But because a triangle is not so much a ring, I usually tend to prefer that same set to be named "(2n-gon, n-antiprism)-wedge".
student91 wrote:but if we all agree on K4.8 being the gyrobicupolic ring, then why doesn't anyone change this in the wiki?
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please help
wendy wrote:Lace prisms have always been about 'progressions', the antiprism has always been part of the mix. The idea that they are exclusively WME figures is hardly correct, since the progressions are usually reckoned in terms of the alternate CF structure as well. Likewise, antitegums of non-regular figures must always involve a catalan, and not a WME. That is the wording used in my earlier post.
Still, the operations like DDG etc are not so much "operations done after the lace prism", since these are actually done before the lace-prism is applied. It's actually a sub-symmetry of the icosahedral group, in exactly the same way that tri-diminished icosahedron is subsymetric to the icosahedron. Most of them exist in Johnson's figures too. For example, K4.168 is DGG but then one of its bases is DGG too. There has never been any issue with admitting K4.168 as a lace-prism, because the icosahedron-vertex is variously diminished or gyrated.
On the other hand, the figures i have difficulty with calling lace-prisms, are things like ike || cube. The triangle || line is a valid segmentotope, but not a lace prism. ...
... Not in that form, because there is no progression from triangle to line. You have to deconstruct the triangle to something that does have a progression to a line: ie pt - line. The triangle || line never occurs where lace-prism faces might appear, but the pt.line.line can, because there are three bases. ...
... Writing, eg so3so4ox&#x, for example, does not make it a 'lace prism', but a parallel set of compounds laced together. ...
... The progression is not obvious. Even at "@3xo * xx 2 o% " which is the currently recieved orbifold, makes me no sense, since i have not untangled orbifold notation well enough to read lace towers. I can construct the thing, but the orbifold figures have lots of active regions, which i have not teased out.
The connection with WM figures seems to go no further back until the time when i was asked to describe the vertex-figures of the WME figures. Some experience and the use of 'vertex-nodes' and the radiant figures etc, allowed me to quickly grasp what was going on. They're lace prisms.
One should note that the lace-prism notation, as applied to segmentotopes, is an accommodation for the purpose. The unit edge do not occur in the vertices of uniform figures.
My earliest use of what we might call a lace prism comes from radiant space and the 'cuboctahedron product' that Jonathan played around with.
You imagine three coordinates, x,y,z. A point (1,0,0) represents the polytope X at the base size. You can scale it up and down accordingly. The point say (1,1,0) represents the prism-product of X,Y at their size. The prism and tegum products were defined as the figures defined by the sum(x)=1, and max(x)=1, which are their normal canonical vertices. The nature of the pyramid product was evaluated at first, from cosidering the figure stretched from (1,0,0) to (0,1,0) to (0,0,1), where the faces are indeed pyramid figures.
Of course, you can stick other figures in here too. Consider the triangle (1,1,0), (1,0,1), and (0,1,1). What does this mean? In modern parlence, we would write this as a lace-prism xxo2xox2xxo&#x. But there are elements of P(X, Pyr(Y,Z)) in there. The square faces at the end are P(x, T(y,z)), etc. But what is this exotic thing in the triangle?
