student91 wrote:[...]
The first two restrictions in my previous post still are true: a triangle can be altered anyhow, a square only in an xyyx-fashion (then it's the sech.prod. of a digonal prism and a x||y-polytope) the triangle in the first restriction can be replaced with any simplex. (this way you can see sechmentotopes are alternations of 1-simplices (except for the highlighted cases in my first post about this) ). those simplices keep bothering me, because I can't find a way to "tame" them (i.e. put them in a system of operations)
student91
... sechmentotope ...
student91 wrote:[...]
you might wonder, why I want to make this aditional system of sechmentotopes, because there is already a sufficient system: Klitzings
This is because, in my opinion, Klitzings way of looking at it provides a good way for sechmentotopes that are based on an (n-1)-symmetry, but it is not very good in describing sechmentotopes based on a less-than-(n-1)-symmetry. (e.g. the ortobicupolic ring: Klitzing sees it as either cube||octagon or 4-cupola||square, whereas I would see it as square square octagon in a triangle fashion, out of which Klitzings views can be derived (as either (square||square)||octagon or square||(square||octagon) )
[...]
Klitzing wrote:If I get it correctly, one of student91's points in his recent post was just the rediscovering of Wendy's lace simplexes.
[...]
--- rk
wendy wrote:[...]
It's hardly a product, though, because it does not come to a algebraic multiply.
wendy wrote:One should not really diss Klitzing's A || B notation. He went out to solve a specific problem for which this is the right notation.
On the other hand, i figured out how to hook multiple vertex nodes to the same dynkin symbol, and devised a notation for it.
It is interesting that student91 uses my notation rather heavily. You can change the lacing edge, the x in #x.. is the same x as in x3o etc.
[...]
Klitzing wrote:No, you are wrong. f-edges etc. are well used in CRFs.
First of all:
x denotes an edge of length 1
f denotes an edge of length 1.618 (tau)
v denotes an edge of length 0.618 (1/tau = tau - 1)
q denotes an edge of length 1.414 (sqrt2)
h denotes an edge of length 1.732 (sqrt3)
u denotes an edge of length 2
w denotes an edge of length 2.414 (1 + sqrt2)
...
x occures as vertex figure of x3o, f occures as vertex figure of x5o, v occures as vertex figure of x5/2o, q occures as vertex figure of x4o, h occures as vertex figure of x6o, and u occurs as vertex figure of the apeirogon (x-infin-o).
It is the invention of lace towers, which makes all these figures occur. Note that then you will consider vertex layers, resp. the structures described by them. But not all edges of the dissected structure will fall in those layers only. In the contrary, those sections would often Need for vertex distances not being described by edge length distances only.
A regular pentagon e.g. then can be described as ofx&#xt, i.e. a tower of 3 layers with a single point in the uppermost, 2 points in a medial layer, which are spaced by tau, and finally 2 points within the 3rd layer, spaced by unity. Vertices of the different layers then are connected by unit lacing edges (...&#xt). Similarily a regular hexagon can be given either as xux&#xt or as ohho&#xt. And a regular octagon as xwwx&#xt. You even could write a square not only as xx&#x, but likewise as oqo&#xt. - All these figures clearly use unit distances for their edges only, but there are other vertex distances within these layers, which not qualify as true edges, which rather require other lengths.
--- rk
Klitzing wrote:Wrt lace rings, those clearly are useful descriptions for CRFs too.
In fact a regular n-gon can be given as oo...oo&#xr (n unringed nodes, i.e. cyclically consecutive vertices).
Likewise the semiregular n-gonal prism then would be xx...xx&#xr (n ringed nodes, i.e. cyclically consecutive orthogonal edges).
The tetrahedron could be given as oox&#xr, the square pyramid as oxx&#xr, and a triangle prism could be given as ooxx&#xr.
But note, that those are just quite simple, low dimensional examples. And therefore might look superfluous.
Consider e.g. the hexadecachoron x3o3o4o. It can be written as segmentochoron tet || dual tet, and therefore as lace prism xo3oo3ox&#x. But as the tetrahedron itself can be given as a segmentohedron o3o || x3o (and the dual one then as o3x || o3o), the whole thing thus could be rewritten as oxoo3ooox&#xr.
