student91 wrote:In the D4.11 & D4.12 topic Klitzing and I were discussing whether or not we would introduce a new lace term U.
The Idea for this term came when I tried to represent D4.11 in it's demitesseractic symmetry. D4.11 can indeed be represented as such. It would then be the "union" of f3x3o *b3o, x3o3f *b3x and o3o3x *b3F. This "union" is derived as the convex hull of these three things when placed around the same origin.
I at first thought this was a degenerate lace simplex, but Klitzing pointed out it was not. He said it was a degenerate lace tower, which is indeed much more accurate.
Nevertheless I think this "union" might need a new lace term, because it's degenerate.
a lace tower can be seen as several polytopes made of the same dynkin graph, with their origins distributed over one dimension. The coordinates can be given as ([dynkin polytope], height)
a lace city can be seen as several polytopes with the same dynkin graph, with their origins distributed over two dimensions. Hence, their coordinates can be given as ([dynkin polytope], [x-position], [y-poisition])
Now the new "union" operator would take several polytopes with the same dynkin graph, and distribute their origins over zero dimensions, i.e. don't distribute it at all. In fact, it is pretty similar to the compound operator, with the additional step that you take the convex hull of the resulting vertices.
When we leave it's definition as such, it would be only usefull for convex polytopes. We could also define it as [take all the vertices of the dynkin polytopes with the same origin and the same orientation] and then [connect all vertices with an x-edge, if possible].
The first definition would give only convex polytopes, thus is very suited for the CRF project, but not for much else.
The second definition can give any polytope, but you have to be really sure that you have everything right.
A notation we could introduce can be &U, for union according to definition 1, or &U(x), for union according to definition two with x-edges. I like the U, because it looks a bit similar to the Union sign of set theory.
If you want it to be defined a bit differently, or have a suggestion for a different notation, or just read the topic, please let me know what you think.
o3x x3x x3o
x3x u3x x3u x3x
x3o x3u oH3Ho u3x o3x
x3x u3x x3u x3x
o3x x3x x3o
wendy wrote:Similar kinds of constructs exist to make the 2_21, the 27-vertex-polypeton.
Indeed the word union does not accurately describe the process I described. I chose this word because of my shallow vocabulary, so if you have a better word for this process, I will eagerly use it. (of course, if we're going to introduce it.)wendy wrote:I'm not really overfussed with the idea of 'unions', since the figure is probably not a union (a union is more like a compound than a lacing), and the whole point of the lace-notation is that the edge length already gives enough information to construct everything.
When they're so common, how do you describe them? do you describe them all as either lace-simplices or lace-towers, or are there some exceptions to this? if so, how are these exceptios described?[...]flat lace-prisms are supprisingly common. Often one pulls these things out of lattices, so one has the compound formed by, eg
I'm very unfamiliar with hyperbolic geometry, so I don't know if my new lace term would ruin those properties of flat lace-prisms./6/ + 3/3, a polytope having 126 vertices, being 70+56, according to the decimal count. This is a lace compound, that might be written as xo3oo3oo3ox3oo3oo3xo&#x. It is, of course, no other than the 2_31, or 5/B.
Many lace cities can be wrapped into flat lace prisms. For example, one has x5x2o5o + fo5of2ox5xo + xo5ox2fo5of + o5o2f5f, which is a presentation of x3o3o5o, against an equatorial girth. You see much more when this is rolled out into a lace-city, since you can see the various sections of the rings it is made of.
The thing is that even something like oo3ox4oo&xt, while it be flat in 3d, comes out as a proper pyramid in hyperbolic 4space, so it's probably no different to ox3oo5oo&x or oo3oo5ox&x, which are also valid pyramids.
1 0 0 0 10.472136
0 1 0 0 37.885438 *
0 0 0 1 54.832815 a 0 1 0 -1
0 0 1 0 82.249223 b 0 0 1 -1
1 1 0 0 86.249223 c 1 1 -1 0
1 0 0 1 109.665631 a
1 0 1 0 147.554175 b
0 1 0 1 181.442719 d 1 -1 1 -1
0 1 1 0 229.803398 b
0 0 1 1 270.164078 a
1 1 0 1 274.164078 c
1 1 1 0 332.996894 b
1 0 1 1 379.829710 a
0 1 1 1 506.439634
1 1 1 1 653.993788
student91 wrote:When they're so common, how do you describe them?
student91 wrote:I'm very unfamiliar with hyperbolic geometry, so I don't know if my new lace term would ruin those properties of flat lace-prisms.
student91 wrote:Anyhow, I still think the lace-tower representation of bilbiro is, although a lot more accurate,
Klitzing wrote:would you mind to elaborate on that one?