xxo2xox2oxx&#x
o..2o..2o.. | 4 * * | 1 1 2 2 0 0 0 0 0 | 1 2 2 1 2 2 1 2 0 0 0 0 0 | 2 1 1 2 1 1 2 2 1 0 0 0 | 1 1 2 1 1 0
.o.2.o.2.o. | * 4 * | 0 0 2 0 1 1 2 0 0 | 0 2 1 2 0 0 0 2 1 2 1 2 0 | 1 2 1 0 0 0 2 1 2 1 2 1 | 1 0 1 2 1 1
..o2..o2..o | * * 4 | 0 0 0 2 0 0 2 1 1 | 0 0 0 0 1 2 2 2 0 1 2 2 1 | 0 0 0 1 1 2 1 2 2 1 1 2 | 0 1 1 1 2 1
---------------+-------+-------------------+---------------------------+-------------------------+------------
x.. ... ... | 2 0 0 | 2 * * * * * * * * | 1 2 0 0 2 0 0 0 0 0 0 0 0 | 2 1 0 2 1 0 2 0 0 0 0 0 | 1 1 2 1 0 0
... x.. ... | 2 0 0 | * 2 * * * * * * * | 1 0 2 0 0 2 0 0 0 0 0 0 0 | 2 0 1 2 0 1 0 2 0 0 0 0 | 1 1 2 0 1 0
oo.2oo.2oo.&#x | 1 1 0 | * * 8 * * * * * * | 0 1 1 1 0 0 0 1 0 0 0 0 0 | 1 1 1 0 0 0 1 1 1 0 0 0 | 1 0 1 1 1 0
o.o2o.o2o.o&#x | 1 0 1 | * * * 8 * * * * * | 0 0 0 0 1 1 1 1 0 0 0 0 0 | 0 0 0 1 1 1 1 1 1 0 0 0 | 0 1 1 1 1 0
.x. ... ... | 0 2 0 | * * * * 2 * * * * | 0 2 0 0 0 0 0 0 1 2 0 0 0 | 1 2 0 0 0 0 2 0 0 1 2 0 | 1 0 1 2 0 1
... ... .x. | 0 2 0 | * * * * * 2 * * * | 0 0 0 2 0 0 0 0 1 0 0 2 0 | 0 2 1 0 0 0 0 0 2 0 2 1 | 1 0 0 2 1 1
.oo2.oo2.oo&#x | 0 1 1 | * * * * * * 8 * * | 0 0 0 0 0 0 0 1 0 1 1 1 0 | 0 0 0 0 0 0 1 1 1 1 1 1 | 0 0 1 1 1 1
... ..x ... | 0 0 2 | * * * * * * * 2 * | 0 0 0 0 0 2 0 0 0 0 2 0 1 | 0 0 0 1 0 2 0 2 0 1 0 2 | 0 1 1 0 2 1
... ... ..x | 0 0 2 | * * * * * * * * 2 | 0 0 0 0 0 0 2 0 0 0 0 2 1 | 0 0 0 0 1 2 0 0 2 0 1 2 | 0 1 0 1 2 1
---------------+-------+-------------------+---------------------------+-------------------------+------------
x.. x.. ... | 4 0 0 | 2 2 0 0 0 0 0 0 0 | 1 * * * * * * * * * * * * | 2 0 0 2 0 0 0 0 0 0 0 0 | 1 1 2 0 0 0
xx. ... ...&#x | 2 2 0 | 1 0 2 0 1 0 0 0 0 | * 4 * * * * * * * * * * * | 1 1 0 0 0 0 1 0 0 0 0 0 | 1 0 1 1 0 0
... xo. ...&#x | 2 1 0 | 0 1 2 0 0 0 0 0 0 | * * 4 * * * * * * * * * * | 1 0 1 0 0 0 0 1 0 0 0 0 | 1 0 1 0 1 0
... ... ox.&#x | 1 2 0 | 0 0 2 0 0 1 0 0 0 | * * * 4 * * * * * * * * * | 0 1 1 0 0 0 0 0 1 0 0 0 | 1 0 0 1 1 0
x.o ... ...&#x | 2 0 1 | 1 0 0 2 0 0 0 0 0 | * * * * 4 * * * * * * * * | 0 0 0 1 1 0 1 0 0 0 0 0 | 0 1 1 1 0 0
... x.x ...&#x | 2 0 2 | 0 1 0 2 0 0 0 1 0 | * * * * * 4 * * * * * * * | 0 0 0 1 0 1 0 1 0 0 0 0 | 0 1 1 0 1 0
... ... o.x&#x | 1 0 2 | 0 0 0 2 0 0 0 0 1 | * * * * * * 4 * * * * * * | 0 0 0 0 1 1 0 0 1 0 0 0 | 0 1 0 1 1 0
ooo2ooo2ooo&#x | 1 1 1 | 0 0 1 1 0 0 1 0 0 | * * * * * * * 8 * * * * * | 0 0 0 0 0 0 1 1 1 0 0 0 | 0 0 1 1 1 0
.x. ... .x. | 0 4 0 | 0 0 0 0 2 2 0 0 0 | * * * * * * * * 1 * * * * | 0 2 0 0 0 0 0 0 0 0 2 0 | 1 0 0 2 0 1
.xo ... ...&#x | 0 2 1 | 0 0 0 0 1 0 2 0 0 | * * * * * * * * * 4 * * * | 0 0 0 0 0 0 1 0 0 1 1 0 | 0 0 1 1 0 1
... .ox ...&#x | 0 1 2 | 0 0 0 0 0 0 2 1 0 | * * * * * * * * * * 4 * * | 0 0 0 0 0 0 0 1 0 1 0 1 | 0 0 1 0 1 1
