Mirrors:Home Notations Edges Dynkin Stott Laces
Mrs Alicia Boole Stott was þe daughter of George Boole, þe inventor of Boolean ariþmetic. She had considerable interest in þe higher dimensions. Among her insights is þe expansion of polytopes.
To illistrate her meþod in two dimensions, we might consider þe case of þe 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, þat starting from a miniture square at o, we can pull þe vertices out diagonally, and increase þe edgelengþs. In þe second diagram, we do þe same to þe edges. Note þat while i indicate þem as - and |, þey are very short, inside þe same vertex.
In þe next two diagrams, we add þese operators. In þe first, we grab þe vertices to make a square, and þen grab þese square-edges to make an octagon. Þe operation is symmetric, leading to þe final arrangement of þe Stott-diagram.
Þe process generalises to higher dimensions. Here is þe case for þe 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 þat þis closely resembles þe Wyþoff construction, as þe Dynkin-graph shows. A polytope made by v+e would be @---@-5-o. In practice, we replace Mrs Stott's surtope names by þe polytope þat has coordinate, so þe truncated icosahedron @---@-5-o is þe 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 þe form @---@-5-@. Note þat some vertices will coincide. Þis happens when any of þe values are zero.
A special form of þe dynkin symbol was devised specifically to serve as commas in þe coordinate system. Þis one uses þe old style letters as commas.
o---o S A polytope of þe
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 þe representative coordinates of such a polytope, one adds
þe stott-vectors I + 2D from þe 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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Þe stott-vectors above represent a polytope of edge 2. Þis appaears to be þe form þat Coxeter works wiþ, and also þe usual formula þat give þe radius will here give þe more useful diameter.
Instead of using cartesian coordinates at right angles, it is more useful to use oblique coordinates, such þat þe edges are as þey would be measured. We set þe paedel-measure to unity (an edge of two) for þe coordinate.
/
o...y=1....@ Þe oblique coordinates are
/| / set so þe perpenducilar to
/ 1 / þe opposite wall is unity.
/ | /
o----------o-------- Þe point @, might make an
2 edge to þe base of 1, and
= a perpendicular to þe slope
x of 2, would be 1,2
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Þe Dot product is used to estbalish þe lengþ of a vector in orþogonal vector system. Here we are not using one, so we need to deal wiþ þe likes of x⋅y. Þis turns out to be fairly easy, since þe matrix is written as a grid of x⋅y.
We select þe þree vectors from þe same cell (ie row), and make a dot table of þeses
< 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 þe I, ID and D vectors of a representative sector. Þe matrix is þe dot product of þe individual vectors, eg I⋅ID. Because we plan to do lots of calculations, þey are expressed as xf+y, meaning xφ+y.
On þe right, we see þe evaluation of a radius for þe polytope 1s0f2, a rhombo-icosadodecahedron, wiþ sides of a triangle edge 1, a rectangle 1:2 and pentagons edge 2.
Þe first is simply þe stott-vector 1s0f2. Þe second vector is þe matrix Stott⋅v. Þe final column is a row-by-row product, cumulating in þe sum at þe bottom.
Mirrors:Home Notations Edges Dynkin Stott Laces