I.e. on a degenerate (= 0 height) lace tower or lace simplex description of "jak"?
student91 wrote:a lace city can be seen as several polytopes with the same dynkin graph, with their origins distributed over two dimensions. Hence, their coordinates can be given as ([dynkin polytope], [x-position], [y-poisition])
wendy wrote:Klitzing wrote:would you mind to elaborate on that one?
I.e. on a degenerate (= 0 height) lace tower or lace simplex description of "jak"?
You can make this figure from, eg xoo3oxo2oxo3oox2oox3xoo&#xt. It's kind of like the cuboctahedron, if you look at it in three hedrices. For example, if x3o is 'w' then -w would be o3x, and the jak comes at (w,-w,0) EP3C (where 3C rotates the triangles by 120 degrees in the plane, a complex version of 'change sign'.
4B1 against 6:.
/4B=2_21 4B/ = 1_22
, C , A , B 27 C 1 4 = 1
A , B , C , 12 . * . C 3+9 2/2 = 20
, C , A , B 3 C 12 /4/ = 30
A , B , C , 0 3+9 2/2 = 20
12 5 8 9 8 5 - /4B cell. 1 4 = 1
/6C, against 7:
5B/ 5/B
50 o x o o o o o 5/ 7
32 o o o o o x o /5 7 2/3 35
18 o o x o c o o 4/1 21 /5/ 42
8 o o o o c o x 1/4 21 3/2 35
2 o o o x o o o 5/ 7 /5 7
0 x o o o o o o
14 6 10 12 12 10 6 28 1.06
Hurraywendy wrote:I have decided that there is room to create student91's request,
I understand that, I hadn't expected my new term to be described perfectly the first time I would try. In fact I made this topic so we can help each other perfecting the new termalthough at a slightly different point. Don't worry, i did this to r.klitzing's notation too, because there are 'global' interests to protect.
I have to admit I don't fully understand what a thatching exactly is as far as I get it, an "edge-thatching" makes the most-convex hull of a set of edges, and a vertex-thatching makes the convex hull of a set of vertices. In that respect, vertex-thatching is exactly the same as taking the convex hull, if I'm right.[...]
'z' is the appropriate symmetry for a point, since something like 'o' implies that it's a zero-length line (but lives in a 1d space). z is a node that here loops onto itself, and destroys any symmetry. In any case &#z can be read as 'zero-height/dimension' (added here).
Thatching
Ever since Hedrondude rolled out the regiments, i have been trying to find a term to make a non-convex hull. For example, the regiment (figures sharing vertices + edges) that contains {5/2,5} and {3,5/2}, the first of these is the 'most convex' of these. But you can't call it convex, and it's hardly a hull, so i decided instead that it needs a new word. That word is 'thatching', here 'edge-thatching', since the edges are kept proud. Vertex-thatching is the default meaning of thatch, so it does make a convex hull.
For the present case, you can use something like &#z as a suffix, since the edges will come from the stott-differences. If you want to further imply an edge, you can add that after this, ie &#zx.
Lace Union
I suppose we could keep this term for a thatched compound, especially if it is expressed as a laced layer. I'm all ears, though.
Of course, but I just thought it to be superfluous to describe something that is more basic than a lace-tower as a lace-tower.Lace projections
[I guess this is only usefull for projection purposes?]
Lace towers
It is still appropriate to designate a series of layers from a tiling, as a lace tower, since we have ample tools now to determine the height.
You're welcome[...]
Thank you for your help.
Wendy
That's why I made this topic, rather that just introducing a new lace-term out of the blueKlitzing wrote:@student91:
You surely are free to start from existing concepts and build thereon your owns. We generally cannot hinder that. But you should be careful in doing so. Because we don't want any 2nd Babylonian confusion. You should always verify whether your ideas are unique and indeed new, and that those cannot be represented by the so far existing system.
