Keiji wrote:I have not included wintersolstice's "Partially-base snubdis octahedron antiprism", because I have no idea what he meant by that name. (if you can provide me Klitzing's index for that shape, that would help0
wintersolstice wrote:Keiji wrote:I have not included wintersolstice's "Partially-base snubdis octahedron antiprism", because I have no idea what he meant by that name. (if you can provide me Klitzing's index for that shape, that would help0
it's the "cube||icosahedron"
"snubdis"= alternation of truncate. if you take the octahedron and truncate it, then alternate it you get the icosahedron, so it's based on the cube||octahedron (i.e octahedral/cube antiprism)
Keiji wrote:Can we just call it the "snubdis antiprism" or somesuch then, to keep it short?
I don't really want to name it after any individual Platonic, because it's pretty much cross-family, and unique in that respect.
quickfur wrote:Klitzing wrote:quickfur wrote:[...]As for what is/isn't a cupola, I think that may be splitting hairs. As Klitzing says, there's no unique generalization of the 3D notion of cupola. IMO the real solution is to recognize that there is no unique 4D analogue of the 3D cupolae instead of trying to decide which of the equally-valid possibilities should be designated as "true" cupolae.
Well, a cupola in 3d has full symmetrical bases. It has triangles and squares connecting those.
So to extrapolate these figures into 4d, you would have to say what each of those components would become. Top-base polygons (the smaller ones) might become platonic solids. (But I extended that even to quasiregular cases after all.) The triangles most obviously generalize to pyramids. The squares should become somthing with axial symmetry again. So they either could become antiprisms (my choice, as there are more segmentochora which can be classified by that case) or prisms. In the latter case you would have additional things occuring within that new dimension: triangular prisms (kind as line atop square). Those cases do not even apply to ike (it would be hyperbolic!), nor to the quasiregulars. But those OTOH are caps again...
[...]
But again, I think this is just splitting hairs. I could, for example, analyse a 3D cupola as a segmentohedron having a bottom face that is the result of marking an unmarked node in the CD diagram of the top face. For example, a square cupola would have a top face x4o, then the bottom face would be x4x. Then in the 4D case, I can allow any such combination: x4o3o || x4x3o can be considered a cupola, and so can x4x3o || x4x3x, or o4x3o || x4x3o, or any other such combination with the same relationship between the top/bottom cells. This would become a third definition of 4D cupola. (And I'm sure you can come up with more, if you wanted to.)
I could argue that this definition is more encompassing than yours, and therefore "better". But that is just a matter of opinion. My point is that there is more than one way to generalize the 3D concept, which aren't all compatible with each other, so maybe we should just call them all lace prisms and be done with it, instead of trying to argue over which definition of cupola is the "correct" one.
oPo || xPo : pyramid
oPo || xPx : (except for P=2 flat or even hyperbolic)
xPo || xPo : prism
xPo || oPx : antiprism
xPo || xPx : cupola
xPx || xPx : prism again
oPoQo || xPoQo : pyramid
oPoQo || oPxQo : (except for P=Q=3 flat or even hyperbolic, but o3x3o = x3o4o!)
oPoQo || xPxQo : (except for P=2 hyperbolic)
oPoQo || xPxQx : (except for 2=P<=Q hyperbolic)
xPoQo || xPoQo : prism
xPoQo || oPxQo : what I considered a cupola
xPoQo || oPoQx : what I considered an antiprism
xPoQo || xPxQo : (except for P=2 or P=Q=3 flat or even hyperbolic)
xPoQo || xPoQx : what I considered a cap
xPoQo || oPxQx : (except for Q=2 flat or even hyperbolic)
xPoQo || xPxQx : (except for Q=2 flat or even hyperbolic)
oPxQo || oPxQo : prism again
oPxQo || xPxQo : no special name
oPxQo || xPoQx : what I considered a cupola too
oPxQo || xPxQx : (except for 2=P<=Q flat or even hyperbolic)
xPxQo || xPxQo : prism again
xPxQo || xPoQx : no special name
xPxQo || oPxQx : no special name
xPxQo || xPxQx : no special name
xPoQx || xPoQx : prism again
xPoQx || xPxQx : no special name
xPxQx || xPxQx : prism again
wintersolstice wrote:Klitzing wrote:You should be careful when applying names to kind of property-extrapolations into 4d, which do not conform with the very meaning of the word itself. This is a great deal esp. of Wendys polygloss, to try to cut all that historically wrong applied even. - The very word gyro just means rotated. Sure a rotated polygon looks like its dual, thus for segmentohedra this would be the same. But in 4d a rotated cube does not become an octahedron!
because a rotated square becomes it's dual this could mean that gyration could be redifined to mean both in 4D (like there being more than one meaning of "antiprism" in 4D) I'm not saying it should only that it could
I created a thread for a proposal to modify the definition of "Gyrated" for both analogies
it is only a suggestion and maybe there is a better one
Keiji wrote:I have not included wintersolstice's "Partially-base snubdis octahedron antiprism", because I have no idea what he meant by that name. (if you can provide me Klitzing's index for that shape, that would help)
[...]
