Lace prism

Lace prisms are an extension of ordinary polytopes, deriving a class of two or more layers laced together. The simplest cases are the Wythoff Lace-Prisms.

A great number of uniform polytopes have a construction in wythoff-mirror-edge form, and can be represented by equal-level markings on the Coxeter-Dynkin graph.

Vertex Nodes
Polytopes constructed by Wythoff's construction are normally represented by marking the corresponding nodes on the Coxeter-Dynkin symbol. Such marks represent an edge striking the mirror perpendicularly, either of zero length, (ie 'o'), or at a non-zero length (eg 'x'). One could represent these connections by use of an extra node, which is not a mirror, but represents a vertex. This node is then connected to the marked nodes of the Coxeter-Dynkin symbol, the non-zero marks are transferred to the branches between the vertex and mirror node.

A pentagon, for example, is usually represented by @--5--o, which is written as x5o. When this is converted into a vertex-node, the vertex-node is written as $, and we have $o--5--o. There is one vertex in the symmetry element, it drops a half-edge to the first node, and nothing to the second. This means that it is on the second mirror.

The number of nodes of any kind represent the dimension of the simplex, or zero-counted, the general dimension. In the example above,


 * $ represents a vertex
 * $o is a mirror edge (literally an edge into a mirror)
 * $---o-5-o is a polygon (the same as a triangle), in this case a pentagon.

Lace Prisms
A wythoff lace prism is two or more wythoff mirror-edge polytopes, laced together. The representation of a simple example, the pentagonal antiprism, might be xo5ox&#x. It still has pentagonal symmetry, but the doubling of the node-symbols designate a double-layered polytope. The bit after the & means that we have added a height, the # means no symmetry, and the x is the lacing. Although it has the symmetry to the left of the &, it has a height or altitude with no symmetry.

The representation using vertex-nodes is to connect the first $ to the first letter in each set, eg x.5o., gives $---o-5-o, and the second to the second vertex-node, ie ,o5.x = o--5--o---$. These are top and bottom layers, the connection is a series of edges $$ which is indicated by the #x' in the symbol.

So we have for $---o--5--o---$


 * $ represents a vertex, There are two different kind of vertex, not joined by mirrors.
 * $---o is an edge on the top layer.  It is formed by the reflection of the top vertex ($) into a mirror. (o).  The half-edge is the --- bit
 * o---$ is an edge on the bottom layer. As above, for the bottom layer/
 * $---$ is a lacing-edge.  This lies in the symmetry element, and runs from top to bottom
 * $---o-5--o is a thing in the triangle-dimension with five edges, ie a pentagon on the top.
 * o-5-o---$  is a pentagon on the bottom.  Note the edge points to a different mirror, so its a dual here.
 * $---o   $   is a triangle, literally an edge-point pyramid.  Vertex-nodes are ultimately connected to the nullitope node, so this is not disconnected,
 * $  o---$   is a triangle pointing upwards.  The edge is on the bottom layer, and the top is a point.
 * $--o-5-o--$ has four nodes (like a tetrahedron), and therefore is a polyhedron.

Lace Towers
A lace tower is a stack of lace prisms, the seperate vertex-nodes address different 'floors' of the tower. It has the effect of being able to represent the layers of a polytope in a linear style. An icosahedron might be represented vertex first, by the layers o5o, o5x, x5o, o5o (point, pentagon, inverted pentagon, point. There are four vertex-nodes, so four letters at each node.  The lacing between layers are all equal to edges, so this is

ooxo5oxoo&#xt, Here the t stands for tower, that the four vertex-nodes are in a tower.

Lace Cities
Lace cities are a representation of a polytope as points on a plane, such that each point is given as a wythoff polytope, perpendicular to the plane of view. This allows a much more benign view of polytopes, because one can 'walk' different alignments that are not present in towers. In the diagrams above are sections of the 24choron {3,4,3} "ico". These are presented vertex-first. The first is down a triangular axis, consisting of triangles x3o, inverted driangles o3x and points o3o. The second is the same figure down a square axis. x4o are squares (such as (±1, ±1), and o4q is an inverted square with vertices (±2,0) and (0,±2). The edge is sqrt(2) of those of the x4o.

In this view, one can look at things like edge-first (first example, columns), or face-first (second example, diagonals) projections of this figure, without having to draw new coordinates.