What I was referring to with "angle" and "structure" was unclear. You could see the bilbiro-thing as a lace-city, with a lacing of x between two things, and a "lacing" of f as third connection. All these distances would become 0, so the things become thrown on a point. So the bilbiro-thing could be seen as both a deflated lace-tower as a deflated lace-city. In a lace-city, you can have an angle beween three things. (so there can be an angle between the lacings themselves, rather than between the egdes. It was this angle I was talking about)[...]Just as the stacking of lace prisms already tells, the lacing edges of either segment then can have any arbitrary angle between them. They definitely are not bound to align. It is just the axis, which is assumed to be continued. In fact any lace prism has 2 parallel bases, each being perpendicular to the axis. Accordingly the bases and those intermediate pseudo facets of a tower too will be all perpendicular to the common axis, or, taken differently, are mutually parallel. - This latter wording also then will hold for degenerate towers too. If the segment heights would calculate as zero you clearly have no true axis. But still a base (or sectioning pseudo facet) normal. And therefore, in that special case, parallelity of bases (or sectioning pseudo facets) just reads as coplanarity (of corresponding dimension for sure).
Ah, okay, so lace towers can be much more complex than just a stack of lace-prisms. I can look at D4.10, and see the lace-tower is much more complex indeed. maybe, to clarify this complexity, we might introduce a "lacing matrix", which would show what things are connected to what. The "lacing matrix" of the bilbiro-compound would the be:But there is also a different feature in real lace towers. It is that not all neighbouring pairs of bases and/or pseudo facets ought to be conected directly by lacings of the assigned length. If the respective segment heights become small enough, there might be edges connecting one vertex layer e.g. to the one but next. And such cases might even intertwine, resulting in cases where neighbouring layers are not being laced at all, each just being connected to other ones. - Such an behaviour for sure becomes more frequent, the lower the respective heights would be. So it is not at all surprising, when you consider degenerate towers, i.e. zero heights.
f2o2o | * | x | f
x2x2f | x | * | x
o2F2x | f | x | *
I saw it as if there is a lacing between f2o2o and o2F2x, but this lacing has length f. This f-edge becomes part of bilbiro's pentagons. I hope that's not too creative use of lacings?[...]
@Wendy:
I first thought about that new suffix "&zx(t)" being nothing but an optional clarification that all according heights there would evaluate as zero. Thus more the kind of being just reader-friendly.
But then I realized that bit mentioned above. Those there mentioned new degenerate structures definitely cannot be described as lace simplices, as not all lacings shall occur. Thus those need to be represented as lace towers at most. But then the 2 base facets ought to be real facets, if it would be a true lace tower, even when having some zero segment heights. But those examples use only pseudo facet layer descriptions!
Thus bilbiro only could be represented (in that advised symmetry representation) as fxo2oxF2ofx&#zxt, where this additional "z" then not only just point out that the heights happen to become zero, but also that all layers in fact are just pseudo facets.
Note that the final "t" still is needed here. Even so the first and third layer are corealmic too, and therefore shall have a height of zero, they are not to be laced. And thus the neglection of that "t" would be wrong.
student91 wrote:As far as I get it, the postfix &#zx means: &# there is a lacing, i.e. things get connected, rather than they are forming a compound, then z, which I think means there won't be a higher-dimensional tower nor a higher-dimensional simplex, but everything stays in the same dimension, and then x, which I'm not totaly sure of, but I guess that means the edge-length is x. Then the question arises, what edge-length, is it the edge-length described by definition [2]?
What I was referring to with "angle" and "structure" was unclear. You could see the bilbiro-thing as a lace-city, with a lacing of x between two things, and a "lacing" of f as third connection. All these distances would become 0, so the things become thrown on a point.
So the bilbiro-thing could be seen as both a deflated lace-tower as a deflated lace-city. In a lace-city, you can have an angle beween three things. (so there can be an angle between the lacings themselves, rather than between the egdes. It was this angle I was talking about)
Ah, okay, so lace towers can be much more complex than just a stack of lace-prisms. I can look at D4.10, and see the lace-tower is much more complex indeed. maybe, to clarify this complexity, we might introduce a "lacing matrix", which would show what things are connected to what. The "lacing matrix" of the bilbiro-compound would the be:
- Code: Select all
f2o2o | * | x | f
x2x2f | x | * | x
o2F2x | f | x | *
You can see this matrix is symmetrical. for this particular case it's not that usefull, but for e.g. D4.10, quickfur could just give us such a matrix, instead of the ACSII-art. This is a tentative thing, though.
