| Site Map | Polytopes | Dynkin Diagrams | Vertex Figures, etc. | Incidence Matrices | Index |
Lace prisms as such are defined here as being monostratic stacks of 2 polytopes, both of which are describable by the same (undecorated) Dynkin diagram (thus providing an overall unified axial symmetry), which are laced somehow together kind of like children drums. Thus in general we have at least 3 edge types involved: x representing the ones of the top layer (possibly being several ones themselves), y representing the ones of the bottom layer (again possibly being several ones), and then z the lacing ones. A typical Dynkin symbol then would be xo3oy4oo&#z. Here each left node symbol at any node position describes the top layer, while the right one describes the bottom layer. Accordingly this so far could be kind of described alternatively by x3o4o || o3y4o. But clearly this latter description no longer provides the additional clue for the lacing edges, which in the former description are given by means of the suffix &#z, which in fact tells that the additional (&) lacing (#) edges have size (or type) z.
When restricting to all unit edges only, the class gets still distinguished by means of positioning. Thus it is enough to use a single edge symbol x throughout. Our example then becomes xo3ox4oo&#x, then fully representing a segmentochoron (here: oct || co). Moreover, this then is quite general here. In fact, every unit-edged lace prism also is a segmentotope. Conversely, every segmentotope, whose 2 base polytopes provide a common (non-degenerate) symmetry, which furthermore is Dynkin describable, also is a lace prism. – For the remainder of this page we will restrict to that set-theoretical intersection.
Visual examples for all the here relevant segmentochora have been displayed explicitly here. These then were given as axial projection images, color coding the 2 layers in blue resp. red, and providing the lacing edges in gold. Those there are given both as further projection to 2D pictures and as true 3D VRMLs.
The main issue in this page however will be a bit different. Of special interest here will be the dimensional series themselves. Consider as a single first example the series of hypercubical pyramids: First we have the square pyramid (squippy), then the cubical pyramid (cubpy), etc. Note that squippy has both height and circumradius being 1/sqrt(2) = 0.707107, whereas the height of cubpy is only 1/2 and its circumradius already is 1. Therefore the tesseractic pyramid (tespy) already becomes degenerate, as it would have a zero height and thus an infinite circumradius. – Quite in a similar fashion most dimensional lace prism series would come to an end too.
In order to outline this most general ceasing behaviour, we note that this is based mainly on the already mentioned 2 measures, the height and the (overall) circumradius (which in turn can be calculated from the height and the 2 circumradii of the base polytopes). – In fact, with R as the radius of the d dimensional circumsphere of a segmentotpe and ri being the d-1 dimensional radius of its i-th base (which for lace prisms never has to use a further shift) and h being the height, we derive the general formula 4R2h2 = (r22-r12)2 + 2(r12+r22)h2 + h4. – And we further note, that for mere Stott expansions (or contractions) wrt. to the overall axial symmetry only those 3 relevant circumradii would change, whereas the height would remain unchanged. Based on this observation it will be enough to abandon in here all those lace prisms, which would have an x node simultanuously at any single position within both layers. Instead we can restrict to the fundamental cases with positional node pairs oo, ox, and xo only! – A full detailed enlisting, however not as high dimensional as given here, is given at the page on lace simplices.
A further restriction on the to be considered cases can be achieved – at least for fully connected diagrams (i.e. whenever the axial symmetry is not reducible) – whenever a connected substring of the diagram bears only o nodes at one layer. For then, at the corresponding part on other layer, there cannot be more than one single x node. (Else you would ask at best for lacing elements which are hippies, copies, spidpies, etc., which all are degenerate themselves and so the overall height is forced to be zero as well.) That is, according to this restriction, the ringed nodes x only can alternate between the layers, i.e. after some ox we first need an xo (and the other way round) with possible node ommissions oo inserted anywhere! (And, as already stated above, the remainder then only are the Stott expansions of those more basic shapes, replacing some of those oo nodes into xx nodes.)
When taking into account bifurcation points of the Dynkin diagram of the axial symmetry, then the same alternation rule holds true there as well. (Again at least when arguing for cases with truely positive heights.) But for prooving this claim for such symmetries we have to distinguish 3 cases now. – The first one assumes that the bifurcation point itself and its 3 nearest neighbours all are unringed in at least one layer. Now simply consider any (sub-)polytope of that other layer, which opposes this (possibly larger) unringed region, and additionally has at least 2 ringed nodes. Any such polytope then already has a (layer-wise) circumradius which is larger than 1. Therefore then an according all unit-edged pyramid, which here would be needed for some lacing (sub-)element in the respective lace prism, ought belong to hyperbolic geometry already. But from this restriction it becomes clear additionally, that only 2 legs, emmanating off from the bifurcation point, are allowed to bear ringed nodes at all! I.e., for not necessarily fundamental lace prisms, the third leg either has to be completely unringed (only oo nodes) or is a Stott expansion thereof (possibly some xx nodes).
Next consider the bifurcation node itself of some reduced fundamental lace prism to be ringed. Then to all 3 emanating legs the above consideration wrt. axially simplicial symmetries applies (with final inclusion of this bifurcation point). Thus again that alternation condition does apply here too. (Sadly, in this case nothing can be said about fully empty legs.)
In the complemental final case (of reduced fundamental lace prisms with bifurcation nodes) we thus can assume that this bifucation point itself is of type oo and that in either layer at least one of its neighbouring nodes is ringed. Thus the here possible (sub-)polytopes are either xo3oo3ox *b3oo&#x (hin = hex || alt. hex, which has height h = 1/sqrt(2) = 0.707107) or xo3oo3ox *b3ox (hexarit = hex || rit, which itself is degenerate with zero height). Therefore just the former of these 2 cases has to be considered further here. But that one then again requires that the third leg has to be fully unringed. This here would follow again by mere application of the axially simplicial result to the first-to-third substring as well as to the second-to-third substring. These elsewise would contradict. But thereby our claim is proved in that final case as well.
On a sideline to this topic of ceasing lace prisms a further note has to be added wrt. the final truely degenerate cases. Lace prisms of height zero a priori just tell nothing but that very property, i.e. the "bases" get aligned within the same hyperspace in the provided symmetry orientation and their vertices then can be connected by unit sized "lacing edges". But nothing can be said from this property alone onto whether the smaller base then comes to lie completely within the boundary of the larger one, or whether in contrary the vertices of the smaller protrude a bit to the outside of the bounding facets of the larger base. In the first case this degenerate lace prism thus would provide a cellular decomposition of the larger base. But in the latter case this lace prism rather provides a tegum sum of these bases. Thus in this latter case we could replace the tailing &#x indeed by &#zx.