... ... .xx&#x | 0 2 2 | 0 0 0 0 0 1 2 0 1 | * * * * * * * * * * * 4 * | 0 0 0 0 0 0 0 0 1 0 1 1 | 0 0 0 1 1 1
... ..x ..x | 0 0 4 | 0 0 0 0 0 0 0 2 2 | * * * * * * * * * * * * 1 | 0 0 0 0 0 2 0 0 0 0 0 2 | 0 1 0 0 2 1
---------------+-------+-------------------+---------------------------+-------------------------+------------
xx. xo. ...&#x | 4 2 0 | 2 2 4 0 1 0 0 0 0 | 1 2 2 0 0 0 0 0 0 0 0 0 0 | 2 * * * * * * * * * * * | 1 0 1 0 0 0 trip
xx. ... ox.&#x | 2 4 0 | 1 0 4 0 2 2 0 0 0 | 0 2 0 2 0 0 0 0 1 0 0 0 0 | * 2 * * * * * * * * * * | 1 0 0 1 0 0 trip
... xo. ox.&#x | 2 2 0 | 0 1 4 0 0 1 0 0 0 | 0 0 2 2 0 0 0 0 0 0 0 0 0 | * * 2 * * * * * * * * * | 1 0 0 0 1 0 tet
x.o x.x ...&#x | 4 0 2 | 2 2 0 4 0 0 0 1 0 | 1 0 0 0 2 2 0 0 0 0 0 0 0 | * * * 2 * * * * * * * * | 0 1 1 0 0 0 trip
x.o ... o.x&#x | 2 0 2 | 1 0 0 4 0 0 0 0 1 | 0 0 0 0 2 0 2 0 0 0 0 0 0 | * * * * 2 * * * * * * * | 0 1 0 1 0 0 tet
... x.x o.x&#x | 2 0 4 | 0 1 0 4 0 0 0 2 2 | 0 0 0 0 0 2 2 0 0 0 0 0 1 | * * * * * 2 * * * * * * | 0 1 0 0 1 0 trip
xxo ... ...&#x | 2 2 1 | 1 0 2 2 1 0 2 0 0 | 0 1 0 0 1 0 0 2 0 1 0 0 0 | * * * * * * 4 * * * * * | 0 0 1 1 0 0 squippy
... xox ...&#x | 2 1 2 | 0 1 2 2 0 0 2 1 0 | 0 0 1 0 0 1 0 2 0 0 1 0 0 | * * * * * * * 4 * * * * | 0 0 1 0 1 0 squippy
... ... oxx&#x | 1 2 2 | 0 0 2 2 0 1 2 0 1 | 0 0 0 1 0 0 1 2 0 0 0 1 0 | * * * * * * * * 4 * * * | 0 0 0 1 1 0 squippy
.xo .ox ...&#x | 0 2 2 | 0 0 0 0 1 0 4 1 0 | 0 0 0 0 0 0 0 0 0 2 2 0 0 | * * * * * * * * * 2 * * | 0 0 1 0 0 1 tet
.xo ... .xx&#x | 0 4 2 | 0 0 0 0 2 2 4 0 1 | 0 0 0 0 0 0 0 0 1 2 0 2 0 | * * * * * * * * * * 2 * | 0 0 0 1 0 1 trip
... .ox .xx&#x | 0 2 4 | 0 0 0 0 0 1 4 2 2 | 0 0 0 0 0 0 0 0 0 0 2 2 1 | * * * * * * * * * * * 2 | 0 0 0 0 1 1 trip
---------------+-------+-------------------+---------------------------+-------------------------+------------
xx. xo. ox.&#x | 4 4 0 | 2 2 8 0 2 2 0 0 0 | 1 4 4 4 0 0 0 0 1 0 0 0 0 | 2 2 2 0 0 0 0 0 0 0 0 0 | 1 * * * * * tepe
x.o x.x o.x&#x | 4 0 4 | 2 2 0 8 0 0 0 2 2 | 1 0 0 0 4 4 4 0 0 0 0 0 1 | 0 0 0 2 2 2 0 0 0 0 0 0 | * 1 * * * * tepe
xxo xox ...&#x | 4 2 2 | 2 2 4 4 1 0 4 1 0 | 1 2 2 0 2 2 0 4 0 2 2 0 0 | 1 0 0 1 0 0 2 2 0 1 0 0 | * * 2 * * * {4} || tet
xxo ... oxx&#x | 2 4 2 | 1 0 4 4 2 2 4 0 1 | 0 2 0 2 2 0 2 4 1 2 0 2 0 | 0 1 0 0 1 0 2 0 2 0 1 0 | * * * 2 * * {4} || tet
... xox oxx&#x | 2 2 4 | 0 1 4 4 0 1 4 2 2 | 0 0 2 2 0 2 2 4 0 0 2 2 1 | 0 0 1 0 0 1 0 2 2 0 0 1 | * * * * 2 * {4} || tet
.xo .ox .xx&#x | 0 4 4 | 0 0 0 0 2 2 8 2 2 | 0 0 0 0 0 0 0 0 1 4 4 4 1 | 0 0 0 0 0 0 0 0 0 2 2 2 | * * * * * 1 tepe
Anyway, i decided that 'radiant space' leads to more dead ends than it solves, and come down to trying to hack Coxeter's exotic CD diagrams. The most reliable source of these is 'regular complex polytopes', which gives explicit examples. ...
