four-dimensional analogues of the deltahedron

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

four-dimensional analogues of the deltahedron

Postby Hodge8 » Mon Mar 11, 2013 4:51 pm

I wonder whether there are convex non-uniform four-dimensional analogues of the deltahedron.

Any references to work on those (possible?) polychora?
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Re: four-dimensional analogues of the deltahedron

Postby Klitzing » Mon Mar 11, 2013 6:33 pm

Deltahedra are named after the shape of the upper case letter Delta, an triangle. Thus those well could be called trianglohedra as well. Then the direct anologon would be tetrahedrochora for 4D. Those are known. Again provided convexity. The respective research in a broader sense was done by a german couple with surename Blind in the last century. In fact they were looking for the analogons of the Johnson solids, extrapolating in that sense, that the bounding facets are asked to be regular polytopes.

For online references cf. e.g. http://bendwavy.org/klitzing/explain/delta.htm and http://bendwavy.org/klitzing/explain/johnson.htm (scroll down).

If you require their papers, I'd have to digg the references out.

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Re: four-dimensional analogues of the deltahedron

Postby Klitzing » Mon Mar 11, 2013 6:42 pm

And here are the mentioned references

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Re: four-dimensional analogues of the deltahedron

Postby Hodge8 » Mon Mar 11, 2013 10:29 pm

Excellent - thank you. Exactly what I was looking for.
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Re: four-dimensional analogues of the deltahedron

Postby Hodge8 » Wed Mar 13, 2013 5:27 pm

Follow-up questions:

I am trying to visualize the tetrahedral dipyramid and the icosahedral dipyramid.

I guess the tetrahedral dipyramid is two pentachorons joined at a face; a possible 3-D projection envelope being a triangular bipyramid;

and the icosahedral dipyramid is either two 20x tetrahedra clusters joined at a face, or two 20x tetrahedra clusters joined at 20 faces (a possible 3-D projection envelope being an icosahedron).

I would be grateful for confirmation or otherwise.
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Re: four-dimensional analogues of the deltahedron

Postby Klitzing » Wed Mar 13, 2013 8:27 pm

Take the 600-cell in vertex to vertex orientation, cut off a vertex pyramid at either side, omit all the remaining middle part, and glue those 2 icosahedral pyramids together.

Either one of those icosahedral pyramids can easily be visualized as an icosahedron, any vertex of it being joined to the bodycenter by an additional edge. Thus any icosahedral triangle becomes a tetrahedron with tip at the center (shortened by projection). - You will have to use 2 such clusters, attach those at the icosahedron, which thus has to be reduced from either pyramid, leaving a total of 40 tetrahedra.

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Re: four-dimensional analogues of the deltahedron

Postby quickfur » Thu Mar 14, 2013 7:31 pm

Hodge8 wrote:Follow-up questions:

I am trying to visualize the tetrahedral dipyramid and the icosahedral dipyramid.

It's just a tetrahedron with two pyramids erected on it, one on one side, and one on the other.

Alternatively, think of it as two 5-cells joined together at one face.

I guess the tetrahedral dipyramid is two pentachorons joined at a face; a possible 3-D projection envelope being a triangular bipyramid;

Correct. Though keep in mind that a pentachoron itself can project to a triangular bipyramid (though with a different internal structure): see here, for example.

And both can project to a tetrahedron: in the 5-cell's case, the apex would project to a point in the middle of the tetrahedron, whereas on the dipyramid, the two opposite apices would project to that same point. One can imagine one point being "smaller" (i.e. farther away in 4th direction) and the other point being "bigger" (closer in 4th direction).

and the icosahedral dipyramid is either two 20x tetrahedra clusters joined at a face, or two 20x tetrahedra clusters joined at 20 faces (a possible 3-D projection envelope being an icosahedron).

I would be grateful for confirmation or otherwise.

The icosahedral dipyramid has 40 cells; 20 cells surrounding each apex, and each group of 20 cells connects with the other group at an icosahedron-shaped interface (i.e. the interface between the two clusters is the 20 faces of an icosahedron, which lies internal to the polytope).