The remainder of this page examines the different symmetry types separately. There the lower dimensional cases are listed each. The respective heights will be given. After some dimensional steps, it can be seen that most cases would fade out. Just very few serial cases can survive beyond the respective limiting point, remaining existent then for any dimension. These few infinite series are characterised for either symmetry.
The dimensionally non-infinite series of symmetries clearly are out of scope here, as the according lace prisms never produce infinite series. On the other hand, the lower dimensional listings, provided below for these infinite ones, clearly are a further good source of reference by their own. And a similar type for access would be missing for the other symmetries. This simply is why those will be supplemented here below as well:
| 2D |
h = 1 |
oo&#x (line = point || point) |
h = sqrt(3)/2 = 0.866025 |
ox&#x ({3} = point || line) | |
| 3D |
h = 1 |
oo3oo&#x (line = point || point) |
h = sqrt(2/3) = 0.816497 |
ox3oo&#x (tet = point || {3}) xo3ox&#x (oct = {3} || dual {3}) | |
| 4D |
h = 1 |
oo3oo3oo&#x (line = point || point) |
h = sqrt(5/8) = 0.790569 |
ox3oo3oo&#x (pen = point || tet) xo3ox3oo&#x (rap = tet || oct) | |
h = 1/sqrt(2) = 0.707107 |
oo3ox3oo&#x (octpy = point || oct) xo3oo3ox&#x (hex = tet || dual tet) ox3xo3ox&#x (octaco = oct || co) | |
| 5D |
h = 1 |
oo3oo3oo3oo&#x (line = point || point) |
h = sqrt(3/5) = 0.774597 |
ox3oo3oo3oo&#x (hix = point || pen) xo3ox3oo3oo&#x (rix = pen || rap) oo3xo3ox3oo&#x (dot = rap || inv rap) | |
h = sqrt(2/5) = 0.632456 |
oo3ox3oo3oo&#x (rappy = point || rap) xo3oo3ox3oo&#x (dihin = pen || inv rap) xo3oo3oo3ox&#x (tac = pen || dual pen) ox3xo3ox3oo&#x (rapasrip = rap || srip) ox3xo3oo3ox&#x (rapaspid = rap || spid) xo3ox3xo3ox&#x (sripa = srip || inv srip) | |
| 6D |
h = 1 |
oo3oo3oo3oo3oo&#x (line = point || point) |
h = sqrt(7/12) = 0.763763 |
ox3oo3oo3oo3oo&#x (hop = point || hix) xo3ox3oo3oo3oo&#x (ril = hix || rix) oo3xo3ox3oo3oo&#x (bril = rix || dot) | |
h = 1/sqrt(3) = 0.577350 |
oo3ox3oo3oo3oo&#x (rixpy = point || rix) xo3oo3ox3oo3oo&#x (hixadot = hix || dot) xo3oo3oo3oo3ox&#x (gee = hix || dual hix) oo3xo3oo3ox3oo&#x (rixa = rix || inv rix) ox3xo3ox3oo3oo&#x (rixasarx = rix || sarx) ox3xo3oo3oo3ox&#x (rixascad = rix || scad) ox3oo3xo3ox3oo&#x (dottaspix = dot || spix) xo3ox3xo3ox3oo&#x (sarxasibrid = sarx || sibrid) xo3ox3oo3xo3ox&#x (spixa = spix || inv spix) | |
h = 1/2 |
oo3oo3ox3oo3oo&#x (dotpy = point || dot) xo3oo3oo3ox3oo&#x (hixalrix = hix || inv rix) ox3xo3oo3ox3oo&#x (rixaspix = rix || spix) ox3oo3xo3oo3ox&#x (dottascad = dot || scad) oo3ox3xo3ox3oo&#x (dottasibrid = dot || sibrid) xo3ox3xo3oo3ox&#x (spixalsarx = sarx || inv spix) ox3xo3ox3xo3ox&#x (sibridacard = sibrid || card) | |
| 7D |
h = 1 |
oo3oo3oo3oo3oo3oo&#x (line = point || point) |
h = 2/sqrt(7) = 0.755929 |
ox3oo3oo3oo3oo3oo&#x (oca = point || hop) xo3ox3oo3oo3oo3oo&#x (roc = hop || ril) oo3xo3ox3oo3oo3oo&#x (broc = ril || bril) oo3oo3xo3ox3oo3oo&#x (he = bril || inv bril) | |
h = sqrt(2/7) = 0.534522 |
oo3ox3oo3oo3oo3oo&#x (rilpy = point || ril) xo3oo3ox3oo3oo3oo&#x (hopabril = hop || bril) xo3oo3oo3oo3oo3ox&#x (zee = hop || dual hop) oo3xo3oo3ox3oo3oo&#x (rilalbril = ril || inv bril) ox3xo3ox3oo3oo3oo&#x (rilasril = ril || sril) ox3xo3oo3oo3oo3ox&#x (rilastaf = ril || staf) ox3oo3xo3ox3oo3oo&#x (brilaspil = bril || spil) ox3oo3oo3xo3ox3oo&#x (brilalscal = bril || inv scal) xo3ox3xo3ox3oo3oo&#x (srila sabril = sril || sabril) xo3ox3oo3xo3ox3oo&#x (spila sibpof = spil || sibpof) xo3ox3oo3oo3xo3ox&#x (scala = scal || inv scal) oo3xo3ox3xo3ox3oo&#x (sabrila = sabril || inv sabril) | |
h = 1/sqrt(7) = 0.377964 |
