The cantellated duocylinder

Higher-dimensional geometry (previously "Polyshapes").

The cantellated duocylinder

Postby quickfur » Wed Sep 17, 2008 7:28 pm

The "official" Grand Antiprism is a uniform polychoron consisting of 20 pentagonal antiprisms in two orthogonal rings, with the rings joined to each other via tetrahedra. These two rings are just like the mutually orthogonal rings in the duoprisms and the duocylinder, and the tetrahedra are laid out along a toroidal 2-manifold just like the duocylinder's ridge.

These likenesses are not coincidental. If we relax the regularity requirement on the tetrahedra, we may obtain a whole series of grand antiprisms, consisting of 2n m-gonal antiprisms in two rings of n antiprisms each, with non-regular tetrahedra connecting them. I'm going to denote these grand antiprisms as n,m-grand antiprisms, where n denotes the number of antiprisms (n must always be even, otherwise the shape is not closed by the sheet of tetrahedra), and m denotes the base polygon of the antiprisms.

Well, as n and m approach infinity, the n,m-grand antiprisms approach a shape with a similar geometry to the duocylinder, except that where the duocylinder's ridge is, this shape has another 3-manifold (basically a thickened version of the duocylinder's ridge), joined to the two tori by two ridges of the kind of shape as the duocylinder's ridge. The thickness of this 3-manifold depends on how much we "stretch" the tetrahedra in the approximating grand antiprisms.

We may think of this shape as the cantellated duocylinder: it's what we get if we truncate the duocylinder along its toroidal ridge. The two bounding tori shrink (but remains topologically the same), and a new 3-manifold is introduced. Or, alternatively, it's what we get if we expand the duocylinder's bounding tori outwards, and fill in the gap with a new 3-manifold (hence the name cantellated). The cantellated duocylinder is the limiting shape of the (non-uniform) grand antiprisms, and thus can be usefully approximated by an n,m-grand antiprism of a high order. (This is for the same reasons we use high-order duoprisms to approximate a duocylinder: so that we don't have to deal with quadratic 4D equations, which are Not Nice(tm) to program.)

The cantellated duocylinder is also the limiting shape of another related series of polychora: the cantellated duoprisms. Basically, you take a duoprism and insert cubes (well, cuboids) where its square faces are. The resulting shape as the same number of prisms as the duoprism, but with additional cuboidal cells. Taking the limit of these shapes as n and m grow to infinity in the source n,m-duoprism, we approach the cantellated duocylinder.
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Re: The cantellated duocylinder

Postby wendy » Thu Sep 18, 2008 8:45 am

I doubt that it's a cantellated duocylinder.

cantellation is 'rectified-rectified', which does not work here. That is, cantellated x = rectified rectified x, by the Conway-Kepler rule.

If we suppose that truncation / rectification is the result of the intersection of duals, (decent of faces), then there is only one kind of figure that arises between the duoprism and the dual (bi-circular tegum), and one might be more correct to call it a truncated bicircular tegum, or truncated duocylinder, as the figure looks more like a bicircular tegum or bicircular prism, accordingly.

For the figure 'grand antiprism', it has little to do with antiprisms, except that such occur in the faces. The designated symbol for this in the PG is j5j2j5j, which suggests a regular construction from a symmetry [p][p]. The actual construction is bitoridal, being a diminished version of xyPyx&yxPxy.

The xyPyx&yxPxy is am intersection of 2P * 2P prisms, the first is x*y, the second y*x. The representing faces consist of 2P prisms, of height x, and the edges of the 2Pgon being of length x, y. This forms around the polar bands. The second set of faces consist of xy&yx&t, this can be made by placing in parallel planes, rectangles of xy, and yx, and connecting the edges. This figure has 4P prisms, and 4P² distorted cubes.

Removing vertices creates two new faces, each a tetrahedron, giving 8P² faces, each a tetrahedron. Thid gives 12P² tetrahedra, and 8P antiprisms.

