Mirrors:Home Edges Dynkin Stott Laces
Mrs Alicia Boole Stott was the daughter of George Boole, the inventor of Boolean arithmetic. She had considerable interest in the higher dimensions. Among her insights is the expansion of polytopes.
To illistrate her method in two dimensions, we might consider the case of the square.
-------o -
| \ ---o ---o
----o | \ \ o \ sq--oct
| | - \ \ \ \ / /
--o | | \ \ ---o o \ o / /
o | | | + | | o | | o o | pt---rh
v e v +e e+ v
square rhomb octagon octagon stott-diagram
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We see here, that starting from a miniture square at o, we can pull the vertices out diagonally, and increase the edgelengths. In the second diagram, we do the same to the edges. Note that while i indicate them as - and |, they are very short, inside the same vertex.
In the next two diagrams, we add these operators. In the first, we grab the vertices to make a square, and then grab these square-edges to make an octagon. The operation is symmetric, leading to the final arrangement of the Stott-diagram.
The process generalises to higher dimensions. Here is the case for the seven icosahedral mirror-edge members.
I ---- tI voo --- veo
/ 3 5' \ / \ / \ v
/ \/ 6 / \/ \ /
/ / \ \ / / \ \ /
pt---ID rID -4- tID ooo - oeo voh -- veh o----e
\ \ / / \ \ / / \
\ /\ 10 \ /\ / \
\ 5 3' / \ / \ / h
D --- tD ooh --- oeh
5
by symbol name by miniture icosa v----e----h
numbers show kinds
of polygon face shared
by pairs of polyhedra
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We note that this closely resembles the Wythoff construction, as the Dynkin-graph shows. A polytope made by v+e would be @---@-5-o. In practice, we replace Mrs Stott's surtope names by the polytope that has coordinate, so the truncated icosahedron @---@-5-o is the sum of an icosahedron @---o-5-o and an icosadodecahedron o---@-5-o.
o-------------o-------5-------o
f . 0 . 1 2f. 0. 0 ff. 1 . 0 2f = 3.223068
f . 0 . 1 ff. f. 1 ff. 1 . 0 ff = 2.618033
f . 0 . 1 ff. f. 1 f. f . f f = 1.618033
f . 0 . 1 f. 1. ff f. f . f 1 = 1.000000
f . 0 . 1 f. 1. ff 1. 0 . ff 0 = 0.000000
cyclic permutation and all change of sign
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We can 1ten establish five representative vertices of any polytope, of the form @---@-5-@. Note that some vertices will coincide. This happens when any of the values are zero.
A special form of the dynkin symbol was devised specifically to serve as commas in the coordinate system. This one uses the old style letters as commas.
o---o S A polytope of the
o-4-o Q form
o-5-o F
o-6-o H 1" 0" 2"
o5/2o V o-----o--5--o = 1 s 0 f 2, meaning an @s@f of sides
1", 0" 2".
To find the representative coordinates of such a polytope, one adds
the stott-vectors I + 2D from the table above, eg
1 I = f,0,1 D2 = 9f + 9
0 ID = 0,0,0 4f + 4
2 D = 2f,2f,2f 8f + 5
-------- ------
3f,2f,f3 21f + 18
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The stott-vectors above represent a polytope of edge 2. This appaears to be the form that Coxeter works with, and also the usual formula that give the radius will here give the more useful diameter.
Instead of using cartesian coordinates at right angles, it is more useful to use oblique coordinates, such that the edges are as they would be measured. We set the paedel-measure to unity (an edge of two) for the coordinate.
/
o...y=1....@ The oblique coordinates are
/| / set so the perpenducilar to
/ 1 / the opposite wall is unity.
/ | /
o----------o-------- The point @, might make an
2 edge to the base of 1, and
= a perpendicular to the slope
x of 2, would be 1,2
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The Dot product is used to estbalish the length of a vector in orthogonal vector system. Here we are not using one, so we need to deal with the likes of x⋅y. This turns out to be fairly easy, since the matrix is written as a grid of x⋅y.
We select the three vectors from the same cell (ie row), and make a dot table of theses
< vectors > < matrix >
vector S⋅v
( f, 0, 1 ) [ f+2 2f+2 2f+1 ] 1 5f+4 5f+4
( 2f, 0 , 0) gives [ 2f+2 4f+4 4f+2 ] 0 8f+6 -
( f2 , 1, 0) [ 2f+1 4f+2 3f+3 ] 2 8f+7 16f+14
------
21f+18
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What we have here is the I, ID and D vectors of a representative sector. The matrix is the dot product of the individual vectors, eg I⋅ID. Because we plan to do lots of calculations, they are expressed as xf+y, meaning xφ+y.
On the right, we see the evaluation of a radius for the polytope 1s0f2, a rhombo-icosadodecahedron, with sides of a triangle edge 1, a rectangle 1:2 and pentagons edge 2.
The first is simply the stott-vector 1s0f2. The second vector is the matrix Stott⋅v. The final column is a row-by-row product, cumulating in the sum at the bottom.
Mirrors:Home Edges Dynkin Stott Laces