Klitzing wrote:Note that the final "t" still is needed here. Even so the first and third layer are corealmic too, and therefore shall have a height of zero, they are not to be laced. And thus the neglection of that "t" would be wrong.
I saw it as if there is a lacing between f2o2o and o2F2x, but this lacing has length f. This f-edge becomes part of bilbiro's pentagons. I hope that's not too creative use of lacings?
As far as I got it now, &#z is something pretty similar to what I wanted, and I can use this for what I need. However, I would still like some explanation on what &#z exactly does, esp. how it differs from &#zx.
wendy wrote:Lace towers
It is still appropriate to designate a series of layers from a tiling, as a lace tower, since we have ample tools now to determine the height. These are to be read as 0=o, 1=x, against o3o3o5o. So the inner shell (marked *), is o3x3o5o. From this to the x3o3x5x, there is a flat lace segmentochoron in &#xt. So the whole 'cage' comes up as a dozen polychora, laced together by unit lacings, but it's still flat. If you replaced it bt &#ft, it would stand upright as a kind of pyramid.
The letters 'a' to 'd' here are differences between layers, is used to verify that thing is flat.
- Code: Select all
1 0 0 0 10.472136
0 1 0 0 37.885438 *
0 0 0 1 54.832815 a 0 1 0 -1
0 0 1 0 82.249223 b 0 0 1 -1
1 1 0 0 86.249223 c 1 1 -1 0
1 0 0 1 109.665631 a
1 0 1 0 147.554175 b
0 1 0 1 181.442719 d 1 -1 1 -1
0 1 1 0 229.803398 b
0 0 1 1 270.164078 a
1 1 0 1 274.164078 c
1 1 1 0 332.996894 b
1 0 1 1 379.829710 a
0 1 1 1 506.439634
1 1 1 1 653.993788
hi || rahi = oo3oo3ox5xo&#x
srahi || xhi = oo3xx3ox5xo&#x
sidpixhi || srix = xx3oo3ox5xo&#x
prahi || grix = xx3xx3ox5xo&#x
rox || hi = oo3xo3oo5ox&#x
xhi || thi = oo3xo3xx5ox&#x
tex || sidpixhi = xx3xo3oo5ox&#x
grix || prix = xx3xo3xx5ox&#x
rahi || sidpixhi = ox3oo3xo5ox&#x
xhi || prahi = ox3xx3xo5ox&#x
rahi || tex = ox3ox3xo5oo&#x
thi || prahi = ox3ox3xo5xx&#x
srix || srahi = xo3ox3xo5ox&#x
xo3xx3oo3oA&#zx
o.3o.3o.3o. | 20 * | 1 3 3 0 | 3 3 3 6 0 0 | 1 6 3 0
.o3.o3.o3.o | * 30 | 0 0 2 4 | 0 0 1 4 2 2 | 0 2 2 1
---------------+-------+-------------+-------------------+----------
x. .. .. .. | 2 0 | 10 * * * | 3 0 3 0 0 0 | 0 6 0 0
.. x. .. .. | 2 0 | * 30 * * | 1 2 0 2 0 0 | 1 2 2 0
oo3oo3oo3oo&#x | 1 1 | * * 60 * | 0 0 1 2 0 0 | 0 2 1 0
.. .x .. .. | 0 2 | * * * 60 | 0 0 0 1 1 1 | 0 1 1 1
---------------+-------+-------------+-------------------+----------
x.3x. .. .. | 6 0 | 3 3 0 0 | 10 * * * * * | 0 2 0 0
.. x.3o. .. | 3 0 | 0 3 0 0 | * 20 * * * * | 1 0 1 0
xo .. .. ..&#x | 2 1 | 1 0 2 0 | * * 30 * * * | 0 2 0 0
.. xx .. ..&#x | 2 2 | 0 1 2 1 | * * * 60 * * | 0 1 1 0
.o3.x .. .. | 0 3 | 0 0 0 3 | * * * * 20 * | 0 1 0 1
.. .x3.o .. | 0 3 | 0 0 0 3 | * * * * * 20 | 0 0 1 1
---------------+-------+-------------+-------------------+----------
.. x.3o.3o. | 4 0 | 0 6 0 0 | 0 4 0 0 0 0 | 5 * * * tet
xo3xx .. ..&#x | 6 3 | 3 3 6 3 | 1 0 3 3 1 0 | * 20 * * tricu
.. xx3oo ..&#x | 3 3 | 0 3 3 3 | 0 1 0 3 0 1 | * * 20 * trip
.o3.x3.o .. | 0 6 | 0 0 0 12 | 0 0 0 0 4 4 | * * * 5 oct
wendy wrote:Richard asks about the facets of 2_31 = 5/B, when this is presented as /6/ + 3/3. ie xo3oo3oo3ox3oo3oo3xo&#xz.
