Mirrors:Home Edges Dynkin Stott Laces
The first practical problem that involved what would become lace prisms, was to place values in the stott matrix for some hyperbolic groups. In essence, one might take something like x4x3o4o3z, and try to determine the diameter of that figure. Since the diagonal values were known, it's just a matter of finding the missing value. And so began a process of evaluating vertex-figures.
A pattern slowly emerged which allowed the vertex figure to be read directly from the graph. The coxeter-dynkin symbol is read differently, and straight comes the vertex figure. It became more apparent after i separated out the lacing edges, and could see the top and bottom solo.
$-4-$ elongated square antiprism. t= x4o | | Zigzag^2 = 3.41421356238 l= $4$ o-4-o Bases^2 = 1.00000000000 b= o4x
Note the use of multiple dollar signs ($). These are vertex-nodes here, and both are pointing to the same symmetry. The idea here is to read the structure as a progression, with multiple layers, arranged in a simplex.
One can simply read off the symbol as by regarding each marked node as a separate wythoff figure, parallel in the space of the unringged nodes. So here we have two figures, square over rotated square, with the zigzag set to the shortchord of an octagon (as x4x).
It turns out to be totally general. The Stott matrices can now be completed.
The dynkin matrix was found when I was working on a problem which involved plane normals to the mirrors. Such vectors have a dot-product of 1 in the normal to the plane, and zero for all vectors in the plane.
These were solved one by one, until i figured out what was being done was simply to invert the matrix. The dual turns out to be the dot products of the normals, but such vectors are at the supplement of the angles of the mirrors. A rexx program was prepared to test this idea.
[D:\save\cdata\SOURCE]regina matrix4 ssf ssf 0.145898033750315455 ! 4D f. ** 0.76393202250021 1.38196601125011 2.00000000000000 1.61803398874989 1.38196601125011 2.76393202250021 4.00000000000000 3.23606797749979 2.00000000000000 4.00000000000000 6.00000000000000 4.85410196624968 1.61803398874989 3.23606797749979 4.85410196624968 4.00000000000000 [D:\save\cdata\SOURCE]regina matrix4 sqsq: sqsq: -7.000000000000000000 ! 4D . ** 2.00000000000000 5.00000000000000 5.65685424949238 4.24264068711929 5.00000000000000 2.00000000000000 4.24264068711929 5.65685424949238 5.65685424949238 4.24264068711929 2.00000000000000 5.00000000000000 4.24264068711929 5.65685424949238 5.00000000000000 2.00000000000000
The dream was starting to come true. We could get values of the stott matrix straigt from the dynkin symbol.
Dr Richard Klitzing wrote a paper on segmentotopes, or equilateral polytopes with vertices in parallel layers. This was a complete listing for four dimensions. The notation for these is to write "top || bottom", using things like reversals etc where needed.
The trouble with loose items is that they can be normalised, for example, triangles as pointing up. Also with no underlying symmetry, it could be possible to place two figures of different symmetry together, such as icoashadron || cube.
The fix for this is to write two (or more) edge-letters at each node, each of the positions connected to a different vertex-node.
I mentioned that the heights of these figures could be found by a spreadsheet, and produced an example in Microsoft "Excel". Neither of the normal spread-sheets (Lotus 1-2-3 or Quartro Pro) support live matrix functions. The process amounts to taking the "sloping ladder" problem, and applying it in an oblique coordinate system.
t The drift is the difference \ between the top and bottom. \ The diagonal is the length \ of lacing by 2. The height ----b is to be found. d
The top and bottom layers are expressed in vectors, which are taken as a matrix dot product against the Stott matrix. Likewise, the drift is calculated as the matrix dot applied to the differences. Since matrix-dot gives a square, it is a simple matter of height^2 = lacing^2 - drift^2.
Richard Klitzing took this spreadsheet, and made it more user-friendly to ordinary mathematicans, and raised the number of dimensions that can be catered for, this is the version at his web site.
The notation needs to be robust enough to allow a great variety of problems to use the same symbol. Yet it must retain its programability. The notion, however must be kept simple. Here we take several polytopes set in the same symmetry, and simply designate them as such, a compound, so to speak. If something more is needed, it is to be added.
The first example derives from Dr Klitzing's segmentotopes, of polytopes laced together. From this comes lacing figures in a tower, rather a graphical version of coordinates beginning with a vertex or something.
