it), a line, has only one type-a line.

In 2D, there is 2-the rectangle, and the square.

In 3D, there is 3-the rectangle extruded into 3D in such a way that the new dimension, "height" is not equal to either side of the rectangle.
-2 - A square extruded into 3D in such a way that the new dimension is not equal to the side of the square (or a rectangle with height equal to one of its sides)
-3 - The cube (all sides equal)

In 4D, there would be 4!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!


Well, irockyou has talked about non-perpendicular prisms, but I think that is not really what you asked. Actually, in four dimensions, there is five analogues of right-angled prisms, not four. You can have all four "heights" different, or two of them the same, or three, or all four (tesseract). So what's the fifth possibility? Why, having two heights the same, AND the two remaining heights the same. This corresponds to having a square prism extruded by its height.
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Re: A few comments, questions, etc. etc.

Postby moonlord » Fri Jul 28, 2006 7:32 am

Kevin_Ahab wrote:Okay - first of all, what do we call the transformation analogous to making something a prism from 2D to 3D. I mean, for instance, a rectangular prism is created by extruding a rectangle into the third dimension, so that the two rectangles do not lie on the same plane, are paralell, and all their perimenters connect. In 4D, I presume, a 3D figure would be extruded into 4D so that the two 3D objects do not lie in the same realm, are paralell, and all their surface areas connect. What is this called?


Well, in the CSG that's called extruding. I see no reason for not calling it that way.

Kevin_Ahab wrote:Also, the pyramid. Any object in 2D is made into a pyramid by placing a point not on the same plane and making all verticies on the 2D object go to it (or the whole of any round sides). A regular pyramid is one in which the point in the other plane is centered. What is the analogy of this in 4D.


You take a point in another 3-realm and repeat the procedure. A 4D pyramid is regular if the perpendicular from the apex vertex to the 3D base intersects it in the geometrical center. This is called tapering to a point. It's definition is not final, thought, so it's better not to ask questions :D.

There are two ways to construct a prism in 4D.

1: Extrude any x (3D body). The result, xE in CSG notation, is what you're looking for.
2: Cartesian product between two 2D shapes. The result is called a duoprism. You can see that cartesian product between a 3D shape and a segment is the same with the first contruction method.
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Re: A few comments, questions, etc. etc.

Postby Keiji » Fri Jul 28, 2006 7:49 am

moonlord wrote:You take a point in another 3-realm and repeat the procedure. A 4D pyramid is regular if the perpendicular from the apex vertex to the 3D base intersects it in the geometrical center. This is called tapering to a point. It's definition is not final, thought, so it's better not to ask questions :D.


Yes it is...

On the Wiki, I wrote:T = Tapers the object to a point such that the new lines are the same length as the original lines.
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Postby moonlord » Fri Jul 28, 2006 8:01 am

Don't be that stubborn, you know there's a thread where people are still arguing about it.
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Postby Keiji » Fri Jul 28, 2006 12:12 pm

Well, that thread died. :P
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