And yes, the projection envelope can be an icosahedron. It can also be a hexagonal dipyramid or a decagonal dipyramid, depending on the orientation of the icosahedral cross-section relative to the 4D viewpoint.
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Re: four-dimensional analogues of the deltahedron

Postby wendy » Fri Mar 15, 2013 7:58 am

It's best to avoid 'bi-pyramid' in four dimensions, since it gives two different meanings.

One meaning, is a pyramid whose base is a pyramid. For example, a 'pentagonal bipyramid' would would be a pyramid with apex at A, of a pyramid with an apex at B, of a pentagon. Since one can easily reverse the roles of A and B, this could be regarded as a bi-pyramid.

The second meaning is to reflect 3d usage, that is a 3d girth in XYZ, tapering away to points in +W and -W. This is the dual of the prism, and ought be called the (base)al tegum, eg 'tetrahedral tegum'. It is the dual of a 'tetrahedral prism', the tegum being everywhere the dual of a prism.

Likewise, we ought not propogate 'rhombus' into 3d. The diverse constructions of 'rhombus' leads to a rhombus in 2D, but quite diverse figures in 3D.

In one instance, we might consider equal lengths at equal angles (eg the sides radiating at 60 degrees from a triangle or tetrahedron vertex). This leads to a rhombohedron and generally a rhombotope, which is generally an oblique cube, or generally, a measure polytope expanded or contracted along its long axis. The figure is in general, a 'simplex anti-tegum', being the dual of a 'simplex anti-prism'.

In the second instance, we might suppose for general values x, y, z, ... that one can mark the points +x,0,0 and -x,0,0, and 0,+y,0, and 0,-y,0 and so forth, and cover it. In 2D, it leads to a rhombus. If one calls a 'prismoid' the general rectangle prism of free sides 1/x, 1/y, 1/z, &c, then the dual is the figure described above is the general 'tegmoid', or right-cross polytope.

So we have here, the tetrahedral and icosahedral tegum, rather than bipyramid, these being dual to a tetrahedral and dodecahedral prism. Likewise, we note that the use of adjectives gives that there is one missing dimension, and that this is a hight-based measure.
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Re: four-dimensional analogues of the deltahedron

Postby quickfur » Fri Mar 15, 2013 2:20 pm

I generally use "bipyramid" to mean tegum, and "pyramid pyramid" to mean the (n+1)-dimensional pyramid of an n-dimensional pyramid. So an icosahedral bipyramid = icosahedral tegum (dual of dodecahedral prism), whereas pentagonal pyramid-pyramid = pyramid of pentagonal pyramid (which happens to be the same as the antiprism of a pentagonal pyramid).
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Re: four-dimensional analogues of the deltahedron

Postby Klitzing » Fri Mar 15, 2013 2:47 pm

quickfur wrote:I generally use "bipyramid" to mean tegum, and "pyramid pyramid" to mean the (n+1)-dimensional pyramid of an n-dimensional pyramid.
[...]
whereas pentagonal pyramid-pyramid = pyramid of pentagonal pyramid (which happens to be the same as the antiprism of a pentagonal pyramid).


:o_o: Quickfur, here you mix up your own, just given definition!

Code: Select all
lace city of (n-py)-py is:
o-n-o   o-n-o
    x-n-o


Code: Select all
lace city of (n-ap)-(dipy = tegum) is:
    x-n-o
o-n-o   o-n-o
    o-n-x

This latter display is being read from left to right. But it well can be read diagonally, then being the lace city of n-py || dual n-py, what also can be called the antiprism of the n-py (as base).

Thus (5-py)-py is not the [ (5-py)-ap = (5-ap)-dipy ]

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Re: four-dimensional analogues of the deltahedron

Postby quickfur » Fri Mar 15, 2013 3:15 pm

haha you're right, the pentagonal pyramid antiprism is a dipyramid of the pentagonal antiprism :) oops. :oops:
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