oo3oo3ox3oo3oo3oo&#x (brilpy = point || bril) xo3oo3oo3ox3oo3oo&#x (hopalbril = hop || inv bril) xo3oo3oo3oo3ox3oo&#x (hopalril = hop || inv ril) oo3xo3oo3oo3ox3oo&#x (rila = ril || inv ril) ox3xo3oo3ox3oo3oo&#x (rilaspil = ril || spil) ox3xo3oo3oo3ox3oo&#x (rilascal = ril || scal) ox3oo3xo3oo3ox3oo&#x (brilascal = bril || scal) ox3oo3xo3oo3oo3ox&#x (brilastaf = bril || staf) oo3ox3xo3ox3oo3oo&#x (brila sabril = bril || sabril) oo3ox3xo3oo3ox3oo&#x (brila sibpof = bril || sibpof) xo3ox3xo3oo3ox3oo&#x (srila sibpof = sril || sibpof) xo3ox3xo3oo3oo3ox&#x (scalalsril = scal || inv sril) xo3ox3oo3xo3oo3ox&#x (scalalspil = scal || inv spil) xo3oo3ox3xo3ox3oo&#x (spilal sabril = spil || inv sabril) xo3oo3ox3xo3oo3ox&#x (spila = spil || inv spil) ox3xo3ox3xo3ox3oo&#x (sabrilacral = sabril || cral) ox3xo3ox3xo3oo3ox&#x (sabrila topal = sabril || topal) ox3xo3ox3oo3xo3ox&#x (sibpoffa topal = sibpof || topal) xo3ox3xo3ox3xo3ox&#x (crala = cral || inv cral) | |
| 8D |
h = 1 |
oo3oo3oo3oo3oo3oo3oo&#x (line = point || point) |
h = 3/4 |
ox3oo3oo3oo3oo3oo3oo&#x (ene = point || oca) xo3ox3oo3oo3oo3oo3oo&#x (rene = oca || roc) oo3xo3ox3oo3oo3oo3oo&#x (brene = roc || broc) oo3oo3xo3ox3oo3oo3oo&#x (brocahe = broc || he) | |
h = 1/2 |
oo3ox3oo3oo3oo3oo3oo&#x (rocpy = point || roc) xo3oo3oo3oo3oo3oo3ox&#x (ek = oca || dual oca) xo3oo3ox3oo3oo3oo3oo&#x (ocabroc = oca || broc) oo3xo3oo3ox3oo3oo3oo&#x (rocahe = roc || he) oo3oo3xo3oo3ox3oo3oo&#x (broca = broc || inv broc) ox3xo3ox3oo3oo3oo3oo&#x (rocasaro = roc || saro) ox3xo3oo3oo3oo3oo3ox&#x (rocasuph = roc || suph) ox3oo3xo3ox3oo3oo3oo&#x (brocaspo = broc || spo) ox3oo3xo3oo3oo3oo3ox&#x (brocasuph = broc || suph) ox3oo3oo3xo3ox3oo3oo&#x (heasco = he || sco) xo3ox3xo3ox3oo3oo3oo&#x xo3ox3oo3xo3ox3oo3oo&#x xo3ox3oo3oo3xo3ox3oo&#x xo3ox3oo3oo3oo3xo3ox&#x oo3xo3ox3xo3ox3oo3oo&#x oo3xo3ox3oo3xo3ox3oo&#x | |
h = 1/4 |
oo3oo3ox3oo3oo3oo3oo&#x (brocpy = point || broc) xo3oo3oo3ox3oo3oo3oo&#x (ocahe = oca || he) xo3oo3oo3oo3oo3ox3oo&#x (ocalroc = oca || inv roc) oo3xo3oo3oo3ox3oo3oo&#x (rocalbroc = roc || inv broc) ox3xo3oo3ox3oo3oo3oo&#x (rocaspo = roc || spo) ox3xo3oo3oo3oo3ox3oo&#x (rocaseto = roc || seto) ox3oo3xo3oo3ox3oo3oo&#x (brocasco = broc || sco) ox3oo3oo3xo3oo3ox3oo&#x (heaseto = he || seto) oo3ox3xo3ox3oo3oo3oo&#x oo3ox3xo3oo3oo3ox3oo&#x oo3ox3oo3xo3ox3oo3oo&#x xo3ox3xo3oo3ox3oo3oo&#x xo3ox3xo3oo3oo3oo3ox&#x xo3ox3oo3xo3oo3ox3oo&#x xo3ox3oo3oo3xo3oo3ox&#x xo3oo3ox3xo3ox3oo3oo&#x xo3oo3ox3xo3oo3oo3ox&#x xo3oo3ox3oo3xo3ox3oo&#x oo3xo3ox3xo3oo3ox3oo&#x ox3xo3ox3xo3ox3oo3oo&#x ox3xo3ox3xo3oo3oo3ox&#x ox3xo3ox3oo3xo3ox3oo&#x ox3xo3ox3oo3oo3xo3ox&#x ox3xo3oo3ox3xo3oo3ox&#x ox3oo3xo3ox3xo3ox3oo&#x xo3ox3xo3ox3xo3ox3oo&#x xo3ox3xo3ox3oo3xo3ox&#x | |
h = 0 (degenerate) |
oo3oo3oo3ox3oo3oo3oo&#x (hepy = point || he) xo3oo3oo3oo3ox3oo3oo&#x (ocalbroc = oca || inv broc) oo3xo3oo3oo3oo3ox3oo&#x (roca = roc || inv roc) ox3xo3oo3oo3ox3oo3oo&#x (rocasco = roc || sco) ox3oo3xo3oo3oo3ox3oo&#x (brocaseto = broc || seto) ox3oo3oo3xo3oo3oo3ox&#x (heasuph = he || suph) oo3ox3xo3oo3ox3oo3oo&#x oo3ox3oo3xo3oo3ox3oo&#x oo3oo3ox3xo3ox3oo3oo&#x xo3ox3xo3oo3oo3ox3oo&#x xo3ox3oo3xo3oo3oo3ox&#x xo3oo3ox3xo3oo3ox3oo&#x xo3oo3ox3oo3xo3oo3ox&#x ox3xo3ox3xo3oo3ox3oo&#x ox3xo3ox3oo3xo3oo3ox&#x ox3xo3oo3ox3xo3ox3oo&#x ox3xo3oo3ox3oo3xo3ox&#x ox3oo3xo3ox3xo3oo3ox&#x oo3ox3xo3ox3xo3ox3oo&#x xo3ox3xo3ox3xo3oo3ox&#x xo3ox3xo3oo3ox3xo3ox&#x ox3xo3ox3xo3ox3xo3ox&#x | |
| 9D |
h = 1 |
oo3oo3oo3oo3oo3oo3oo3oo&#x (line = point || point) |
h = sqrt(5)/3 = 0.745356 |
ox3oo3oo3oo3oo3oo3oo3oo&#x (day = point || ene) xo3ox3oo3oo3oo3oo3oo3oo&#x (reday = ene || rene) oo3xo3ox3oo3oo3oo3oo3oo&#x (breday = rene || brene) oo3oo3xo3ox3oo3oo3oo3oo&#x (treday = brene || trene) oo3oo3oo3xo3ox3oo3oo3oo&#x (trena = trene || inv trene) | |
h = sqrt(2)/3 = 0.471405 |
