One fruitful sorce of laminatruncates are polytopes of the from `{p,q,r}`,
where `{p,q}` is spheric, and `{q,r}` is hyperbolic. In practice,
the `{p,q,r}` has its verticies at infinity, but the circle that is
centred on it is a equidistant plane.

For some conditions, this plane may end up being perfectly flat, and
thence, the `{p,q,r}` turns out to be inscribed in a laminahedron.
When this happens, we can use the laminate faces as mirrors, to produce a
honeycomb of `t{p,q}`.

The vertex figure of a `t{p,q,r}` is a *r*-gonal pyramid, the
sloping edges being the `{2p}` and the base of edges to match the short
chord of a `{q}`. The vertex figure has an equatorial `{q,r}` and
`{2p,4}` on the lines of longitude. These must be equal.

This happens in the following cases.

`t{p,2p,4}`, which comes from`{2p,4}`=`{2p,4}``t{2,3,6}`, which comes from`{3,6}`=`{4,4}``t{2,6,3}`, which comes from`{6,3}`=`{4,4}``t{4,3,8}`, which comes from`{3,8}`=`{8,4}`

In all of these cases, there is a flat face `{q,r}` which can
be used as a mirror. These may be used to fill space with the required
cells. Because this is a new operation, it is deserving of a new prefix:
to this end, I use `lt{}` for **laminatruncated**

The `lt{p,2p,4}` gives space filled with `t{2q,3}`, four at
an edge. It is, the regular honeycomb `t{2p,3,4}`.

The `lt{2,3,6}` and `t{2,6,3}` give rise to space filled with
uniform triangular prisms, and uniform hexagonal prisms. These two honeycombs
are *not* dual, since the heights of the prisms are undisturbed in the
duality, but the bases change.

The `lt{4,3,8}` gives rise to space filled with truncated cubes.
The vertex figure is an octagon dipyramid, this being the convex hull of the
octahedron rotated 45 deg on one of its axis. That is, the `lt{4,3,8}`
contains the verticies and edges of a `{8,3,4}`

This is a class of figure that exists in four dimensions. It is designated
as `xt{p,q,r,s}`, but is related to the `bt{q,p,q,r}` and
`bt{r,s,r,q}`. Interestingly, the dual of the `xt{p,q,r,s}`
is a `xt{s,r,q,p}`

The `xt{p,2p,4,2}` is a `{2p,3,3,4}`, since this is what the
`bt{2p,p,2p,4}` resolves to.

One would suspect `bt{4,2,4,2p}` would reolve to `{4,3,3,2p}`.
It certianly looks like it will, but I am not yet sure of the implications of
this happening.

The `xt{2,3,6,2}` resolves to 4-space filled with bitriangular prisms,
and `xt{2,6,3,2}` give rise to 4-space and filled with hexagon-hexagon prisms.

These *are* duals in the same symmetries. That is, the centres of the
cells of a uniform honeycomb of `{3}{3}` are the verticies of a uniform
honeycomb of `{6}{6}`. This does not happen all the time: a uniform
honeycomb of `{3}{6}` does not give rise to another one on dualing.

The `xt{4,3,8,2}` gives rise to space filled with a bitruncated
24-cell. The edges of this match that of the `{3,8}` or `{8,4}`.
The vertex figure consists of an octagon-octagon cross, such as might be had
by rotating a 16-cell, 45 degrees in one plane, and 45 degrees in the vertical
plane. Therefore, this figure contains the verticies and edges of a `{8,3,3,4}`.
The `bt{3,4,3}` has truncated cubes as facets, these appear as cells of
the `lt{8,3,4}`, as discussed above.

The cells of the `xt{4,3,8,2}` are the same size as those of the
`bt{3,4,3,8}`. The diagram below represents the torus that lies
half way between the two octagons in the vertex figure. Each square
represents a `bt{3,4,3}`, and each column or row is a `bt{3,4,3,8}`.
This means that the `xt{4,3,8,2}` consists at each vertex of eight
distinct sets of `bt{3,4,3,8}` in two different ways. The `lt{4,3,8}`
appear as the lines of edges that appear between the square cells.

+---+---+---+---+---+---+---+---+ | | | | | | | | x<+--- bt{3,4,3} +---+---+---+---+---+---+---+---+ [just the square] | | | | | | | | | +---+---+---+---+---+---+---+---+ <- lt{4,3,8} | | | | | | | | | [just the line] +---+---+---+---+---+---+---+---+ \ | | | | | | | | | > bt{3,4,3,8} +---+---+---+---+---+---+---+---+ / [whole row] | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+

The `xt{2,4,3,8}` give rise to space tiled with bi-octagon prisms.
The vertex figure of this is the convex hull of two 24-cells, placed in dual
positions. It will be seen, therefore that this figure contains the verticies,
edges and faces of an `{8,3,4,3}`.

The vertex figure has 2.48 *(dec: 288)* faces. These are tetrahedra that
lie between the edges of the pair of incscribed 23-cells `{3,4,3}`.
Since the faces of these tetrahedra do not fall in a plane, there is no
through planes as in the other lattices.

The edges of the vertex figure form hexagons and octagons, these become
the vertex figures of `{8,6}` and `{4,8}`.

Copyright 2002 Wendy Krieger