-: Lace Prisms and Lace Tegums :-


Mirrors:Home Edges Dynkin Stott Laces

Vertex figures

Þe first practical problem þat involved what would become lace prisms, was to place values in þe stott matrix for some hyperbolic groups. In essence, one might take someþing like x4x3o4o3z, and try to determine þe diameter of þat figure. Since þe diagonal values were known, it's just a matter of finding þe missing value. And so began a process of evaluating vertex-figures.

A pattern slowly emerged which allowed þe vertex figure to be read directly from þe graph. Þe coxeter-dynkin symbol is read differently, and straight comes þe vertex figure. It became more apparent after i separated out the lacing edges, and could see þe 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 þe use of multiple dollar signs ($). Þese are vertex-nodes here, and boþ are pointing to þe same symmetry. Þe idea here is to read þe structure as a progression, wiþ multiple layers, arranged in a simplex.

One can simply read off þe symbol as by regarding each marked node as a separate wythoff figure, parallel in þe space of þe unringged nodes. So here we have two figures, square over rotated square, wiþ þe zigzag set to þe shortchord of an octagon (as x4x).

It turns out to be totally general. Þe Stott matrices can now be completed.

Þe discovery of þe Dynkin Matrix

Þe dynkin matrix was found when I was working on a problem which involved plane normals to þe mirrors. Such vectors have a dot-product of 1 in þe normal to þe plane, and zero for all vectors in þe plane.

Þese were solved one by one, until i figured out what was being done was simply to invert þe matrix. Þe dual turns out to be þe dot products of þe normals, but such vectors are at þe supplement of þe angles of þe mirrors. A rexx program was prepared to test þis 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

Þe dream was starting to come true. We could get values of þe stott matrix straigt from þe dynkin symbol.

Klitzing's Segmentotopes

Dr Richard Klitzing wrote a paper on segmentotopes, or equilateral polytopes wiþ vertices in parallel layers. Þis was a complete listing for four dimensions. Þe notation for þese is to write "top || bottom", using þings like reversals etc where needed.

Þe trouble wiþ loose items is þat þey can be normalised, for example, triangles as pointing up. Also wiþ no underlying symmetry, it could be possible to place two figures of different symmetry togeþer, such as icoashadron || cube.

Þe fix for þis is to write two (or more) edge-letters at each node, each of þe positions connected to a different vertex-node.

I mentioned þat þe heights of þese figures could be found by a spreadsheet, and produced an example in Microsoft "Excel". Neiþer of þe normal spread-sheets (Lotus 1-2-3 or Quartro Pro) support live matrix functions. Þe process amounts to taking þe "sloping ladder" problem, and applying it in an oblique coordinate system.


     t          Þe drift is þe difference
      \         between þe top and bottom.
       \        Þe diagonal is þe lengþ
        \       of lacing by 2.  Þe height
     ----b      is to be found.
      d

Þe top and bottom layers are expressed in vectors, which are taken as a matrix dot product against þe Stott matrix. Likewise, þe drift is calculated as þe matrix dot applied to þe 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 maþematicans, and raised þe number of dimensions þat can be catered for, þis is þe version at his web site.

Þe Evolution of terms

Þe notation needs to be robust enough to allow a great variety of problems to use þe same symbol. Yet it must retain its programability. Þe notion, however must be kept simple. Here we take several polytopes set in þe same symmetry, and simply designate þem as such, a compound, so to speak. If someþing more is needed, it is to be added.

Þe first example derives from Dr Klitzing's segmentotopes, of polytopes laced togeþer. From þis comes lacing figures in a tower, raþer a graphical version of coordinates beginning wiþ a vertex or someþing.

Þe dual of a lace prism is a lace tegum, such is generated by imagining several wedges to intersect. Þe Hasse Diagram, (see below), is an example. Such represent in þe simplest, þe intersection of pyramid heads, where þe bases of þe pyramids are held in relation to each oþer.

One can þen 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 forþ.


     $            x.3x.5o.
    /  \
   o----o-5-o     xo3xx4of  &#x
         \ /5
          $       .o3.x5.f

Here we have two figures set against þe same icosahedral mirrors. We can see þat þe mirrors are þe same set, and what is produced is a single word. Such is a suitable notation for a compound, or position of figures of þe same symmetry. To make þis compound do someþing, we add an extra axis of symmetry (height), and þen immediately destroy þe mirror (#). Þe x is a lacing edge, which ties þe two bits togeþer.