The dual of a lace prism is a lace tegum, such is generated by imagining several wedges to intersect. The Hasse Diagram, (see below), is an example. Such represent in the simplest, the intersection of pyramid heads, where the bases of the pyramids are held in relation to each other.
One can then describe points on a plane etc, as polytope bases, and derive lace cities, or marked versions when a figure is projected onto two dimensions. And so forth.
$ x.3x.5o. / \ o----o-5-o xo3xx4of &#x \ /5 $ .o3.x5.f
Here we have two figures set against the same icosahedral mirrors. We can see that the mirrors are the same set, and what is produced is a single word. Such is a suitable notation for a compound, or position of figures of the same symmetry. To make this compound do something, we add an extra axis of symmetry (height), and then immediately destroy the mirror (#). The x is a lacing edge, which ties the two bits together.
.5o $t t.......t.......t ./ \ / \ / \ o-5-o \ / \ / .o5o= &# \ /= \ / \ / $b ====b=======b==== o5=
The above is a pentagonal antiprism, viewed from the side. The top and bottom vertices are labelled $t and $b in the symbol. Top and bottom edges by wythoff's construction are marked .... and ==== respectively, and the nodes to produce these edges marked with a single letter "." or "=".
The notion of lacing is that the top and bottoms on an antiprism resemble skins on a drum, and that the zigzag between them is the lacing necessary to hold these tight. The base lace is then let spread into any connections between bases of this nature.
Lace cones are how to form the dual of lace prisms. In essence, a lace cone is a pyramid apex, which is intended to intersec with other cones.
\ \ - / A tegum as an o-----o---- / - intersection of |\ |\ / - lace cones. \ | \ | \ a b \| a--+--o---- \ - o--+--b | \ - |\ | \ | - \ | \| \| o-----o----- | |
The diagram on the right shows the cube as an intersection of lace cones, the apices marked "a" and "b". The lines on the planes including 'a' are continued past the edge of the figure, to show the individual elements are larger than the intersection.
One can demonstrate this by holding a cube with three faces at a vertex in one hand, and the second hand covers the other three vertices.
When the dimension is not solid, the apex is multiplied by cartesian product by an orthogonal space to make it solid. One can see this on the simplex, by holding two faces in one hand, and the other two in the other hand. Note that the intersection shapes start off as a pair of lines, like a V or orthogonal A, yet the top of these has been extended.
Likewise, a tetrahedron can be shown as the intersection of a triangular prism and a point. Here the point is made 'solid' by extending it into a plane. The intersection is then between a point-pyramid and a plane.
The sections perpendicular to the axis of an antitegum, represents the intersection of a polytope and its dual. This is the series of truncate by decent of duals. The last point of a surtope, just before it disappears, represents the centre of it, and thus the vertices of an antitegum represent the surtope.
The series starts as 0-truncate and 0-rectate as the first figure increases unfettled by the second. At the point of contact, the vertices of the first start to disappear, and new faces appear as the dual gets smaller. The vertices of the 0-rectate or base figure are the surtope points of both the vertices of the base and the faces of the dual.
The first truncate and rectate causes the vertex of intersection to move along the edge of the base, until such point as the edges disappear. (rectate). This is repeated until the nth truncate and rectate are the vanishing dual, and the final point of disappearance.
The hasse diagram of a polytope is a representation of the direct incidences of the constituent surtopes. These are incident surtopes which differ by a rank of one. A property of the hasse diagram, is that between the bottom and any other node, is a hasse diagram of a surtope.
Since the polytope has a dual, there is an upwards path to the content, and these also represent a kind of around-surtope. But it is also a dual of the surtope of the dual, and hence also a surtope. So between any two points of representing incident surtopes of different ranks, there is a hasse diagram.
The hasse diagram can be represented topologically, by the intersection of lace cones of matching duals. The point representing a surtope is then the furtherest point from the exterior.
The polytope thus constructed is a special kind of lace tegum, called an antitegum. Since every surtope of this figure is itself a hasse diagram, constructed over the surtope in the vertex figure and its dual, the surtopes of a lace tegum are themselves lace tegums.
As with incidence itself, the up-incidence of a surtope is the down-incidence of its verge (or surtope-figure).
