Wythoff Polytopes

Basically, this is any polytope written in a dynkin symbol, using node-marks that represent directly, the size of the edge. so F3x is a WP. This distinction means that you can use the coordinates as a vector, and use the fancy vector arithmetic to calculate size and distances. In essence, a polytope like x5o3x is just a fancy coordinate (1,0,1), "AICO" (all icosahedral coordinates).

We can directly make a matrix that allows you to do a dot product between two vectors specified as AICO, or even find the length of a given vector.

Wythoff Lace Prisms

These are wythoff polytopes "laced together". This seems to be the mainstay of the polytope project, and part of the hassle is finding coordinates from lacing lengths. The first bit is relatively straight-forward: L² = H² + D². L is given in the "&#xt" bit. H is sought, and we shall find D from our vector project.

The relevant vector we are going to look at, is then (T-B), where these are the polytope vector at the top, less the similar one at the bottom. So we have, eg for xo3ox5of, t = xoo = (1,0,0) b = o,x,f = (0,1,f), d = (1,-1,-f). We then calculate d·d, and subtract this from L², to get the height. ie

H² = L² - (t-b)·(t-b).

Because the coordinates are oblique, we can not do the usual dot product, but must either set the thing into an orthogonal schema, or do a "matrix-dot" product, ie D² = S

_{ij}d

_{i}d

_{j}, where S

_{ij}is the relevant stott matrix.

Stott Matrices

The two ways to get the stott matrix are to use the relation S

_{ij}D

_{ij}= I, where D

_{ij}is the dynkin matrix, or the time-honoured way of writing the stott matrix out from the Schläfli vector and the resident animal. The latter is very efficient if you are using hand calculations.

The Dynkin matrix D<sub>ij</sub> is simply a matrix, where the main diagonal D

_{ii}is filled in with '1', and the rest of the elements are filled in with D

_{ij}= -cos( pi/[ij]), where [ij] is the branch-mark between nodes i, j (counting unconnected nodes as '2'. It's a dynkin matrix, because you can load it directly from the dynkin symbol. When i do this by hand, i normally double all the entries, and put a leading '1/2' out the front. This saves having to remember both 1.61803398875 and 0.809017&c.

You can do it by hand: just write out the required matrix by using the Schläfli vector and the resident animal. When you do this in front of someone, like 'just use this technique', it looks very cool etc.

The vector is written in the first column, starting at the bottom. It always takes the same pattern for [3],3, etc, so once you learnt the general pattern, it's pretty easy to write these things down. The order is icosahedral, which means we put the non-threes bit down the bottom end (ie we do 3,3,5, not 5,3,3).

The schläfli vector is derived pretty much from a kind of 'continued fraction' derived from the schläfli symbol. For the first few, you can see that the animal is a single cell, occupied by the dimension-number. Tilings are the surface of an n+1 polytope, so {3,6} is a polyhedron, not a 2d thing, so n=3 here.

- 3...3 1, 2, 3, 4, 5, 6, ... Animal = [n]
- 3...4 q, 2, 2, 2, 2, 2, ... Animal = [n]
- 3...5 f, 2, 3-f, 4-2f, 5-3f, 6-4f, Animal = [n]
- 3...6 h, 2, 1, 0 Animal = [n]
- 3...P a, 2, 3-b, 4-2b, 5-3b, ... Animal = [n] a=shortchord, b=third-chord = a²-1.
- 3...3 q, 2q, 3, 2, 1, 0
- 3...A 2, 2, 4, 4, 4, 4, 4, 4, ...
- 3...B 3, 2, 4, 6, 5, 4, 3, 2, 1, 0

The animals for the larger symmetries like {3,4,3} and {3,3,3,3,B} = 2_21 are given here. There's only three of them.

- Code: Select all
`Animals for the non-regular symmetries`

3..4,3 3...A 3.....B

k_11 k_21

[ 2n-2 n-1 ] [ n n-2 ] [ 2n-2 n-1 2n-6 ]

[ n-1 2 ] [ n-2 n ] [ n-1 4 n-3 ]

[ 2n-6 n-3 n ]

The rest of the S

_{ij}is filled out by S

_{ij}= j S

_{i1}, as long as j <= i. The matrix is symmetric, so S

_{ij}= S

_{ji}.

Here are some worked examples, of writing these things directly. The order is to fill in the animal first, then the schlafli vector (which must be found to get the correct denominator).

- Code: Select all
`{3,3,5}`

5-3f

[ 4-2f ] [ 4-2f 3-f 2 f ]

2 [ 3-f ] 2 [ 3-f 6-2f 4 2f ]

-- [ 2 ] -> --- [ 2 4 6 3f ]

[ f . . 4 ] 5-3f [ f 2f 3f 4 ]

Schlafli Animal

2_21 = .3.3.3.3.B. n=6

3 (overall demom)

[ 4 5 6 4 2 3 ]

2 [ 5 10 12 8 4 6 ]

- [ 6 12 18 12 6 9 ]

3 [ 4 8 12 ( 10 5 6 ) ]

[ 2 4 6 ( 5 4 3 ) ]

[ 3 6 9 ( 6 3 6 ) ]

S.v ( animal )

Canonical Polytopes etc

We can write a prism of size l*b*h as (l)2(b)2(h). The brackets mean that the letters are just algebraic variables, without any special meaning. In any case, while this is a wythoff-polytope representing a prism of l long, b broad, and h high, the standard coordinate gives l,b,h ACS (all change of sign). In other words, putting l directly into the coordinate, gives a prism whose size is 2l, 2b, 2h.

Since we often just feed the 'size-as-written' from the dynkin symbol into a program, there are the odd 2 and 4 to be met as we eliminate this doubling.

So one has to be wary of this when one finds the value of d from lacing and height.

For example, the subtraction of eg (1,1,0) from (0,1,1) gives a vector (1,0,1), but this applies to a polytope whose side is *2*. (it gets confusing, and i need to think this one through, its always one of my trouble-spots).

APACS and EPACS etc

The task of finding coordinates from a wythoff polytope, involves passing a single coordinate say (1,1,0) AICO, through a number of filters to get a larger number out the bottom. Much can be saved by way of expanding filters. But basically, AICO represents a corresponding vector in every element of the icosahedral symmetry. You get one of those black-and-white symmetry groups, and think, the white ones are even octants, the black ones are odd quadrants.

Most polytopes, and all Wythoff polytopes, have vertices in every quadrant.

The routine AICO -> EPACS replaces a single point with up to five points.

The routine EPACS -> ACS replaces a single epac point (1,0,0) with three points (1,0,0), (0,1,0), (0,0,1)

The routine ACS -> ALL replaces a point of acs with up to eight standard points.

There are other coordinate systems.

EICO is just the white sectors of the icosahedral. It converts five-fold into EPECS

AICO is the icosahedral group, * 2 3 5 it five-folds down EPACS.

EICO is the rotary icosahedral group 2 3 5 (snub dodeca), it is five-fold to EPECS

APACS is the ordinary cubic symmetry. *2 3 4

EPACS is the pyritohedral symmetry 3*2

APECS is the tetrahedral symmetry *2 3 4

EP&CS is the octahedral-rotary group 2 3 4 (ie snub cube)

EPECS is the tetrahedral-rotary group 2 3 3

ACS is the rectangular group * 2 2 2

In a lace tower, the bit after the &# is another coordinate axis. So xo3oo5ox&#x actually is 4d, three over the base by AICO, and the fourth by height.