... After many attempts (including reading x3o4x3o4z as a tiling of rhombic dodecahedra!), ...
x3o4x3o4*a (N → ∞)
. . . . | 12N | 4 4 | 2 4 2 2 2 | 2 1 2 1
-----------+-----+---------+-----------------+--------
x . . . | 2 | 24N * | 1 1 1 0 0 | 1 1 1 0
. . x . | 2 | * 24N | 0 1 0 1 1 | 1 0 1 1
-----------+-----+---------+-----------------+--------
x3o . . | 3 | 3 0 | 8N * * * * | 1 1 0 0
x . x . | 4 | 2 2 | * 12N * * * | 1 0 1 0
x . . o4*a | 4 | 4 0 | * * 6N * * | 0 1 1 0
. o4x . | 4 | 0 4 | * * * 6N * | 1 0 0 1
. . x3o | 3 | 0 3 | * * * * 8N | 0 0 1 1
-----------+-----+---------+-----------------+--------
x3o4x . | 24 | 24 24 | 8 12 0 6 0 | N * * * sirco
x3o . o4*a | 12 | 24 0 | 8 0 6 0 0 | * N * * co
x . x3o4*a | 24 | 24 24 | 0 12 6 0 8 | * * N * sirco
. o4x3o | 12 | 0 24 | 0 0 0 6 8 | * * * N co
... I introduced a thing called the vertex-node.
pt 8 3 3(3)3 d0 = 1:60 = 3/2
3 3 8 G=24 d1 = 0:60 = 1/2
pt 27 8 8 Hess Polyhedron d0 = 2
3 3 72 3 d1 = 1
3(3)3 8 8 27 G=548 = dec 648 d2 = 0:60 = 1/2
pt 200 27 72 27 Whytting Polychoron d0 = 2:80 = 8/3
3 3 1800 8 8 200 = dec 240 d1 = 1:80 = 5/3
3(3)3 8 8 1800 3 1800 = dec 2160 d2 = 1:20 = 7/6
3(3)3(3)3 27 72 27 200 G=10.9600 = dec 155520 d3 = 0:80 = 2/3.
pt (1) 200 1800 1800 200 Tiling of WP
3 3 (80) 27 72 27 230 = dec 270
3(3)3 8 8 (230) 8 8
3(3)3(3)3 27 72 27 (80) 3
Wytting 200 1800 1800 200 (1)
quickfur wrote:Yesterday I finally implemented a function in polyview to compute dichoral angles, and I was going through the bicupolic rings pages on the wiki to add dichoral angles to them, when I noticed something interesting: the n-gonal magnabicupolic rings are basically Stott expansions of the n-gonal prism pyramids, ...
x-n-o o-n-o
= Stott exp. of
x-n-x x-n-x o-n-x o-n-x
... and the n-gonal orthobicupolic rings are basically Stott expansions of the n-gonal pyramid pyramids. All corresponding dichoral angles are the same, including the segmentochoron heights.
x-n-x o-n-x
= Stott exp. of
x-n-o x-n-o o-n-o o-n-o
Klitzing wrote:[...]Both immediate and known.
--- rk
quickfur wrote:Klitzing wrote:[...]Both immediate and known.
--- rk
Yes. But I was more interested in the gyrobicupolic ring case. Does a Stott-contracted version exist? and if so, what would it be?
x-n-x
x-n-o o-n-x
o-n-x
o-n-o (-x)-n-x
o-n-o
o-n-(-x) (-x)-n-o
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