Goto: Home | Geometry | Lamina | Caps

Laminate Caps

This is a construction based on a class of figures that have laminate sides. Although the class exists in all dimensions, the most important group is in two dimensions.

The basic idea is that one makes a figure with a aperigonal face, and then a second one, and then joins the two together. A simple example is the honeycomb of alternate squares and triangles.

               oo                oo                  |     |     |
                                                     |     |     |
    o-----o-----o----     ----o-----o-----o--       -o-----o-----o--
     \   / \   / \            |     |     |        /  \   / \   / \
      \ /   \ /   \ /         |     |     |       /    \ /   \ /   \
    ---o-----o-----o-      ---o-----o-----o--     ------o-----o-----o
                                                        |     |     |
               oo                oo                     |     |     |
         | 2 2 oo              2 oo | 2

                                 o                      o
           oo                 4 / \ 4                4 / \ 4
        o=======o              /   \                  /   \
       3 \     / 3            o=====o                o=====o
          o---o                 oo                  3 \    / 3
            3                                          o--o

In the figure above, the two figures are presented separately as having an infinite face, marked oo. Underneath these is the Wythoff figure for them. Below this is the vertex figure for each set. The aperigon is left in the last to show it as an internal chord of the pentagonal vertex figure.

Hyperbolic Figures

The same can be done in hyperbolic geometry as well. Except here, there is a richer group of figures to work with. Even if we restrict ourselves to those that lie in the same edges as {3,p}, there is an amazingly divers set of figures to be found. Note that this same restriction will only find extra, the pentagonal antiprism, in normal geometry.

How to procede

The {3,p} is most useful, since the cluster of triangles around a vertex is a cell of {p,p/2}, and removal of selected verticies of the {3,p} will leave a uniform honeycomb consisting entirely of triangles and p-gons. Of course, we want to remove some verticies, otherwise we revert back to a straight {3,p}.

We need to be able to build objects up that represent one, two or more triangles at the vertex figure.

triangle: One side of the vertex figure represents a triangle.

polygon: Two sides of the vertex figure is a p-gon.

petrie laminate: This is a laminate that is bounded by alternate verticies of a Petrie polygon of the {3,p}.

branched laminae: Higher than 3, we find branched laminae. As with all laminae, we can inscribe different kinds of faces in each segment. There are two modes with "4": a s{3,oo} and a t{p/2,oo}

five segments: Five segments can represent several different things: (p,3,p,oo) from "3 oo | p/2", "| 2 p oo", "|3 3 oo".

six segments: Six segments brings into play lots more. It can be (p p p oo), from "p oo | p/2", or a super-snub.


The presentation used here is that a snub face has three sides that it shares with different kinds of faces. The snub faces all have the same (clockwise) progression. That is, one can number the edges of such triangles clockwise "1,2,3", and arrange it so that "all sides marked "3" face pentagons, and "all sides marked "2" face unmarked triangles".

At least one of each kind of vertex must appear at a corner, since both ends of the "3" edge appear at the vertex of the pentagon in the above example. This gives rise to a cycle of alternating snub faces, and the faces or edges that are marked with the same number.

One extention of this is to use larger polygons, such as squares or pentagons. One of the non-convex uniform polyhedra is an example that uses squares in this way.

Another method is to put multiple cycles in the snub faces. Instead of writing "1 2 3 4 5 6 7 8 9" around the faces of the snub enneagon, we write "1,2,3,1,2,3,1,2,3", and treat the enneagon as an overgrown triangle.

Non-unique vertex figures

This allows for non-unique vertex figures. One can have the same vertex figure applied to what is visiably different figures.

         o-----o            o-----o
        / \ s  /\          / \ 3 / \
       /         \        /         \
      o           o      o           o
      |  8      8 |      |  8     8  |
      |           |      |           |
      o--       --o      o===========o
       \ s     s /        \ s     s /
        \ / 3 \ /          \ / s \ /
         o-----o            o-----o

The figure at the left shows a | 3 8 8, a snub figure related to the {8,6}. The triangles form little clusters of four, the three snub triangles surround the triangle marked "3". All triangles belong to these little clusters of four.

The thing on the right has exactly the same vertex figure, but an entirely different grouping of triangles. The triangles at the bottom of the vertex figure now form a long corridor that surrounds a Petrie polygon. The triangle at the top is now surrounded by three octagons, and no triangles. The top half represents a 3 oo | 4, while the bottom half represents a | 2 2 oo.


The notation I use for these figures is largely derived from extentions to Don Hatch's notation. In essence, I accomidate both the extended snubs and the embedded laminae by additions of devices to Don Hatch's notation.

The notation that Don Hatch uses is transliterated into the new style (without change), and then additional devices added. Don Hatch starts his notation off at a polygon: ours can start anywhere, the style being to bring together all elements of a figure.

      Hatch  ( 3 | 3   3   4 )     ( 3 || 3 || 3 || 3 || 3 || 5 || )

     Krieger z 3 x 3 o 3 o 4 o     z 3  s  3 s 3  s 3  s  3 s 3  s

The z-node normally is used to link the current node to the first node. Here it is in the first node, and thus it changes the meaning of what follows from a skew Petie-polygon of a n-1 dimensions to a normal flat polygon, typically in hyperbolic space. A numeral implies that the mirrors on either side are at an angle of pi/n. Capital letters can also be used. An unmarked node. An edge is bisected by this mirror, and is perpendicular to it. There are snub faces formed by removal of alternate verticies of the corresponding x form.

alt-snub face: What appears before the $ sign is what is used for the "s" after it. The number of "s" faces must be a divisor of the thing before the $ mark.

laminate face: This is one of the chords. It appears as a face, and what's on the other side appears between the Z and O.

For example, the two octagon thingies are as follows.

The left hand one is z8s8s3s.

The right hand one is a composite of z3x4xZOo, and zZOs2s2s. The second one is inverted, as "Zs2s2sO", which counts from the interior of the aperigon. This ZO is inserted into the first as z3x4xZs2s2sOo

An example of a supersnub is a figure that has a vertex figure of (3,9,9,9,9), and is derived from removing selected verticies of the {3,9}. The vertex figure of this resembles a little stick-person, with the triangle between the legs. The one enneagon that does not neighbour a triangle will appear always over the same sholder of this stick-person. The notation for this is z9$s2s3s9s.

Copyright 2002 Wendy Krieger

Goto: Home | Geometry | Lamina | Caps