oo3ox3oo3oo3oo3oo3oo3oo&#x (renepy = point || rene) xo3oo3ox3oo3oo3oo3oo3oo&#x (eneabrene = ene || brene) xo3oo3oo3oo3oo3oo3oo3ox&#x (vee = ene || dual ene) oo3xo3oo3ox3oo3oo3oo3oo&#x (reneatrene = rene || trene) oo3oo3xo3oo3ox3oo3oo3oo&#x (brenealtrene = brene || inv trene) ox3xo3ox3oo3oo3oo3oo3oo&#x (reneasrene = rene || srene) ox3xo3oo3oo3oo3oo3oo3ox&#x (renea soxeb = rene || soxeb) ox3oo3xo3ox3oo3oo3oo3oo&#x (breneaspene = brene || spene) ox3oo3oo3xo3ox3oo3oo3oo&#x (trenea secane = trene || secane) ox3oo3oo3oo3xo3ox3oo3oo&#x ox3oo3oo3oo3oo3xo3ox3oo&#x xo3ox3xo3ox3oo3oo3oo3oo&#x xo3ox3oo3xo3ox3oo3oo3oo&#x xo3ox3oo3oo3xo3ox3oo3oo&#x xo3ox3oo3oo3oo3xo3ox3oo&#x xo3ox3oo3oo3oo3oo3xo3ox&#x oo3xo3ox3xo3ox3oo3oo3oo&#x oo3xo3ox3oo3xo3ox3oo3oo&#x oo3xo3ox3oo3oo3xo3ox3oo&#x oo3oo3xo3ox3xo3ox3oo3oo&#x | |
h = 0 (degenerate) |
oo3oo3ox3oo3oo3oo3oo3oo&#x (brenepy = point || brene) xo3oo3oo3ox3oo3oo3oo3oo&#x (eneatrene = ene || trene) xo3oo3oo3oo3oo3oo3ox3oo&#x (enealrene = ene || inv rene) oo3xo3oo3oo3ox3oo3oo3oo&#x (renealtrene = rene || inv trene) oo3oo3xo3oo3oo3ox3oo3oo&#x (brena = brene || inv brene) ox3xo3oo3ox3oo3oo3oo3oo&#x (reneaspene = rene || spene) ox3xo3oo3oo3oo3oo3ox3oo&#x (renea supane = rene || supane) ox3oo3xo3oo3ox3oo3oo3oo&#x (brenea secane = brene || secane) ox3oo3xo3oo3oo3oo3oo3ox&#x (brenea soxeb = brene || soxeb) ox3oo3oo3xo3oo3ox3oo3oo&#x (trenea sotane = trene || sotane) ox3oo3oo3oo3xo3oo3ox3oo&#x oo3ox3xo3ox3oo3oo3oo3oo&#x oo3ox3xo3oo3oo3oo3ox3oo&#x oo3ox3oo3xo3ox3oo3oo3oo&#x oo3ox3oo3oo3xo3ox3oo3oo&#x xo3ox3xo3oo3ox3oo3oo3oo&#x xo3ox3xo3oo3oo3oo3oo3ox&#x xo3ox3oo3xo3oo3ox3oo3oo&#x xo3ox3oo3oo3xo3oo3ox3oo&#x xo3ox3oo3oo3oo3xo3oo3ox&#x xo3oo3ox3xo3ox3oo3oo3oo&#x xo3oo3ox3xo3oo3oo3oo3ox&#x xo3oo3ox3oo3xo3ox3oo3oo&#x xo3oo3ox3oo3oo3xo3ox3oo&#x xo3oo3oo3ox3xo3oo3oo3ox&#x oo3xo3ox3xo3oo3ox3oo3oo&#x oo3xo3ox3oo3xo3oo3ox3oo&#x oo3xo3oo3ox3xo3ox3oo3oo&#x ox3xo3ox3xo3ox3oo3oo3oo&#x ox3xo3ox3xo3oo3oo3oo3ox&#x ox3xo3ox3oo3xo3ox3oo3oo&#x ox3xo3ox3oo3oo3xo3ox3oo&#x ox3xo3ox3oo3oo3oo3xo3ox&#x ox3xo3oo3ox3xo3oo3oo3ox&#x ox3xo3oo3oo3ox3xo3oo3ox&#x ox3oo3xo3ox3xo3ox3oo3oo&#x ox3oo3xo3ox3oo3xo3ox3oo&#x ox3oo3oo3xo3ox3xo3ox3oo&#x xo3ox3xo3ox3xo3ox3oo3oo&#x xo3ox3xo3ox3oo3xo3ox3oo&#x xo3ox3xo3ox3oo3oo3xo3ox&#x xo3ox3oo3xo3ox3xo3ox3oo&#x xo3ox3oo3xo3ox3oo3xo3ox&#x oo3xo3ox3xo3ox3xo3ox3oo&#x | |
(others are only possible within according
| ||
| 10D |
h = 1 |
oo3oo3oo3oo3oo3oo3oo3oo3oo&#x (line = point || point) |
h = sqrt(11/20) = 0.741620 |
ox3oo3oo3oo3oo3oo3oo3oo3oo&#x (ux = point || day) xo3ox3oo3oo3oo3oo3oo3oo3oo&#x (ru = day || reday) oo3xo3ox3oo3oo3oo3oo3oo3oo&#x (bru = reday || breday) oo3oo3xo3ox3oo3oo3oo3oo3oo&#x (tru = breday || treday) oo3oo3oo3xo3ox3oo3oo3oo3oo&#x | |
h = 1/sqrt(5) = 0.447214 |
oo3ox3oo3oo3oo3oo3oo3oo3oo&#x xo3oo3ox3oo3oo3oo3oo3oo3oo&#x xo3oo3oo3oo3oo3oo3oo3oo3ox&#x (ka = day || dual day) oo3xo3oo3ox3oo3oo3oo3oo3oo&#x oo3oo3xo3oo3ox3oo3oo3oo3oo&#x oo3oo3oo3xo3oo3ox3oo3oo3oo&#x (tredaya = treday || inv treday) ox3xo3ox3oo3oo3oo3oo3oo3oo&#x ox3xo3oo3oo3oo3oo3oo3oo3ox&#x ox3oo3xo3ox3oo3oo3oo3oo3oo&#x ox3oo3oo3xo3ox3oo3oo3oo3oo&#x ox3oo3oo3oo3xo3ox3oo3oo3oo&#x ox3oo3oo3oo3oo3xo3ox3oo3oo&#x ox3oo3oo3oo3oo3oo3xo3ox3oo&#x xo3ox3xo3ox3oo3oo3oo3oo3oo&#x xo3ox3oo3xo3ox3oo3oo3oo3oo&#x xo3ox3oo3oo3xo3ox3oo3oo3oo&#x xo3ox3oo3oo3oo3xo3ox3oo3oo&#x xo3ox3oo3oo3oo3oo3xo3ox3oo&#x xo3ox3oo3oo3oo3oo3oo3xo3ox&#x oo3xo3ox3xo3ox3oo3oo3oo3oo&#x oo3xo3ox3oo3xo3ox3oo3oo3oo&#x oo3xo3ox3oo3oo3xo3ox3oo3oo&#x oo3xo3ox3oo3oo3oo3xo3ox3oo&#x oo3oo3xo3ox3xo3ox3oo3oo3oo&#x oo3oo3xo3ox3oo3xo3ox3oo3oo&#x | |
(others are only possible within according
| ||
This simple table shows already, that beyond 9D there survive just 8 families out of all dimensional series in the realm of axially simplicial symmetry. These are:
