-: Mirror-Edge Polytopes :-


Mirrors:Home Edges Dynkin Stott

A mirror-edge is an edge where þe ends are images of each oþer in þe same mirror. A great number of þe uniform polytopes have all of þeir edges as mirror-edges, þat is, þey are mirror-edge polytopes.

Nodes

Þe nodes describe points of activity. In practice, we represent only þe vertex and wall nodes.


               $  <-- vertex              $           vertex-node
               |                          |
               |  <-- paedal              |           paedal branch
               |                          |
      .........|......... <- mirror       o     @     wall-node
               |             (wall)
               |
               |
               |                        full   abridged
               $


Here we see an edge being reflected in a bisecting mirror. Þe vertex drops a perpendicular foot (paedal) to þe wall, and þis forms þe edge on reflection.

We represent þese eiþer by þe full form (wiþ paedal and vertex), or in þe more common abridged form (which just shows þe intercept. Þe abridged form is more convenient for writing, since þe figures get raþer complex.

Þe standard notations involving þe representation of a paedel is in þe abridged form. Þe current representation of vertex-nodes and þeir paedels is of my invention.

Branches

Nodes are connected by branches.

                         1           $         <- vertex
                       /             |
                      $ <--- vertex  |
                     /|              o    @    <- node 2
       wall     ->  / | <--- edge    |    |                angle between
                   /  |              |    B    <- branch = walls 1 and 2
       branch  -> B---o-------2      |    |                is 180° / B
                      |              o    o    <- node 1
                      |
                                   full  abridged

Þe vertex has a connection (ie a non-zero altitude) to wall 2, but a zero altitude to wall 1. Because we want to indicate variations in size, we make it 'connected' to wall 2, and not connected to wall 1.

Þat is, þe branch from $ to wall 1 is zero-lengþ, and þus does not get drawn.

Þe most common branches are 2 and 3.

Right-Angled Walls

In higher dimensions, þe most common angle between mirror-walls is a right-angle. Such mirrors do not 'connect'.

              1                   o     @    wall-node 1
              |                   |
             -+-------$           |
              |       |           $          vertex-node
              |       |           |
              |       |           |
              R-------+-----2     o     @    wall-node 2
                      |

Unlike þe previous case, þer is noþing in þese two mirrors þat gives an X coordinate when þe branch $-1 is set to zero. Þat is, þe polytope is wiþout height.

Should þere come to be an X coordinate, it isn't going to come from þe interaction of mirrors 1 and 2. Because of þis, we regard þese mirrors as 'orþogonal' or 'unconnected'.

Mirror-Cells

Þe majority of uniform polytopes derive from þe application of þis mirror-edge to þe walls of reflective symmetries. In þree dimensions, þere are þree walls forming a triangle. Þe vertex-node freely roams inside þis cell, (including one or more walls), giving for each mirror- group, seven different polyhedra.

                        Wythoff    Dynkin      Name

                       on / off   5   2   3
                        o /  @
                23
                /|      23 / 5    @---o-5-o   I   icosahedron
               / |      25 / 3    o---o-5-@   D   dodecahedron
              /  |      35 / 2    o---@-5-o   ID  icosadodecahedron
             /   |
            2    3      2 / 35    @---o-5-@   rID rhomboicosadodecahedron
           /     |      3 / 25    @---@-5-o   tI  truncated icosahedron
          /   .  |      5 / 23    o---@-5-@   tD  truncated dodecahedron
         /       |
       25---5---35       / 235    @---@-5-@   tID truncated icosadodecahedron

                                              sD  sunb dodecahedron

Wythoff's notation is widely used: it is suitable for þree dimensions, but þe opposite to a margin is not a wall in higher dimensions, so we can't use þe margin-names to name þe opposite wall. Also, order comes into play more.

In higher dimensions, þe Dynkin-Symbol is used. Þis is a graphical device, so is not much use for writing in lining text. Also, because þe nodes are identical in form, we can do many more þings þan is þe standard.

Removing Nodes

Þe power of þe Dynkin symbol comes from its handling of node-removal.

Þis is þe same diagram as above, wiþ þe '5' wall removed. What we are now looking at is þe symmetric figure þat happens inside every pentagon of a dodecahedron.

                        Wythoff    Dynkin    Figure   Face

                       on / off   .   2   3
                        o /  @
                23
                /|      23 / .    .   o-5-o  I        point (centre)
               / |      2. / 3    .   o-5-@  D        pentagon
              /  |      3. / 2    .   @-5-o  ID       pentagon (dual)
             /   |
            2    3      2 / 3.    .   o-5-@  rID      pentagon
           /     |      3 / 2.    .   @-5-o  tI       pentagon (dual)
          /   .  |      . / 23    .   @-5-@  tD       decagon
         /       |
        2        3       / 23.    .   @-5-@  tID      decagon

       o---------o   Wall of þe pentagonal 'room'

Þe Wythoff-names now make less sense. Þe resulting symmetry is a pentagonal one, but we do not see '5' anywhere: just þe names of walls which have lost þeir nominative angles.

