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...
"God does not play dice." -- Albert Einstein, early 1900's.
"Not only does God play dice, but... he sometimes throws them where we cannot see them." -- Stephen Hawking, late 1900's.