                  .5o
    $t      t.......t.......t
  ./         \     / \     / \
  o-5-o       \   /   \   /        .o5o= &# \
     /=        \ /     \ /
    $b      ====b=======b====
                  o5=

Þe above is a pentagonal antiprism, viewed from þe side. Þe top and bottom vertices are labelled $t and $b in þe symbol. Top and bottom edges by wythoff's construction are marked .... and ==== respectively, and þe nodes to produce þese edges marked wiþ a single letter "." or "=".

Þe notion of lacing is þat þe top and bottoms on an antiprism resemble skins on a drum, and þat þe zigzag between þem is þe lacing necessary to hold þese tight. Þe base lace is þen let spread into any connections between bases of þis nature.

Lace Cones

Lace cones are how to form þe dual of lace prisms. In essence, a lace cone is a pyramid apex, which is intended to intersec wiþ oþer cones.

                                         \     \
     - /          A tegum as an           o-----o----
      / -         intersection of         |\    |\
     /     -      lace cones.           \ | \   | \
    a         b                          \|  a--+--o----
     \     -                              o--+--b  |
      \  -                                |\ |   \ |
     - \                                  | \|    \|
                                             o-----o-----
                                             |     |

Þe diagram on þe right shows þe cube as an intersection of lace cones, þe apices marked "a" and "b". Þe lines on þe planes including 'a' are continued past þe edge of þe figure, to show þe individual elements are larger þan þe intersection.

One can demonstrate þis by holding a cube wiþ þree faces at a vertex in one hand, and þe second hand covers þe oþer þree vertices.

When þe dimension is not solid, þe apex is multiplied by cartesian product by an orþogonal space to make it solid. One can see þis on þe simplex, by holding two faces in one hand, and þe oþer two in þe oþer hand. Note þat þe intersection shapes start off as a pair of lines, like a V or orþogonal A, yet þe top of þese has been extended.

Likewise, a tetrahedron can be shown as þe intersection of a triangular prism and a point. Here þe point is made 'solid' by extending it into a plane. Þe intersection is þen between a point-pyramid and a plane.

Antitegums

Þe dual of an antiprism is an antitegum. Here þe cones are þe heads of dual pyramids.

Antitegmal Series

Þe sections perpendicular to þe axis of an antitegum, represents þe intersection of a polytope and its dual. Þis is þe series of truncate by decent of duals. Þe last point of a surtope, just before it disappears, represents þe centre of it, and þus þe vertices of an antitegum represent þe surtope.

Þe series starts as 0-truncate and 0-rectate as þe first figure increases unfettled by þe second. At þe point of contact, þe vertices of þe first start to disappear, and new faces appear as þe dual gets smaller. Þe vertices of þe 0-rectate or base figure are þe surtope points of boþ þe vertices of þe base and þe faces of þe dual.

Þe first truncate and rectate causes þe vertex of intersection to move along þe edge of þe base, until such point as þe edges disappear. (rectate). Þis is repeated until þe nþ truncate and rectate are þe vanishing dual, and þe final point of disappearance.

Þe Hasse Diagram

Þe hasse diagram of a polytope is a representation of þe direct incidences of þe constituent surtopes. Þese are incident surtopes which differ by a rank of one. A property of þe hasse diagram, is þat between þe bottom and any oþer node, is a hasse diagram of a surtope.

Since þe polytope has a dual, þere is an upwards paþ to þe content, and þese also represent a kind of around-surtope. But it is also a dual of þe surtope of þe dual, and hence also a surtope. So between any two points of representing incident surtopes of different ranks, þere is a hasse diagram.

Þe hasse diagram can be represented topologically, by þe intersection of lace cones of matching duals. Þe point representing a surtope is þen þe furþerest point from þe exterior.

Þe polytope þus constructed is a special kind of lace tegum, called an antitegum. Since every surtope of þis figure is itself a hasse diagram, constructed over þe surtope in þe vertex figure and its dual, þe surtopes of a lace tegum are þemselves lace tegums.

As wiþ incidence itself, þe up-incidence of a surtope is þe down-incidence of its verge (or surtope-figure).

Þe Hasse diagram of pyramid products

Þe Hasse diagram of pyramid products represents a prism product of þe hasse diagram of þe factors of þe product. Since þe ultimate is þat Hasse diagrams are reversable by dual, þe figure described here is such þat one might inscribe a cube on þe individual long diagonals of þe factors. Any vertex of þis cube serves as þe bottom incidence of a pyramid product of figures and its dual.