(where the notational ((...)*) here is used informal in the reg-ex sense of informatics, stating that the contained subdiagramm may be repeated any number of times, from 0 to infinity; and d represents the number of node positions + 1, i.e. the embedding dimensional number). But in contrast to a mere abstract theorem it furthermore provides access to all the individual cases underneath that threshold.
More general, under this symmetry the respecive occuring heights are quantized as follows:
d | 2 3 4 5 6 7 8 9 10 11 12 | +1
----------+-----------------------------------------------+-----
2d*(h(d))2 | 4 6 8 10 12 14 16 18 20 22 24 | +2
| 3 4 5 6 7 8 9 10 11 12 13 | +1
| 4 4 4 4 4 4 4 4 4 | 0
| . 3 2 1 0 -1 -2 -3 | -1
| . 0 -2 -4 -6 -8 | -2
| . -5 -8 -11 | -3
| . -12 | -4
(Although in the left lower triangle no further numbers occur, the dots give a clue, where hypothetical further numbers "4" ought be placed. From these positions thus the respective sequence can be expanded to the right towards the then finally occuring numbers. The respective increment or decrement then is provided at the right.)
As an aside, just because the different monostratic segments of the n-rectified cross-polytopes (under axially simplicial symmetry) occured within the above listings, here is how their segments can be deduced systematically. Consider some o3((o3)*)x((3o)*)3o4o, having n>1 unringed nodes before the x and m>1 unringed nodes between the x and the 4-link. That is, the dimensionality of that n-rectified cross-polytope (and therefore for the to be obtained lace prisms) thus is d=n+m+1. Then we get generally the following lace tower with n+2 vertex layers (i.e. with n+1 monostratic segments)
o3o3o...o3x3o3o3o...o3o (with m unringed nodes at the left before the x and n unringed nodes thereafter at the right) x3o3o...o3o3x3o3o...o3o (the second x occurs at the left) o3x3o...o3o3o3x3o...o3o (each iteration both x'es move one node positon to the right) ....................... o3o...o3x3o3o3o...o3x3o o3o...o3o3x3o3o...o3o3x (the other x disappears at the right) o3o...o3o3o3x3o...o3o3o (with n unringed nodes at the left before the x and m unringed nodes thereafter at the right)
| E.g. | 4D |
ico |
= oct || co || oct |
| 5D |
rat nit |
= inv rap || spid || rap = rap || srip || inv srip || inv rap | |
| 6D |
rag brag brox |
= inv rix || scad || rix = dot || spix || inv spix || dot = rix || sarx || sibrid || inv sarx || inv rix | |
| 7D |
rez barz sez bersa |
= inv ril || staf || ril = inv bril || scal || inv scal || bril = bril || spil || sibpof || inv spil || inv bril = ril || sril || sabril || inv sabril || inv sril || inv ril | |
| etc. | |||
And, quite similar, here comes the systematical derivation of the monostratic segments of the demihypercube (under axially simplicial symmetry). Consider some x3o3o *b((3o)+) with n>1 unringed nodes. That is, the dimensionality of that demihypercube (and therefore for the to be obtained lace prisms) thus is d=n+1. Then we get generally the following lace tower descriptions, which come out to be the same for n even resp. are different representation possibilities for n odd.
x3o3o3o...o3o (with n-1 unringed nodes at the right – simplex first representation) o3o3x3o...o3o (each iteration the x moves two node positon to the right) .............
| E.g. | 3D |
tet |
= {3} || pt
|
| 4D |
hex |
= tet || dual tet | |
| 5D |
hin |
= pen || inv rap || pt | |
| 6D |
hax |
= hix || dot || dual hix | |
| 7D |
hesa |
= hop || bril || inv ril || pt | |
| 8D |
hocto |
= oca || broc || inv broc || dual oca | |
| 9D |
henne |
= ene || brene || inv trene || inv rene || pt | |
| etc. | |||
o3o3o3o3o...o3o (all unringed nodes – vertex first representation) o3x3o3o3o...o3o (the x occurs at the left) o3o3o3x3o...o3o (each iteration the x moves two node positon to the right) ...............
