flat tapertopes

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

flat tapertopes

Postby PWrong » Thu Aug 03, 2006 1:29 pm

Let's try to count the number of tapertopes, not including any round shapes. A formula for this list could lead to a formula for the tapertopes in general.

1D:
line = 1

2D:
square = 11
triangle = 1'

3D:
tetrahedron = 1''
square pyramid = 11' (Rob's notation) or 1'1' (my notation)
triangular prism = 1'1
111 = cube

4D:
1''' = pentachoron
11'' or 1''1'' = square dipyramid = 11 -> 0 -> 0
111' or 1'1'1' = cube pyramid = 111 -> 0
1'11 = triangular diprism = 1' x 1 x 1= 111 -> 11
1'1' = triangular prism pyramid = (1' x 1) -> 0 = 1'' -> 1' = (11->0) -> 1
1''1 = tetrahedral prism = 1'' x 1 = (1' x 1)
1111 = tetracube

There might be more 4D shapes. We should probably check that none of the 4D shapes are curved like the wedge. So far, our sequence is 1, 2, 4, 7.
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Postby Keiji » Thu Aug 03, 2006 1:40 pm

Why isn't this in "cone-like objects"?

Anyhow, if you haven't noticed the group notation is preferred:

line = x
square = xy
triangle = x<sup>y</sup>

etc.
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Postby PWrong » Thu Aug 03, 2006 1:46 pm

The objects in this list aren't cone-like. But you can move it there if you want. I didn't use the group notation because I don't like writing hundreds of sup tags.
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Postby Nick » Thu Aug 03, 2006 3:45 pm

From what I understand, " ' " represents the object before it being tapered to a point. I understand Rob's viewpoint of 11', a square tapering to a point, but I don't understand your view 1'1'. All I see are two triangles.
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Postby Keiji » Thu Aug 03, 2006 10:31 pm

PWrong wrote:The objects in this list aren't cone-like. But you can move it there if you want. I didn't use the group notation because I don't like writing hundreds of sup tags.


No, I can't move it there if I want. Annoyingly enough, PHPBB has a split function, but not a merge function. It would have been better if this was posted in that topic to start with and it could have been split if someone decided if it shouldn't be there. :roll:

And you can get the notation for every tapertope, complete with sup tags, from here.

irockyou:

xy<sup>z</sup> = line, extruded to square, tapered to square pyramid
x<sup>y</sup>z<sup>w</sup> = line, tapered to triangle, extruded to triangular prism, tapered to triangular prismidal pyramid.

Understand now? ;)
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Postby Nick » Thu Aug 03, 2006 10:42 pm

Oh, yeah, I get it now. Cool.

Something just occured to me, though. Why do we use the number "1"? When would we ever use a 2?
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Postby PWrong » Fri Aug 04, 2006 8:19 am

xy<sup>z</sup> = line, extruded to square, tapered to square pyramid
x<sup>y</sup>z<sup>w</sup> = line, tapered to triangle, extruded to triangular prism, tapered to triangular prismidal pyramid.

In my notation, square pyramid is x<sup>z</sup>y<sup>z</sup>. But that's confusing when we use numbers instead of letters, so we'll stick to Rob's notation.
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Postby Keiji » Fri Aug 04, 2006 9:11 am

irockyou wrote:Oh, yeah, I get it now. Cool.

Something just occured to me, though. Why do we use the number "1"? When would we ever use a 2?


Numbers higher than 1 refer to curved objects. As you can see, this topic is about non-curved objects. ;)
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Postby PWrong » Fri Aug 04, 2006 10:54 am

I'll reorder the objects so it's easier to find patterns. We start with objects that have the most taperings. Taperings on the left come before taperings on the right.

2D: 2 shapes
triangle = 1'
square = 11

3D: 4 shapes
tetrahedron = 1''
triangular prism = 1'1
square pyramid = 11' (Rob's notation) or 1'1' (my notation)
111 = cube

4D: 1 + 3 + 3 + 1 = 8 shapes
1''' = pentachoron

1''1 = tetrahedral prism
1'1' = triangular prism pyramid
11'' = square dipyramid

1'11 = triangular diprism
11'1 = square pyramid prism (I missed this one)
111' = cube pyramid

1111 = tetracube

5D: 1 + 4 + 6 + 4 + 1 = 16 shapes
1''''

1'''1
1''1'
1'1''
11'''

1''11
1'1'1
1'11'
11''1
11'1'
111''

1'111
11'11
111'1
1111'

11111

Can anyone guess the formula? :lol:
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Postby Nick » Fri Aug 04, 2006 11:09 am

Basically, we're simply adding " ' " to different spots. *lol*
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Postby PWrong » Fri Aug 04, 2006 11:29 am

Pretty much. And the formula is of course, 2^(n-1).

btw I added square pyramid prism to the wiki, but I haven't written anything about it.
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Postby papernuke » Sat Aug 05, 2006 2:24 am

Whats a tapertop?

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Postby PWrong » Sat Aug 05, 2006 4:41 am

Whats a tapertop?

It's an object formed by cartesian products, torus products and tapering.

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