Klitzing wrote:wintersolstice wrote:[...]
because a rotated square becomes it's dual this could mean that gyration could be redifined to mean both in 4D (like there being more than one meaning of "antiprism" in 4D) I'm not saying it should only that it could
[...]
it is only a suggestion and maybe there is a better one
No! gyro is greek and means rotating. Consider a gyroscope, which is spinning. Nothing within this stem which would somehow relate to dualizing. (The latter is a mere topological concept.)
--- rk
Klitzing wrote:[...]
So either we coin a special name for any class, just as in 3d, or we could cummulate every lace prism except of prisms, antiprisms, or pyramids into a large class of copoloids... (being thus even larger than your proposal). [...]
Klitzing wrote:There is nothing like a snubdis-operator!!!
Klitzing wrote:So either we coin a special name for any class, just as in 3d, or we could cummulate every lace prism except of prisms, antiprisms, or pyramids into a large class of copoloids... (being thus even larger than your proposal).
Keiji wrote:I shall point you to the table I have added at CRF polychora discovery project#Cupolae_of_regular_polyhedra. I believe this covers all the listed cases. They are all named as either cupolae or antiprisms, but with prefixes, mostly reused from my uniform polytope naming conventions. This does not waste roots, and upon seeing the name you have an idea of the kind of shape it is, because you already know what cupolae and antiprisms are.
Keiji wrote:This is getting a bit silly now, so I'd like to just step in and say this:
The focus of this thread was to classify the segmentochora, not to name them.
Names are useful tools, but I really don't care whether you call it K4.21, the cube||icosahedron, the snubdis antiprism, the syncopated whatsit or anything else.
The important thing is to find out which segmentochora are similar to others, so that we can group things together (like we did a while ago for the bicupolic rings), understand what things are more easily, and overall make it easier to see important errors such as counting things twice or thinking something can be CRF when it can't.
I don't know, maybe everyone else on this forum is able to just picture one polyhedron alongside another, imagine connecting it up and go "Ah, this is one of those!", but I sure can't. And I would hope that this exercise was not just for my own benefit.
[...]
On the gyro vs dual vs syncopated issue though, this is absolutely unnecessary. gyro means rotated and has nothing to do with dual except for the (perhaps misleading to some) overlap in 2D. We already have a word for describing duals, this word is dual and there's nothing wrong with it. Why write "the syncopated X" when I can just write "the dual of X"?
With regards to the post endingKlitzing wrote:So either we coin a special name for any class, just as in 3d, or we could cummulate every lace prism except of prisms, antiprisms, or pyramids into a large class of copoloids... (being thus even larger than your proposal).
I shall point you to the table I have added at CRF polychora discovery project#Cupolae_of_regular_polyhedra. I believe this covers all the listed cases. They are all named as either cupolae or antiprisms, but with prefixes, mostly reused from my uniform polytope naming conventions. This does not waste roots, and upon seeing the name you have an idea of the kind of shape it is, because you already know what cupolae and antiprisms are.
quickfur wrote:
Because wintersolstice was using "gyro/gyrated" as a prefix/adjective. A gyrated rhombicuboctahedron is not a dual rhombicuboctahedron; the word "gyrated" implies that a subpart of the polyhedron was modified by rotation. In 4D, you can cut off some bicupolic rings off some uniform polychora X, and stick them back rotated the "wrong" way -- so those would sensibly be called "gyrated X". But wintersolstice's idea was that not only you can stick the part you cut off back rotated; you can also replace the part with something generated from a dual polytope. So it's not the case that X becomes its dual, nor that X has a part P cut off and have the dual of P glued back on, but that P itself is generated from an underlying shape Q, and now we use the dual of Q to generate a (adjective) version of P to glue back onto X. The issue at hand is, what adjective should we apply to X to indicate such a modification? Simply calling it "dual X" is obviously wrong: we're not taking the dual of X. Calling it "gyrated" is misleading, because nothing is being rotated. Hence the need for a new adjective to describe cutting off a subpart generated from Q and gluing back a subpart generated from the dual of Q.
wendy wrote:I still think of ike || cube as a 'crown jewel'. It's the only one that has two pyritohedral figures in its figure.