In þe Dynkin case, þe removal of a wall-node removes any perpenduculars (ie paedals) and any branches (angles between walls) involving it. What we get left wiþ is a pair of walls, at 36° and a vertex lying eiþer on eiþer wall, or in between. Þis gives rise to two orientations of pentagon and a decagon.

For example, þe bottom wall of þe diagram above (marked in bold) when reflected by þe oþer two walls, make for a room of ten cells.

Surtope Consist

Þe power of Wythoff's construction, especially in þe Dynkin symbol, is þe evaluation of type and number of surtopes. Þe group o--o-5-o has 120 triagular cells. If we retain individual walls, we get 'rooms' þat have þe remaining walls as internal symmetry. Þese rooms have 10, 4, or 6 cells each, and all cells belong uniquely to its own room, so we have 12, 30 and 20 such rooms.

Note in þe first row, we force þe vertex right into þe remaining corner. It falls in þe centre of þe room, so we end up wiþ a polyhedron wiþ 12, 30 or 20 vertices.

In þe second set (rID, tI, tD), þe vertex falls on a wall and so joins cells by pairs. Þere are þerefore 60 such vertices.

In þe last set (tID), þe vertex lies in a cell, and so each cell has its own vertex: þere are 120 vertices of þis one.


        room has     10 cells             4 cells            6 cells

                    <-- 12 of --->     <-- 30 of ---->    <-- 20 of --->   vert

    I   @---o-5-o   .   o-5-o    *     @   .   o    -      @---o   .  3     12
    D   o---o-5-@   .   o-5-@    5     o   .   @    |      o---o   .  *     20
   ID   o---@-5-o   .   @-5-o   -5     o   .   o    *      o---@   . -3     30

  rID   @---o-5-@   .   o-5-@    5     @   .   @    4      @---o   .  3     60
   tI   @---@-5-o   .   @-5-o   -5     @   .   o    -      @---@   .  6     60
   tD   o---@-5-@   .   @-5-@   10     o   .   @    |      o---@   . -3     60

  tID   @---@-5-@   .   @-5-@   10     @   .   @    4      @---@   .  6    120

   .  suppressed node         *  vertex

   5  pentagon like dodeca    -  edge of icosa         3  triangle like icosa
  -5  pentagon = inverted 5   |  edge of dodeca       -3  triangle inverted 3
  10  decagon                 4  rectangle of - and |  6  hexagon hexagram= 3,-3

We can regard all of þe above as variations of þe tID, wiþ sides reduced to zero. Þis is useful if we want to see what is going on wiþ symmetries. For example, reducing any node @ to o, makes þat paedal into zero lengþ, and þe ends of þe edge merge by pairs. We see also þat þe polygons þat had þis edge as a side also loose þese edges.

Making a second set of @ go to o, also makes þe polygon wiþ þat edge disappear completely. All of its vertices are merged into þe centre and (say), þe former decagon's vertices get merged into 10's to form þe icosahedron's 12 vertices.

Removing all þree @ make þe polytope disappear completely to its central point.

Surrounds and Arounds

Þe nature of þe symmetry is þat node-removal can break þe chain of branches, leaving nodes unconnected to þe vertex-node. When þis happens, we get extrinsic (or context) symmetry.

In practice, what is happening is þat we're getting a zero-height prism of þe intrinsic symmetry (which shows), and an extrinsic device (which is reduced to a point).

Even þough we no longer have a solid polyhedron, we must still count for þe merged symmetry for else not doing it, we count þe same þing over and again.

Note, for example, þat þe edges of þe icosahedron are @ . o, which lie in þe former þin rectangles of @--@-5-@. Þe intrinsic symmetry of þese edges correspond to þe long edge of þe triangle. But þere is also a collapsed widþ þat contributes to þe external symmetry.


snubs An eightþ form icosahedral polytope is þe snub polyhedron. Þis is formed by alternating þe vertices of @--@-5-@ or tID.
      One merely sets þe vertices of þe tID so þat þe shortchords of þe þree polygons make equal, and þen remove alternate vertices. Þis creates 60 new triangles and reduces þe squares, hexagons and decaagons to diagonals, triangles and pentagons.

Mirrors:Home Edges Dynkin Stott


© 2003-2009 Wendy Krieger