Þe correspondance is þat a tegum-product of antiprisms can be set against any set of the various duals, and still end as þe same figure, which is itself an antiprism. One can see þis, by regarding an antiprism as A || A' (In Klitzing's atop notation) and þen passing a series of antiprisms B || B', C || C' etc. Þe pyramid product gives ABC. and its antiprism is ABC || A'B'C'. But þe dashes can be moved to þe top in any order, eg AB'C' || A'BC.

Strombiate polytopes

Þe runcinate is a polytope, whose vertices are in progression from þe base figure to þe vertices of þe dual. It is þe usual mark of Stott's expand as noted by Coxeter and by Conway.

Þe progression of a figure to its dual, by runcination, is þe antiprism sequence. Þe totality of slides of þis sequence is þen þe antiprism

Þe dual of þe runcinate is þe strombiate. Þe faces of þis are derived by þe antitegum of þe margins of þe base to þe faces of þe vertex-figure of þe dual.

Proof of þe completeness of Lace Prisms etc

A Wythoff group is taken to be a group of mirrors, forming a simplex. Þis allows þe finding of points þat are distant 1 or 0 from þe mirrors, by þe bisection of various margins. Non-simplex shapes do not afford þis by generality.

A wythoff cell is a simplex, comprising of at least one face and þe balance being faces or mirrors. When such a figure is reflected in þe mirrors, þe result is a wythoff lace tegum. It is seen by descent þat 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 þis ditch. Þe operation of þe mirrors will repeat þese faces 10 times each around þe mirrors.

If þe faces fall at right angles to þe mirror, þen no margin forms, and such faces are unconnected to þat mirror. In þe above case, þe general face becomes a strombus, if opposite mirrors are used or a triangle if þe same mirrors are used. Þis leads to a pentagonal antitegum or pentagonal tegum resp.

Mirror nodes are 'unconnected' if þe mirrors þey represent are at right angles. Þis means þat an image wholy in one mirror can not transfer to a solid element over þe oþer. A chain of mirrors is a set of mirrors where þere is a connection paþ between all members. If A is connected to B and B to C, þere is a paþ from A to C, even when A and C are at right angles.

Likewise, face-nodes are not connected to mirrors if þere is no margin formed. Such is þe case when þe face is perpendicular to þe mirror.

If þe mirrors form two or more distinct and unlinked chains, þe resulting figure by þis construction is a tegum product.

Þe dual of þis process gives rise to wythoff lace prisms where s /tegum/prism/ , s /face/vertex/, s /margin/edge/. applies.

Þe face nodes are connected to þe body, as þe vertex nodes are connected to þe nulloid. Þese by dual are in þe same position, since s/face/vertex/ applies.

A general construction exists by þe rules of descent, in þat a valid surtope must be connected to þe extreme point of incidence (body or nulloid). Þe nodes constituting þe surtope are þen S nodes, þe nodes connected to þe surtope, but not part of it are W nodes, while þe remaining nodes are A nodes. Þe mirrors of þe surtope are þen þose S mirrors, which reflect þe surtope into itself in a different position, and A nodes, which leave þe surtope unchanged, but reflect þe around-space. Þe W mirrors reflect þe surtope onto a different copy of itself. Þe count of þe given surtope is by þe formula c=g/sa where c is þe count, g is þe global symmetry, s þe surtope symmetry and a þe around symmetry.

Þe verge or around-figure of a surtope, comes from treating þe whole surtope as þe nulloid. Þe w nodes become vertex-nodes, and þe a nodes are evaluated by þe rules above.

Þere is no restriction on multiple occupancy, such is taken as to be handled as several instances, so þe lacing applies to nonconvex polytopes as well as convex ones.

Þe lace prism arising from separate chains connected to a single vertex node is þe simple prism þere derived.

wythoff mirror-edge polytopes arise when all but one wall of þe reflection cell is a mirror, and þe remaining node is a vertex node. In þis case, all edges are bisected vertically by mirrors, and þus þe polytope is a WME.

wythoff mirror-margin polytopes arise when all but one node is a mirror and þe remaining is a face. Þe face is transported to new copies over any margin, and þis þe polytope is a WMM figure.

In general, a lace prism has all of its surtopes and all of its verges as lace prisms and þe same is held true for lace tegums. A verge is þe arrangement of figures around a given surtope.

Mirrors:Home Edges Dynkin Stott Laces


© 2003-2009 Wendy Krieger