--- rk
student91 wrote:But those rings (%..%P%..%Q%..%...&#xr), do they occur with more than 4 %'s? Because I think those are either impossible or derived from either a cartesian producted polygon, a prism or a lace simplex. (look, for example, to the triaugmented trigonal prism. it can be seen as oxoxox&#r, but the o's can be cut of, and hence it's not a "new" thing. the same with the triaugmented hexagonal prism oxxoxxoxx&#xr). I think this, because if the ring is bigger than 4, and it exists of more than 1 %P%Q%.., (and it's not a prism) you have to connect a diagonal. if diagonals cross (like oxox4xxxx&#xr in my previous post), you get a degenerate higher-dimensional polytope. If diagonals don't cross, you could cut the shape in two at this diagonal. if you keep cutting in this way, you always get either a cartesian producted polygon, a prism or a lace simplex (because something else could be cut again)
student91 wrote:ah, I see, so they're frequently used. Are they also used for the lacing distance in CRF's? (so &#ut for example?)
Klitzing wrote:A unit-edged cuboctahedron e.g. can be given as xox4oqo&#xt (i.e. as x4o || pseudo o4q || x4o). A unit-edged truncated cube as xwwx4xoox&#xt. The unit-edged great rhombicuboctahedron becomes xxwwxx4xuxxux&#xt. Etc.
student91 wrote:the thing you've written (oxoo3ooox&#xr) can be decomposed in this way to oxo3xoo&#xt and xoo3oxo&#xr, joined at their xo3ox&#x-edge, and the ooo3ooo&#xr and xxx3xxx&#xr are just cartesian products (and a little superfluous, as you said)
o3o x3o
o3x o3o
o3o o3o
o3o
indeedKlitzing wrote:Probably a typo: you'd mean oxo3xoo&#xr and xoo3oxo&#xr.
ooo3ooo&#xr is not at all a cartesian product. It is just a complicated description of a triangle:
- Code: Select all
o3o o3o
o3o
i.e. a lace city from 3 points.
wendy wrote:So so3so4ox&#x is not a valid WLP. Nor is o3o4x || x3o5o.
alternated faceting {starting figure = xo3xo4ox&#y; alternated elements = ..3..4o.; subsequent relaxing = true}
wendy wrote:I'm not sure what is meant by 'segmentopic product'.
Klitzing wrote:wendy wrote:I'm not sure what is meant by 'segmentopic product'.
Me neither fully.
But as far as I got it, he is aiming just for a reversion of operations.
Consider 2 segmentotopes b1 || b2 and b3 || b4.
Then he calls (b1||b2) x (b3||b4) := (b1xb3) || (b2xb4).
But I already argued that then it would be essential to know in advance the preconditions under which those results would be valid or not. (In the sense of kind a theorem.) Not only that all asked for subelements would exist (i.e. connect correctly) but that moreover the required heights do come out positive, resp. taken the other way round, that the lacing edges can be chosen as unity too.
@student91: please verify.
--- rk
student91 wrote:[...]
let's consider the same sechmentotopes as you used: b1||b2 (with height h1), and b3||b4 (with height h3).
furthermore there's a vertex v1 that belongs to b1, and that has a connection (in b1||b2) with v2, a vertex of b2.
now if we product both b1 and b2 with b3, we'd get b1×b3||b2×b3 = b3×(b1||b2).
I see this the following way: you place a b3 at every vertex of b1||b2.
All the edges of b1||b2 (the "base"-polytope) have length 1. Because of this, all the b3's will connect properly into b3-prisms where there's an edge connecting two b3's.
now the true sechmentotopical product: b1×b3||b2×b4. I'll look at this from the viewpoint of b1||b2 with b3's on the vertices of b1, and b4's at the vertices of b2. We could color the vertices where a b3 will be placed red, and the vertices where a b4 will be placed blue. the edges of b1||b2 connection a red vertex with a red one now will yield b3||b3-polytopes, connecting blue with blue yields b4||b4, and connecting blue with red yields b3||b4.
In order to connect properly, the edges of b1||b2 that connect blue with red sould be shrunk to h3.
Those edges are the edges connecting b1 with b2. if we change the distance between b1 and b2 (i.e. h1), those edges will get a different length as well, and thus, it might be possible to make the red-blue edges of length h3. we'll get to whether this is possible or not later.
Now when we've shrunken h1 to the required size, all the b3's should connect correctly to the b4's, and thus, everything connects properly
quickfur wrote:[...]