By the new notation: 3xo*xx2o%&t
Ok, here's the pyritohedral group. It has an orbifold of 3*2. This is a kind of symmetry group akin to the dynkin graph, but is the handy work of conway and thurston (no, not the cowboy's footballer).
There is an extra kind of edge, a 'swallowed edge'. This is an edgekin that is reabsorbed as an internal chord of a polygon, say like the diagonal of a square.
3x*o2o Octahedron
3x*%2o Cuboctahedron (the % is down the diagonal of the square)
3o * x2% Cube. (the squares are actually x % rectangles, the edges are both x and % coinciding
3x*x2o Icosahedron (there's triangles 3x, and iscolese triangles x*x
3x*x2% tCO. The octagon is divided into three faces, by a pair of parallel %, which are parallel edges
3x*x2x rCO The cube faces are the x2x (A,B). There are the rD faces, which generalised to trapeziums ACBC. The triangles are CCC always regular
None of the others stack up like this...
3 4 5 6 8 10 P
xxoPoox P // Pap (6) (14) 22 46 58 93 174
xxxPooo P // Pp 10 (18) 34 47 59 94 175
xoPox2xx Pap // Pap 11 19 (39) 53 65 96 176
xxPoo2xx Pp // Pp (18) (20) 42 (54) (63) 97 177
3,X 35 57 60 74 75 76 89 95 98 110 125 126 127 130 173
54 55 56 ; 128 129 ; 150 151
p ic do
O icosa 36 84 78 111 131 137 (*) 152 159
OD py+ap 37 85 79 116 132 138 140 153 164
OX py 38 86 117 .133 139 154 .165
DD ap 39 80 81 121 134 142 144 155 169
DX (5) 146
ODD meta- 40 87 82 122 135 143 158 156 170
DDD tri- 41 88 83 124 136 147 158 157 172
* **
OG GG OGG GGG DG ODG DGG DDG
* 112 113 114 115 118 119 120 123
** 160 161 162 163 166 167 168 171
<-- o4x -->
O ox xx xx ox 100 107 66 71 61
OG ox xx xx xo 101 67
GG xo xx xx xo 102
OD ox xx xx 103 108 68 72 62
GD xo xx xx 104
OX xo xx 105 109 69 73 63
DD xx xx 106 70
XG xx 64
+ 99
xo3xo4ox tO // C
A s a count always A // B
/ \ v A 3 24 = 48/1/2 top vertices
1---2====3 B 1,2 8 = 48/1/6 bottom vert
| e AB - 48 lacing
B A1 3 12 octa hex-tri edge
A2 - 24 octa hex-hex edge
B1 3 12 cube edge
h AB1 - 24 triangle = line // point
AB2 - 24 triangle = line // pt
AB3 - 24 triangle = pt // line
A12 - 8 hexagon
A23 - 6 square
B32 - 6 square
c A123 - 1 tO
A12B - 8 hex pyr = hexa // point
A13B - 12 tetra = line x2o // line o2x
A23B - 6 square = x4o // o4x
B321 - 1 cube
Klitzing wrote:Do you calculate all those lace prism heights manually? This would become teddious. It would be easier to have some calculation routine by hand, inserting the top resp. bottom Dynkin diagram, and getting out the appropriate height.
In fact this calculation "engine" once was done by Wendy. This "engine" moreover can calculate the circumradius of any polytope, provided its Dynkin symbol.
I'll attach the excel spreadsheet (zipped, as "xls" is a not allowed format to be uploaded) I once got from her (with some minor changes).
(6 (6 (6 5 4 3 2 1 )
(5 (5 10 (5 10 8 6 4 2 )
2 (4 2 (4 8 12 2 (4 8 12 9 6 3 )
- (3 - (3 6 9 12 - (3 6 9 12 8 4 )
7 (2 7 (2 4 6 8 10 7 (2 4 6 8 10 5 )
(1 (1 2 3 4 5 6 (1 2 3 4 5 6 )
wendy wrote:[...]The spreadsheat that richard klitzing put up does this up to about six dimensions, but the same idea can be pushed out to 19 or 20 dimensions without difficulty. The recriprocal of the stott matrix is the dynkin matrix. The main diagonal of this consists of '2', and the value a_ij is -2 cos(pi / p). What you do is enter these values into the input field, it fills in a_ji for you, and finds the value of the stott matrix, and then the matrix-dot of the entered vector.
wendy wrote:You could point the lacing-length to a cell, and then make it a free variable.
Users browsing this forum: No registered users and 1 guest