| E.g. | 3D |
tet |
= pt || {3}
|
| 4D |
hex |
= pt || oct || pt | |
| 5D |
hin |
= pt || rap || dual pen | |
| 6D |
hax |
= pt || rix || inv rix || pt | |
| 7D |
hesa |
= pt || ril || inv bril || dual hop | |
| 8D |
hocto |
= pt || roc || he || inv roc || pt | |
| 9D |
henne |
= pt || rene || trene || inv brene || dual ene | |
| etc. | |||
| 2D |
h = 1 |
oo&#x (line = point || point) |
h = sqrt(3)/2 = 0.866025 |
ox&#x ({3} = point || line)
| |
| 3D |
h = 1 |
oo4oo&#x (line = point || point) |
h = 1/sqrt(sqrt(2)) = 0.840896 |
xo4ox&#x (squap = {4} || dual {4}) | |
h = 1/sqrt(2) = 0.707107 |
ox4oo&#x (squippy = point || {4}) | |
| 4D |
h = 1 |
oo3oo4oo&#x (line = point || point) |
h = sqrt[sqrt(2)-3/4] = 0.814993 |
oo3xo4ox&#x (cubaco = cube || co) | |
h = 1/sqrt(2) = 0.707107 |
ox3oo4oo&#x (octpy = point || oct) xo3ox4oo&#x (octaco = oct || co) | |
h = sqrt[2 sqrt(2)-1]/2 = 0.676097 |
xo3oo4ox&#x (octacube = oct || cube) ox3xo4ox&#x (coasirco = co || sirco) | |
h = 1/2 |
oo3oo4ox&#x (cubpy = point || cube) | |
h = 0 (degenerate) |
oo3ox4oo&#x (copy = point || co) | |
| 5D |
h = 1 |
oo3oo3oo4oo&#x (line = point || point) |
h = sqrt[3(sqrt(2)-1)/2] = 0.788239 |
oo3oo3xo4ox&#x (tessarit = tes || rit) | |
h = 1/sqrt(2) = 0.707107 |
ox3oo3oo4oo&#x (hexpy = point || hex) xo3ox3oo4oo&#x (hexaico = hex || ico) oo3xo3ox4oo&#x (icarit = ico || rit) | |
h = sqrt[sqrt(2)-1] = 0.643594 |
oo3xo3oo4ox&#x (icates = tes || ico) ox3oo3xo4ox&#x (rita sidpith = rit || sidpith) xo3ox3xo4ox&#x (ricasrit = rico || srit) | |
h = sqrt[(sqrt(2)-1)/2] = 0.455090 |
xo3oo3oo4ox&#x (tessap = hex || tes) ox3xo3oo4ox&#x (ica sidpith = ico || sidpith) oo3ox3xo4ox&#x (ritasrit = rit || srit) | |
h = 0 (degenerate) |
oo3ox3oo4oo&#x (icopy = point || ico) oo3oo3oo4ox&#x (tespy = point || tes) xo3oo3ox4oo&#x (hexarit = hex || rit) ox3xo3ox4oo&#x (icarico = ico || rico) | |
(others are only possible within according
| ||
| 6D |
h = 1 |
oo3oo3oo3oo4oo&#x (line = point || point) |
h = sqrt[8 sqrt(2)-9]/2 = 0.760544 |
oo3oo3oo3xo4ox&#x (pentarin = pent || rin) | |
h = 1/sqrt(2) = 0.707107 |
ox3oo3oo3oo4oo&#x (tacpy = point || tac) xo3ox3oo3oo4oo&#x (taccarat = tac || rat) oo3xo3ox3oo4oo&#x (ratanit = rat || nit) oo3oo3xo3ox4oo&#x (nitarin = nit || rin) | |
h = sqrt[6 sqrt(2)-7]/2 = 0.609361 |
oo3oo3xo3oo4ox&#x (pentanit = pent || nit) ox3oo3oo3xo4ox&#x (rinascant = rin || scant) xo3ox3oo3xo4ox&#x (spataspan = spat || span) oo3xo3ox3xo4ox&#x (sirna sibrant = sirn || sibrant) | |
h = sqrt[sqrt(2)-5/4] = 0.405233 |
oo3xo3oo3oo4ox&#x (ratapent = rat || pent) ox3oo3xo3oo4ox&#x (nitascant = nit || scant) oo3ox3oo3xo4ox&#x (rinaspan = rin || span) xo3ox3xo3oo4ox&#x (sartaspan = sart || span) xo3oo3ox3xo4ox&#x (spatasirn = spat || sirn) ox3xo3ox3xo4ox&#x (sibranta carnit = sibrant || carnit) | |
h = 0 (degenerate) |
oo3ox3oo3oo4oo&#x (ratpy = point || rat) xo3oo3ox3oo4oo&#x (taccanit = tac || nit) oo3xo3oo3ox4oo&#x (ratarin = rat || rin) ox3xo3ox3oo4oo&#x (ratasart = rat || sart) ox3oo3xo3ox4oo&#x (nitaspat = nit || spat) xo3ox3xo3ox4oo&#x (sarta sibrant = sart || sibrant) | |
(others are only possible within according
| ||
| 7D |
h = 1 |
oo3oo3oo3oo3oo4oo&#x (line = point || point) |
h = sqrt[(5 sqrt(2)-6)/2] = 0.731802 |
oo3oo3oo3oo3xo4ox&#x (axarax = ax || rax) | |
h = 1/sqrt(2) = 0.707107 |
ox3oo3oo3oo3oo4oo&#x (geepy = point || gee) xo3ox3oo3oo3oo4oo&#x (garag = gee || rag) oo3xo3ox3oo3oo4oo&#x (ragabrag = rag || brag) oo3oo3xo3ox3oo4oo&#x (bragabrox = brag || brox) oo3oo3oo3xo3ox4oo&#x (broxarax = brox || rax) | |
h = sqrt[(4 sqrt(2)-5)/2] = 0.573086 |
oo3oo3oo3xo3oo4ox&#x (axabrox = ax || brox) ox3oo3oo3oo3xo4ox&#x (raxastoxog = rax || stoxog) xo3ox3oo3oo3xo4ox&#x (scagascox = scag || scox) oo3xo3ox3oo3xo4ox&#x (spoxa sobpoxog = spox || sobpoxog) oo3oo3xo3ox3xo4ox&#x (sroxa saborx = srox || saborx) | |
h = sqrt[(3 sqrt(2)-4)/2] = 0.348311 |
oo3oo3xo3oo3oo4ox&#x (axabrag = ax || brag) ox3oo3oo3xo3oo4ox&#x (broxastoxog = brox || stoxog) oo3ox3oo3oo3xo4ox&#x (raxascox = rax || scox) xo3ox3oo3xo3oo4ox&#x (spogascox = spog || scox) xo3oo3ox3oo3xo4ox&#x (scagascox = scag || scox) oo3xo3ox3xo3oo4ox&#x (siborga spox = siborg || spox) oo3xo3oo3ox3xo4ox&#x (sobpoxoga srox = sobpoxog || srox) ox3xo3ox3oo3xo4ox&#x (sobpoxoga tapox = sobpoxog || tapox) ox3oo3xo3ox3xo4ox&#x (saborxa topag = saborx || topag) xo3ox3xo3ox3xo4ox&#x (cragacrax = crag || crax) | |
h = 0 (degenerate) |