So basically your segmentotopic product (A||B) # (C||D) is basically just (A x C)||(B x D) where x denotes the Cartesian product, and # is a placeholder symbol for your "segmentotopic product"? In other words, you take the Cartesian product of the "top" layer of the segmentotope A||B with the "top" layer of the segmentotope C||D, and likewise take the Cartesian product of the respective bottom layers, and then construct a new segmentotope out of the results. Right? And what you're claiming then is, if A||B and C||D exist, then (A x C)||(B x D) must also exist?
Seems reasonable to me, though it doesn't really add anything new. It's just a way of analysing a segmentotope by decomposition into Cartesian products (or conversely, of constructing a higher-dimensional segmentotope from two lower-dimensional ones). Or am I missing something?
student91 wrote:let's consider the same sechmentotopes as you used: b1||b2 (with height h1), and b3||b4 (with height h3).
quickfur wrote:So basically your segmentotopic product (A||B) # (C||D) is basically just (A x C)||(B x D) where x denotes the Cartesian product, and # is a placeholder symbol for your "segmentotopic product"? In other words, you take the Cartesian product of the "top" layer of the segmentotope A||B with the "top" layer of the segmentotope C||D, and likewise take the Cartesian product of the respective bottom layers, and then construct a new segmentotope out of the results. Right? And what you're claiming then is, if A||B and C||D exist, then (A x C)||(B x D) must also exist?
Seems reasonable to me, though it doesn't really add anything new. It's just a way of analysing a segmentotope by decomposition into Cartesian products (or conversely, of constructing a higher-dimensional segmentotope from two lower-dimensional ones). Or am I missing something?
A different variation of this idea, is if we start with an even polytope E (where "even" means it can be alternated) -- say we call its two alternated forms E^{+} and E^{-} -- and a segmentotope F||G, then we take the convex hull of (E^{+} x F) U (E^{-} x G), where x denotes the Cartesian product and U denotes set union. Under what conditions would the result contain F||G as facets? Is it possible to make the result CRF? For example, one could start with a cube and, say, a pentagonal cupola (pentagon||decagon). Since the cube alternates into two dual tetrahedra, we form the 5D Cartesian products tetrahedron x pentagon and dual_tetrahedron x decagon, and take their convex hull. The result should contain pentagonal cupolae as surtopes. Is it possible to make the result CRF?
Klitzing wrote:student91 wrote:let's consider the same sechmentotopes as you used: b1||b2 (with height h1), and b3||b4 (with height h3).
Once more: its a segmentotope!
Klitzing wrote:quickfur wrote:[...]
Union would not work, as then those would all remain within the same Hyperspace and you'd get rather a compound than a pair of stacked bases. But you could like to consider that in the sense of student91's segmentotopal product instead:
E @ (F||G) := (E^{+}||E^{-}) # (F||G) = (E^{+}xF) || (E^{-}xG).
But be aware then: the exisatance of such a new quickfur-operation ( ) would depend on the existance of E^{+}||E^{-}! (And on that height restriction too.) E.g. tet || dual tet (in the sense of alternated cube, i.e. s2s2s) does exist (it is hex = x3o3o4o). But neither r-snic || l-snic nor r-snid || l-snid do exist! - Thus this "@" seems to be kind of rather restricted after all...
--- rk
student91 wrote:[...]
every segmentotope is either:
a lace simplex of wythoffian polytopes with the same symmetry
a segmentotopical product of multiple segmentotopes
multiple lace simplices placed base-to-base
a diminishing of those.
an augmention of a diminishing
A cartesian product of a segmentotope and another CRF-polytope
student91
Klitzing wrote:Some rather easy observations on student91's segmentotopal product "#":
Definition:
(A||B) # (C||D) := (AxC) || (BxD)
Observations:
if D = C then (A||B) # (C||C) = (AxC) || (BxC) = (A||B) x C
esp. C = D = point then (A||B) # line = (A||B) # (pt||pt) = ... = (A||B) x pt = A||B
and C = D = line then (A||B) # square = (A||B) # (line||line) = ... = (A||B) x line = (A||B)-prism
Further:
because of PxQ = QxP (up to orientation in coodinate space) we too have P#Q = Q#P
--- rk
student91 wrote:we should also include the snubs into the wythoffians.
(the last one can be seen as a segmentotopical product of a CRF-prism and another segmentotope, so it can be discarted)
do you think this might be provable? (at least all 4d-segmentotopes are of one of these categories)
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