xo3oo3oo3ox3oo3oo3xo&#x → height = 0
(pseudo suph || pseudo he)
o.3o.3o.3o.3o.3o.3o. | 56 * | 12 20 0 | 30 120 90 0 | 40 120 360 120 0 | 30 40 180 120 240 30 0 | 12 30 120 30 60 | 2 12 30
.o3.o3.o3.o3.o3.o3.o | * 70 | 0 16 16 | 0 48 144 48 | 0 32 288 288 32 | 0 8 96 144 288 96 8 | 0 24 96 36 96 | 0 12 32
---------------------------+-------+--------------+--------------------+-------------------------+---------------------------------+-----------------------+----------
x. .. .. .. .. .. .. & | 2 0 | 336 * * | 5 10 0 0 | 10 20 30 0 0 | 10 10 30 10 20 0 0 | 5 10 20 5 5 | 1 6 5
oo3oo3oo3oo3oo3oo3oo&#x | 1 1 | * 1120 * | 0 6 9 0 | 0 6 36 18 0 | 0 2 18 18 36 6 0 | 0 6 18 9 12 | 0 6 6
.. .. .. .x .. .. .. | 0 2 | * * 560 | 0 0 9 6 | 0 0 18 36 6 | 0 0 6 18 36 18 2 | 0 6 12 9 18 | 0 6 6
---------------------------+-------+--------------+--------------------+-------------------------+---------------------------------+-----------------------+----------
x.3o. .. .. .. .. .. & | 3 0 | 3 0 0 | 560 * * * | 4 4 0 0 0 | 6 4 6 0 0 0 0 | 4 6 4 0 0 | 1 4 1
xo .. .. .. .. .. ..&#x & | 2 1 | 1 2 0 | * 3360 * * | 0 2 6 0 0 | 0 1 6 3 6 0 0 | 0 3 6 3 2 | 0 4 2
.. .. .. ox .. .. ..&#x | 1 2 | 0 2 1 | * * 5040 * | 0 0 4 4 0 | 0 0 2 4 8 2 0 | 0 2 4 4 4 | 0 4 2
.. .. .o3.x .. .. .. & | 0 3 | 0 0 3 | * * * 1120 | 0 0 0 6 2 | 0 0 0 3 6 6 1 | 0 3 2 3 6 | 0 4 2
---------------------------+-------+--------------+--------------------+-------------------------+---------------------------------+-----------------------+----------
x.3o.3o. .. .. .. .. & | 4 0 | 6 0 0 | 4 0 0 0 | 560 * * * * | 3 1 0 0 0 0 0 | 3 3 0 0 0 | 1 3 0
xo3oo .. .. .. .. ..&#x & | 3 1 | 3 3 0 | 1 3 0 0 | * 2240 * * * | 0 1 3 0 0 0 0 | 0 3 3 0 0 | 0 3 1
xo .. .. ox .. .. ..&#x & | 2 2 | 1 4 1 | 0 2 2 0 | * * 10080 * * | 0 0 1 1 2 0 0 | 0 1 2 2 1 | 0 3 1
.. .. oo3ox .. .. ..&#x & | 1 3 | 0 3 3 | 0 0 3 1 | * * * 6720 * | 0 0 0 1 2 1 0 | 0 1 1 2 2 | 0 3 1
.. .o3.o3.x .. .. .. & | 0 4 | 0 0 6 | 0 0 0 4 | * * * * 560 | 0 0 0 0 0 3 1 | 0 3 0 0 3 | 0 3 1
---------------------------+-------+--------------+--------------------+-------------------------+---------------------------------+-----------------------+----------