oo3ox3oo3oo3oo4oo&#x (ragpy = point || rag) xo3oo3ox3oo3oo4oo&#x (gabrag = gee || brag) oo3xo3oo3ox3oo4oo&#x (ragabrox = rag || brox) oo3oo3xo3oo3ox4oo&#x (bragarax = brag || rax) ox3xo3ox3oo3oo4oo&#x (ragasrog = rag || srog) ox3oo3xo3ox3oo4oo&#x (bragaspog = brag || spog) ox3oo3oo3xo3ox4oo&#x (broxascag = brox || scag) xo3ox3xo3ox3oo4oo&#x (srogasiborg = srog || siborg) xo3ox3oo3xo3ox4oo&#x (spoga sobpoxog = spog || sobpoxog) oo3xo3ox3xo3ox4oo&#x (siborga saborx = siborg || saborx) | |
(others are only possible within according
| ||
| 8D |
h = 1 |
oo3oo3oo3oo3oo3oo4oo&#x (line = point || point) |
h = 1/sqrt(2) = 0.707107 |
ox3oo3oo3oo3oo3oo4oo&#x (zeepy = point || zee) xo3ox3oo3oo3oo3oo4oo&#x (zarez = zee || rez) oo3xo3ox3oo3oo3oo4oo&#x (rezabarz = rez || barz) oo3oo3xo3ox3oo3oo4oo&#x (barzasez = barz || sez) oo3oo3oo3xo3ox3oo4oo&#x (sezabersa = sez || bersa) oo3oo3oo3oo3xo3ox4oo&#x (bersa arasa = bersa || rasa) | |
h = sqrt[12 sqrt(2)-15]/2 = 0.701884 |
oo3oo3oo3oo3oo3xo4ox&#x (heptarasa = hept || rasa) | |
h = sqrt[10 sqrt(2)-13]/2 = 0.534354 |
oo3oo3oo3oo3xo3oo4ox&#x (heptabersa = hept || bersa) ox3oo3oo3oo3oo3xo4ox&#x xo3ox3oo3oo3oo3xo4ox&#x oo3xo3ox3oo3oo3xo4ox&#x oo3oo3xo3ox3oo3xo4ox&#x oo3oo3oo3xo3ox3xo4ox&#x | |
h = sqrt[8 sqrt(2)-11]/2 = 0.280048 |
oo3oo3oo3xo3oo3oo4ox&#x (heptasez = hept || sez) ox3oo3oo3oo3xo3oo4ox&#x oo3ox3oo3oo3oo3xo4ox&#x xo3ox3oo3oo3xo3oo4ox&#x xo3oo3ox3oo3oo3xo4ox&#x oo3xo3ox3oo3xo3oo4ox&#x oo3xo3oo3ox3oo3xo4ox&#x oo3oo3xo3ox3xo3oo4ox&#x oo3oo3xo3oo3ox3xo4ox&#x ox3xo3ox3oo3oo3xo4ox&#x ox3oo3xo3ox3oo3xo4ox&#x ox3oo3oo3xo3ox3xo4ox&#x xo3ox3xo3ox3oo3xo4ox&#x xo3ox3oo3xo3ox3xo4ox&#x oo3xo3ox3xo3ox3xo4ox&#x | |
h = 0 (degenerate) |
oo3ox3oo3oo3oo3oo4oo&#x xo3oo3ox3oo3oo3oo4oo&#x oo3xo3oo3ox3oo3oo4oo&#x oo3oo3xo3oo3ox3oo4oo&#x oo3oo3oo3xo3oo3ox4oo&#x ox3xo3ox3oo3oo3oo4oo&#x ox3oo3xo3ox3oo3oo4oo&#x ox3oo3oo3xo3ox3oo4oo&#x ox3oo3oo3oo3xo3ox4oo&#x xo3ox3xo3ox3oo3oo4oo&#x xo3ox3oo3xo3ox3oo4oo&#x xo3ox3oo3oo3xo3ox4oo&#x oo3xo3ox3xo3ox3oo4oo&#x oo3xo3ox3oo3xo3ox4oo&#x oo3oo3xo3ox3xo3ox4oo&#x | |
(others are only possible within according
| ||
This simple table shows already, that we have to be more careful in this case. In fact we have to distinguish between
For A) we can deduce that the quantized heights of the lace prisms in this subset from the axially hypercubic symmetrics always have the following form:
that is, any member from A), i.e. the cases trailing with ..4ox&#x, always will die off beyond d = 19 at most. Cf. the squared height formula as evaluated below:
h^2 | k = 0 1 2 3 4
------+--------------------------------------------------------------------
d = 2 | 0.75
3 | 0.707106781 0.5
4 | 0.664213562 0.457106781 0.25
5 | 0.621320344 0.414213562 0.207106781 0
6 | 0.578427125 0.371320344 0.164213562 -0.042893219 -0.25
7 | 0.535533906 0.328427125 0.121320344 -0.085786438 -0.292893219
8 | 0.492640687 0.285533906 0.078427125 -0.128679656 -0.335786438
9 | 0.449747468 0.242640687 0.035533906 -0.171572875 -0.378679656
10 | 0.406854249 0.199747468 -0.007359313 -0.214466094 -0.421572875
11 | 0.363961031 0.156854249 -0.050252532 -0.257359313 -0.464466094
12 | 0.321067812 0.113961031 -0.093145751 -0.300252532 -0.507359313
13 | 0.278174593 0.071067812 -0.136038969 -0.343145751 -0.550252532
14 | 0.235281374 0.028174593 -0.178932188 -0.386038969 -0.593145751
15 | 0.192388155 -0.014718626 -0.221825407 -0.428932188 -0.636038969
16 | 0.149494937 -0.057611845 -0.264718626 -0.471825407 -0.678932188
17 | 0.106601718 -0.100505063 -0.307611845 -0.514718626 -0.721825407
18 | 0.063708499 -0.143398282 -0.350505063 -0.557611845 -0.764718626
19 | 0.020815280 -0.186291501 -0.393398282 -0.600505063 -0.807611845
20 |-0.022077939 -0.229184720 -0.436291501 -0.643398282 -0.850505063
For the cases of B), i.e. those with trailing ..4oo&#x, the height always is h = sqrt[(2-k)/2] for some integral 0 ≤ k ≤ d-2. – Therefore in total, for non-hyperbolical geometries, the only 7 families of infinite dimensional series of axial hypercubical symmetry are the following ones:
In fact, for any quasiregular figure of the hypercubical symmetry (except the hypercube itself) one derives for any n≥0 and m≥0 the equation: ((o3)n+1)x((3o)m+1)4o = ((ooo3)n)xox3oxo((3ooo)m)4ooo&#xt. Further it even could be expanded to n=−1, when at the right side of the equation the part ((ooo3)n)xox3oxo...&#xt would reduce to just oxo...&#xt. (This then is nothing but the mentioned case of cross-polytopal pyramids.) And conversely it can be expanded to m=−1, when the part ...((3ooo)m)4ooo&#xt becomes ...4oqo&#xt. (This then clearly does not occur above, as the according rotundae would not be CRF anymore.)
