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.
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