We then can add the equatorial 2Pgons, to get another figure, formed with 24P² faces, so designated.

from the 2P gon (edge) to a top/bottom edge of the antiprism in the array: 2P²
from the vertex on the 2Pgon to the faces of the antiprism 2P*2P = 4P²
from an antiprism face to a vertex in the second torus 4P²
the lacing edge of an antiprism to the matching opposite, 4P²
& repeat torii 3, 2, 1 10P²

This gives 24P², for P=5, some dec 600 = twe 500 faces.

The designation of this class of figures is in the 'laminates'. This is a class of figure formed by placing layers one on another, where the only restriction is the 'etching' on the surface of the layer.
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Re: The cantellated duocylinder

Postby quickfur » Thu Sep 18, 2008 4:15 pm

wendy wrote:I doubt that it's a cantellated duocylinder.

cantellation is 'rectified-rectified', which does not work here. That is, cantellated x = rectified rectified x, by the Conway-Kepler rule.

Well, I looked up "cantellate" in Wikipedia, and it seems that it has a different usage than I intended here. I guess maybe "expand" ("expansion"/"expanded") might be a better term here. The basic idea is to "explode" the bounding 3-manifolds outward and fill in the gaps created. In 3D it's cantellation; in 4D, it's runcination; in 5D, it's sterication. Although, in the case of the duocylinder, you could also call it face-truncation, since it has no edges or vertices.

[...]For the figure 'grand antiprism', it has little to do with antiprisms, except that such occur in the faces. The designated symbol for this in the PG is j5j2j5j, which suggests a regular construction from a symmetry [p][p]. The actual construction is bitoridal, being a diminished version of xyPyx&yxPxy.
[...]

You're right, there isn't very much of an antiprismic character in the grand antiprism besides the fact that it has antiprismic cells surrounded by tetrahedra. It is much closer to the duoprisms than a true antiprism (in terms of being bitoridal).

What might be a better name for this category of polychora?
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Re: The cantellated duocylinder

Postby Keiji » Thu Sep 18, 2008 5:38 pm

quickfur wrote:Well, I looked up "cantellate" in Wikipedia, and it seems that it has a different usage than I intended here. I guess maybe "expand" ("expansion"/"expanded") might be a better term here. The basic idea is to "explode" the bounding 3-manifolds outward and fill in the gaps created. In 3D it's cantellation; in 4D, it's runcination; in 5D, it's sterication. Although, in the case of the duocylinder, you could also call it face-truncation, since it has no edges or vertices.


On HDDB, the conventional name for this process is called "cantellation" under any dimension.

You could refer to the Dx number, which in this case is always 1+2n-1 for n-dimensional objects.

The only reason I can see for why the standard names change is because standard (Wikipedia &c) cantellation keeps the same Dx number for any dimension (but is a different process), whereas conventional (HDDB &c) cantellation keeps the same process for any dimension but has a different Dx number.
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Re: The cantellated duocylinder

Postby wendy » Sun Sep 21, 2008 8:15 am

Norman Johnson introduced the terms 'cantellated' and 'runcinated' for describing four-dimensional polytopes.

By his usage, cantellated corresponds to the third node, and runcinate to the fourth node (where the zero node is marked).

The use i made for these is

cantellation = rectified + n-rectified (nodes connected to a given node are marked)
runcination = stott's expand. (first and last node marked)

For 'expand', one moves the existing faces outwards, but retain size. This makes then new faces that correspond to prisms of one surtope (eg edge) by the dual's margin. In 3d, this gives rectangles (rhombo-CO), but in 4d, it gives things like pentagonal prism, etc.

@--5-o---o---@ first = tetrahedron (o---o--@), second = triangle prism (@ o---@). third = pentagonal prism @--5-o @ fourth = dodecahedron @-5-o---o

The general term here is in the PG as 'runcinate', the matching dual is 'strombiate'.

The rectates correspond to any single node marked, eg o--5--o---@---o Taken as a separate figure, its edge-centres give a rectified n-rectate, this can be generated by the single step of marking the nodes attached to the marked node, eg o-5-@--o--@, as 'cantellate'. The corresponding truncated n-rectate is a cantetruncate, which keeps the original marked node o--5--@---@---@.

The confusion in 3d is that the cantellate and runcinate coincide.
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