x.3o.3o.3o. .. .. .. & | 5 0 | 10 0 0 | 10 0 0 0 | 5 0 0 0 0 | 336 * * * * * * | 2 1 0 0 0 | 1 2 0
xo3oo3oo .. .. .. ..&#x & | 4 1 | 6 4 0 | 4 6 0 0 | 1 4 0 0 0 | * 560 * * * * * | 0 3 0 0 0 | 0 3 0
xo3oo .. ox .. .. ..&#x & | 3 2 | 3 6 1 | 1 6 3 0 | 0 2 3 0 0 | * * 3360 * * * * | 0 1 2 0 0 | 0 2 1
xo .. oo3ox .. .. ..&#x & | 2 3 | 1 6 3 | 0 3 6 1 | 0 0 3 2 0 | * * * 3360 * * * | 0 1 0 2 0 | 0 3 0
xo .. .. ox3oo .. ..&#x & | 2 3 | 1 6 3 | 0 3 6 1 | 0 0 3 2 0 | * * * * 6720 * * | 0 0 1 1 1 | 0 2 1
.. oo3oo3ox .. .. ..&#x & | 1 4 | 0 4 6 | 0 0 6 4 | 0 0 0 4 1 | * * * * * 1680 * | 0 1 0 0 2 | 0 2 1
.o3.o3.o3.x .. .. .. & | 0 5 | 0 0 10 | 0 0 0 10 | 0 0 0 0 5 | * * * * * * 112 | 0 3 0 0 0 | 0 3 0
---------------------------+-------+--------------+--------------------+-------------------------+---------------------------------+-----------------------+----------
x.3o.3o.3o.3o. .. .. & | 6 0 | 15 0 0 | 20 0 0 0 | 15 0 0 0 0 | 6 0 0 0 0 0 0 | 112 * * * * | 1 1 0 hix
xo3oo3oo3ox .. .. ..&#x & | 5 5 | 10 20 10 | 10 30 30 10 | 5 20 30 20 5 | 1 5 10 10 0 5 1 | * 336 * * * | 0 2 0 tac
xo3oo .. ox3oo .. ..&#x & | 3 3 | 3 9 3 | 1 9 9 1 | 0 3 9 3 0 | 0 0 3 0 3 0 0 | * * 2240 * * | 0 1 1 hix
xo .. oo3ox3oo .. xo&#zx | 4 6 | 4 24 12 | 0 24 48 8 | 0 0 48 32 0 | 0 0 0 16 16 0 0 | * * * 420 * | 0 2 0 tac
xo .. .. ox3oo3oo ..&#x & | 2 4 | 1 8 6 | 0 4 12 4 | 0 0 6 8 1 | 0 0 0 0 4 2 0 | * * * * 1680 | 0 1 1 hix
---------------------------+-------+--------------+--------------------+-------------------------+---------------------------------+-----------------------+----------
x.3o.3o.3o.3o.3o. .. & | 7 0 | 21 0 0 | 35 0 0 0 | 35 0 0 0 0 | 21 0 0 0 0 0 0 | 7 0 0 0 0 | 16 * * hop
xo3oo3oo3ox3oo .. xo&#zx & | 12 15 | 36 120 60 | 40 240 360 80 | 30 120 540 360 30 | 12 30 120 180 240 60 6 | 2 12 40 15 30 | * 56 * jak
xo3oo .. ox3oo3oo ..&#x & | 3 4 | 3 12 6 | 1 12 18 4 | 0 4 18 12 1 | 0 0 6 0 12 3 0 | 0 0 4 0 3 | * * 560 hop
wendy wrote:... The lace tower 4/ || 1/3 || 4/ or ooo3oxo3ooo3ooo3xox&#xt makes the 2_21. There are 28 on each tetrahedra, and because the 2_21 /4B has 72 simplexs in 36 opposite pairs, ...