The Hasse diagram of pyramid products represents a prism product of the hasse diagram of the factors of the product. Since the ultimate is that Hasse diagrams are reversable by dual, the figure described here is such that one might inscribe a cube on the individual long diagonals of the factors. Any vertex of this cube serves as the bottom incidence of a pyramid product of figures and its dual.
The correspondance is that a tegum-product of antiprisms can be set against any set of the various duals, and still end as the same figure, which is itself an antiprism. One can see this, by regarding an antiprism as A || A' (In Klitzing's atop notation) and then passing a series of antiprisms B || B', C || C' etc. The pyramid product gives ABC. and its antiprism is ABC || A'B'C'. But the dashes can be moved to the top in any order, eg AB'C' || A'BC.
The runcinate is a polytope, whose vertices are in progression from the base figure to the vertices of the dual. It is the usual mark of Stott's expand as noted by Coxeter and by Conway.
The progression of a figure to its dual, by runcination, is the antiprism sequence. The totality of slides of this sequence is then the antiprism
The dual of the runcinate is the strombiate. The faces of this are derived by the antitegum of the margins of the base to the faces of the vertex-figure of the dual.
A Wythoff group is taken to be a group of mirrors, forming a simplex. This allows the finding of points that are distant 1 or 0 from the mirrors, by the bisection of various margins. Non-simplex shapes do not afford this by generality.
A wythoff cell is a simplex, comprising of at least one face and the balance being faces or mirrors. When such a figure is reflected in the mirrors, the result is a wythoff lace tegum. It is seen by descent that a wythoff lace tegum is bounded by WLTs,
A simple example is to set two mirrors at a rational angle, eg c/10, and to place two faces across this ditch. The operation of the mirrors will repeat these faces 10 times each around the mirrors.
If the faces fall at right angles to the mirror, then no margin forms, and such faces are unconnected to that mirror. In the above case, the general face becomes a strombus, if opposite mirrors are used or a triangle if the same mirrors are used. This leads to a pentagonal antitegum or pentagonal tegum resp.
Mirror nodes are 'unconnected' if the mirrors they represent are at right angles. This means that an image wholy in one mirror can not transfer to a solid element over the other. A chain of mirrors is a set of mirrors where there is a connection path between all members. If A is connected to B and B to C, there is a path from A to C, even when A and C are at right angles.
Likewise, face-nodes are not connected to mirrors if there is no margin formed. Such is the case when the face is perpendicular to the mirror.
If the mirrors form two or more distinct and unlinked chains, the resulting figure by this construction is a tegum product.
The dual of this process gives rise to wythoff lace prisms where s /tegum/prism/ , s /face/vertex/, s /margin/edge/. applies.
The face nodes are connected to the body, as the vertex nodes are connected to the nulloid. These by dual are in the same position, since s/face/vertex/ applies.
A general construction exists by the rules of descent, in that a valid surtope must be connected to the extreme point of incidence (body or nulloid). The nodes constituting the surtope are then S nodes, the nodes connected to the surtope, but not part of it are W nodes, while the remaining nodes are A nodes. The mirrors of the surtope are then those S mirrors, which reflect the surtope into itself in a different position, and A nodes, which leave the surtope unchanged, but reflect the around-space. The W mirrors reflect the surtope onto a different copy of itself. The count of the given surtope is by the formula c=g/sa where c is the count, g is the global symmetry, s the surtope symmetry and a the around symmetry.
The verge or around-figure of a surtope, comes from treating the whole surtope as the nulloid. The w nodes become vertex-nodes, and the a nodes are evaluated by the rules above.
There is no restriction on multiple occupancy, such is taken as to be handled as several instances, so the lacing applies to nonconvex polytopes as well as convex ones.
The lace prism arising from separate chains connected to a single vertex node is the simple prism there derived.
wythoff mirror-edge polytopes arise when all but one wall of the reflection cell is a mirror, and the remaining node is a vertex node. In this case, all edges are bisected vertically by mirrors, and thus the polytope is a WME.
wythoff mirror-margin polytopes arise when all but one node is a mirror and the remaining is a face. The face is transported to new copies over any margin, and this the polytope is a WMM figure.
In general, a lace prism has all of its surtopes and all of its verges as lace prisms and the same is held true for lace tegums. A verge is the arrangement of figures around a given surtope.
Mirrors:Home Edges Dynkin Stott Laces