Mirrors:Home Edges Dynkin Stott
A mirror-edge is an edge where the ends are images of each other in the same mirror. A great number of the uniform polytopes have all of their edges as mirror-edges, that is, they are mirror-edge polytopes.
$ <-- vertex $ vertex-node | | | <-- paedal | paedal branch | | .........|......... <- mirror o @ wall-node | (wall) | | | full abridged $
Here we see an edge being reflected in a bisecting mirror. The vertex drops a perpendicular foot (paedal) to the wall, and this forms the edge on reflection.
We represent these either by the full form (with paedal and vertex), or in the more common abridged form (which just shows the intercept. The abridged form is more convenient for writing, since the figures get rather complex.
The standard notations involving the representation of a paedel is in the abridged form. The current representation of vertex-nodes and their paedels is of my invention.
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
The 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.
That is, the branch from $ to wall 1 is zero-length, and thus does not get drawn.
The most common branches are 2 and 3.
In higher dimensions, the 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 the previous case, ther is nothing in these two mirrors that gives an X coordinate when the branch $-1 is set to zero. That is, the polytope is without height.
Should there come to be an X coordinate, it isn't going to come from the interaction of mirrors 1 and 2. Because of this, we regard these mirrors as 'orthogonal' or 'unconnected'.
The majority of uniform polytopes derive from the application of this mirror-edge to the walls of reflective symmetries. In three dimensions, there are three walls forming a triangle. The vertex-node freely roams inside this 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 three dimensions, but the opposite to a margin is not a wall in higher dimensions, so we can't use the margin-names to name the opposite wall. Also, order comes into play more.
In higher dimensions, the Dynkin-Symbol is used. This is a graphical device, so is not much use for writing in lining text. Also, because the nodes are identical in form, we can do many more things than is the standard.
The power of the Dynkin symbol comes from its handling of node-removal.
This is the same diagram as above, with the '5' wall removed. What we are now looking at is the symmetric figure that 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 the pentagonal 'room'
The Wythoff-names now make less sense. The resulting symmetry is a pentagonal one, but we do not see '5' anywhere: just the names of walls which have lost their nominative angles.
In the Dynkin case, the removal of a wall-node removes any perpenduculars (ie paedals) and any branches (angles between walls) involving it. What we get left with is a pair of walls, at 36° and a vertex lying either on either wall, or in between. This gives rise to two orientations of pentagon and a decagon.
For example, the bottom wall of the diagram above (marked in bold) when reflected by the other two walls, make for a room of ten cells.
Note in the first row, we force the vertex right into the remaining corner. It falls in the centre of the room, so we end up with a polyhedron with 12, 30 or 20 vertices.
In the second set (rID, tI, tD), the vertex falls on a wall and so joins cells by pairs. There are therefore 60 such vertices.
In the last set (tID), the vertex lies in a cell, and so each cell has its own vertex: there are 120 vertices of this 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 the above as variations of the tID, with sides reduced to zero. This is useful if we want to see what is going on with symmetries. For example, reducing any node @ to o, makes that paedal into zero length, and the ends of the edge merge by pairs. We see also that the polygons that had this edge as a side also loose these edges.
Making a second set of @ go to o, also makes the polygon with that edge disappear completely. All of its vertices are merged into the centre and (say), the former decagon's vertices get merged into 10's to form the icosahedron's 12 vertices.
Removing all three @ make the polytope disappear completely to its central point.
The nature of the symmetry is that node-removal can break the chain of branches, leaving nodes unconnected to the vertex-node. When this happens, we get extrinsic (or context) symmetry.
In practice, what is happening is that we're getting a zero-height prism of the intrinsic symmetry (which shows), and an extrinsic device (which is reduced to a point).
Even though we no longer have a solid polyhedron, we must still count for the merged symmetry for else not doing it, we count the same thing over and again.
Note, for example, that the edges of the icosahedron are @ . o, which lie in the former thin rectangles of @--@-5-@. The intrinsic symmetry of these edges correspond to the long edge of the triangle. But there is also a collapsed width that contributes to the external symmetry.
Mirrors:Home Edges Dynkin Stott