xox3ooo3ooo3oxo3ooo&#xt → both heights = 1/2 = 0.5
(hix || inv pseudo rix || hix)
o..3o..3o..3o..3o.. & | 12 * | 5 10 1 0 | 10 30 30 10 0 0 | 10 30 60 20 10 30 0 | 5 10 30 20 20 5 20 10 0 | 1 5 10 5 5
.o.3.o.3.o.3.o.3.o. | * 15 | 0 8 0 8 | 0 12 48 4 12 4 | 0 8 48 48 24 24 8 | 0 2 16 24 24 16 24 12 2 | 0 4 8 6 8
--------------------------+-------+-------------+---------------------+---------------------------+-------------------------------+--------------
x.. ... ... ... ... & | 2 0 | 30 * * * | 4 6 0 0 0 0 | 6 12 12 0 0 0 0 | 4 6 12 4 4 0 0 0 0 | 1 4 4 1 0
oo.3oo.3oo.3oo.3oo.&#x & | 1 1 | * 120 * * | 0 3 6 1 0 0 | 0 3 12 6 3 6 0 | 0 1 6 6 6 2 6 3 0 | 0 2 3 3 2
o.o3o.o3o.o3o.o3o.o&#x | 2 0 | * * 6 * | 0 0 0 10 0 0 | 0 0 0 0 0 30 0 | 0 0 0 0 0 0 20 10 0 | 0 0 0 5 5
... ... ... .x. ... | 0 2 | * * * 60 | 0 0 6 0 3 1 | 0 0 6 12 6 3 3 | 0 0 2 6 6 6 6 3 1 | 0 2 2 3 3
--------------------------+-------+-------------+---------------------+---------------------------+-------------------------------+--------------
x..3o.. ... ... ... & | 3 0 | 3 0 0 0 | 40 * * * * * | 3 3 0 0 0 0 0 | 3 3 3 0 0 0 0 0 0 | 1 3 1 0 0
xo. ... ... ... ...&#x & | 2 1 | 1 2 0 0 | * 180 * * * * | 0 2 4 0 0 0 0 | 0 1 4 2 2 0 0 0 0 | 0 2 2 1 0
... ... ... ox. ...&#x & | 1 2 | 0 2 0 1 | * * 360 * * * | 0 0 2 2 1 1 0 | 0 0 1 2 2 1 2 1 0 | 0 1 1 2 1
ooo3ooo3ooo3ooo3ooo&#xt | 2 1 | 0 2 1 0 | * * * 60 * * | 0 0 0 0 0 6 0 | 0 0 0 0 0 0 6 3 0 | 0 0 0 3 2
... ... .o.3.x. ... | 0 3 | 0 0 0 3 | * * * * 60 * | 0 0 0 4 0 0 2 | 0 0 0 2 0 4 2 0 1 | 0 2 0 1 2
... ... ... .x.3.o. | 0 3 | 0 0 0 3 | * * * * * 20 | 0 0 0 0 6 0 0 | 0 0 0 0 6 0 0 3 0 | 0 0 2 3 0
--------------------------+-------+-------------+---------------------+---------------------------+-------------------------------+--------------
x..3o..3o.. ... ... & | 4 0 | 6 0 0 0 | 4 0 0 0 0 0 | 30 * * * * * * | 2 1 0 0 0 0 0 0 0 | 1 2 0 0 0
xo.3oo. ... ... ...&#x & | 3 1 | 3 3 0 0 | 1 3 0 0 0 0 | * 120 * * * * * | 0 1 2 0 0 0 0 0 0 | 0 2 1 0 0
xo. ... ... ox. ...&#x & | 2 2 | 1 4 0 1 | 0 2 2 0 0 0 | * * 360 * * * * | 0 0 1 1 1 0 0 0 0 | 0 1 1 1 0
... ... oo.3ox. ...&#x & | 1 3 | 0 3 0 3 | 0 0 3 0 1 0 | * * * 240 * * * | 0 0 0 1 0 1 1 0 0 | 0 1 0 1 1
... ... ... ox.3oo.&#x & | 1 3 | 0 3 0 3 | 0 0 3 0 0 1 | * * * * 120 * * | 0 0 0 0 2 0 0 1 0 | 0 0 1 2 0
... ... ... oxo ...&#x | 2 2 | 0 4 1 1 | 0 0 2 2 0 0 | * * * * * 180 * | 0 0 0 0 0 0 2 1 0 | 0 0 0 2 1
... .o.3.o.3.x. ... | 0 4 | 0 0 0 6 | 0 0 0 0 4 0 | * * * * * * 30 | 0 0 0 0 0 2 0 0 1 | 0 2 0 0 1
--------------------------+-------+-------------+---------------------+---------------------------+-------------------------------+--------------
x..3o..3o..3o.. ... & | 5 0 | 10 0 0 0 | 10 0 0 0 0 0 | 5 0 0 0 0 0 0 | 12 * * * * * * * * | 1 1 0 0 0
xo.3oo.3oo. ... ...&#x & | 4 1 | 6 4 0 0 | 4 6 0 0 0 0 | 1 4 0 0 0 0 0 | * 30 * * * * * * * | 0 2 0 0 0
xo.3oo. ... ox. ...&#x & | 3 2 | 3 6 0 1 | 1 6 3 0 0 0 | 0 2 3 0 0 0 0 | * * 120 * * * * * * | 0 1 1 0 0
xo. ... oo.3ox. ...&#x & | 2 3 | 1 6 0 3 | 0 3 6 0 1 0 | 0 0 3 2 0 0 0 | * * * 120 * * * * * | 0 1 0 1 0
xo. ... ... ox.3oo.&#x & | 2 3 | 1 6 0 3 | 0 3 6 0 0 1 | 0 0 3 0 2 0 0 | * * * * 120 * * * * | 0 0 1 1 0
... oo.3oo.3ox. ...&#x & | 1 4 | 0 4 0 6 | 0 0 6 0 4 0 | 0 0 0 4 0 0 1 | * * * * * 60 * * * | 0 1 0 0 1
... ... ooo3oxo ...&#x | 2 3 | 0 6 1 3 | 0 0 6 3 1 0 | 0 0 0 2 0 3 0 | * * * * * * 120 * * | 0 0 0 1 1
... ... ... oxo3ooo&#x | 2 3 | 0 6 1 3 | 0 0 6 3 0 1 | 0 0 0 0 2 3 0 | * * * * * * * 60 * | 0 0 0 2 0
.o.3.o.3.o.3.x. ... | 0 5 | 0 0 0 10 | 0 0 0 0 10 0 | 0 0 0 0 0 0 5 | * * * * * * * * 6 | 0 2 0 0 0
--------------------------+-------+-------------+---------------------+---------------------------+-------------------------------+--------------
x..3o..3o..3o..3o.. & | 6 0 | 15 0 0 0 | 20 0 0 0 0 0 | 15 0 0 0 0 0 0 | 6 0 0 0 0 0 0 0 0 | 2 * * * * hix
xo.3oo.3oo.3ox. ...&#x & | 5 5 | 10 20 0 10 | 10 30 30 0 10 0 | 5 20 30 20 0 0 5 | 1 5 10 10 0 5 0 0 1 | * 12 * * * tac
xo.3oo. ... ox.3oo.&#x & | 3 3 | 3 9 0 3 | 1 9 9 0 0 1 | 0 3 9 0 3 0 0 | 0 0 3 0 3 0 0 0 0 | * * 40 * * hix
xox ... ooo3oxo3ooo&#xt | 4 6 | 2 24 2 12 | 0 12 48 12 4 4 | 0 0 24 16 16 24 0 | 0 0 0 8 8 0 8 8 0 | * * * 15 * tac
... ooo3ooo3oxo ...&#x | 2 4 | 0 8 1 6 | 0 0 12 4 4 0 | 0 0 0 8 0 6 1 | 0 0 0 0 0 2 4 0 0 | * * * * 30 hix
wendy wrote:The real question now is how the group of 36 diameters of 4B/ works.
o -- o3o3o3o3o (point)
r R -- o3o3x3o3o (dot)
P S p -- x3o3o3o3x (scad)
r R -- o3o3x3o3o (dot)
o -- o3o3o3o3o (point)
\ \ \
\ \ +- x3o3o *b3o3o (hin)
\ +------- o3o3o *b3x3o (rat)
+------------- o3o3x *b3o3o (alt. hin)
where:
o = o3o3o3o (point)
r = o3x3o3o (rap)
R = o3o3x3o (inv. rap)
p = x3o3o3o (pen)
P = o3o3o3x (dual pen)
S = x3o3o3x (spid)
o
o A V o o = o3o2o3o point
V + A A = o3x2o3x bitriangle prism,
V x3o2x3o bitriangle prism, centrally inverted
o A V o
+ m3m2m3m bihexagonal tegum
o
A V
V A A = o3x o3x = triddip,
V = x3o x3o = bidual triddip
A V
wendy wrote:The figure xo3ox xo3ox xo3ox &#xz + tri-hexagon tegum = 1_22.
Klitzing wrote:This appealing representation of mo = 1_22 = o3o3o3o3o *c3x can be given as a very interesting 6-dim tegum sum:
mo = hull( x3o x3o x3o, o3x o3x o3x, x3x o3o o3o, o3o x3x o3o, o3o o3o x3x ) = xoxoo3oxxoo xooxo3oxoxo xooox3oxoox&#zx
Users browsing this forum: No registered users and 1 guest