| E.g. | 2D |
dual {4}
|
= pt || q-line || pt |
| 3D |
oct co |
= pt || {4} || pt
= dual {4} || q-{4} || dual {4}
| |
| 4D |
hex ico rit |
= pt || oct || pt = oct || co || oct = co || q-cube || co | |
| 5D |
tac rat nit rin |
= pt || hex || pt = hex || ico || hex = ico || rit || ico = rit || q-tes || rit | |
| 6D |
gee rag brag brox rax |
= pt || tac || pt = tac || rat || tac = rat || nit || rat = nit || rin || nit = rin || q-pent || rin | |
| 7D |
zee rez barz sez bersa rasa |
= pt || gee || pt = gee || rag || gee = rag || brag || rag = brag || brox || brag = brox || rax || brox = rax || q-ax || rax | |
| etc. | |||
| 4D |
h = 1 |
oo3oo3oo&#x (line = point || point) |
h = sqrt(5/8) = 0.790569 |
ox3oo3oo&#x (pen = point || tet) xo3ox3oo&#x (rap = tet || oct) | |
h = 1/sqrt(2) = 0.707107 |
oo3ox3oo&#x (octpy = point || oct) xo3oo3ox&#x (hex = tet || dual tet) ox3xo3ox&#x (octaco = oct || co) | |
| 5D |
h = 1 |
oo3oo3oo *b3oo&#x (line = point || point) |
h = 1/sqrt(2) = 0.707107 |
ox3oo3oo *b3oo&#x (hexpy = point || hex) xo3ox3oo *b3oo&#x (hexaico = hex || ico) xo3oo3ox *b3oo&#x (hin = hex || gyro hex) ox3xo3ox *b3oo&#x (icarit = ico || rit) | |
h = 0 (degenerate) |
oo3ox3oo *b3oo&#x (icopy = point || ico) xo3oo3ox *b3ox&#x (hexarit = hex || rit) ox3xo3ox *b3ox&#x (icarico = ico || rico) | |
| 6D |
h = 1 |
oo3oo3oo *b3oo3oo&#x (line = point || point) |
h = 1/sqrt(2) = 0.707107 |
oo3oo3oo *b3oo3ox&#x (tacpy = point || tac) xo3oo3ox *b3oo3oo&#x (hax = hin || gyro hin) oo3xo3oo *b3ox3oo&#x (ratanit = rat || nit) oo3oo3oo *b3xo3ox&#x (taccarat = tac || rat) ox3xo3ox *b3oo3oo&#x (nitarin = nit || rin) | |
h = sqrt(3/8) = 0.612372 |
ox3oo3oo *b3oo3oo&#x (hinpy = point || hin) xo3ox3oo *b3oo3oo&#x (hinanit = hin || nit) xo3oo3oo *b3ox3oo&#x (hinro = hin || rat) xo3oo3oo *b3oo3ox&#x (dijak = tac || hin) ox3xo3oo *b3ox3oo&#x (nitasirhin = nit || sirhin) ox3xo3oo *b3oo3ox&#x (nitasiphin = nit || siphin) ox3oo3oo *b3xo3ox&#x (ratasiphin = rat || siphin) xo3ox3oo *b3xo3ox&#x (sirhinasart = sirhin || sart) | |
h = 0 (degenerate) |
oo3oo3oo *b3ox3oo&#x (ratpy = point || rat) oo3xo3oo *b3oo3ox&#x (taccanit = tac || nit) ox3oo3ox *b3xo3oo&#x (ratarin = rat || rin) ox3oo3xo *b3oo3ox&#x (hinalsiphin = hin || gyro siphin) oo3ox3oo *b3xo3ox&#x (ratasart = rat || sart) xo3ox3xo *b3oo3xo&#x (nitaspat = nit || spat) ox3xo3ox *b3ox3xo&#x (sarta sibrant = sart || sibrant) | |
(others are only possible within according
| ||
| 7D |
h = 1 |
oo3oo3oo *b3oo3oo3oo&#x (line = point || point) |
h = 1/sqrt(2) = 0.707107 |
oo3oo3oo *b3oo3oo3ox&#x (geepy = point || gee) xo3oo3ox *b3oo3oo3oo&#x (hesa = hax || gyro hax) oo3xo3oo *b3ox3oo3oo&#x (bragabrox = brag || brox) oo3oo3oo *b3xo3ox3oo&#x (ragabrag = rag || brag) oo3oo3oo *b3oo3xo3ox&#x (garag = gee || rag) ox3xo3ox *b3oo3oo3oo&#x (broxarax = brox || rax) | |
h = 1/2 |
ox3oo3oo *b3oo3oo3oo&#x (haxpy = point || hax) xo3ox3oo *b3oo3oo3oo&#x (haxabrox = hax || brox) xo3oo3oo *b3ox3oo3oo&#x (haxabrag = hax || brag) xo3oo3oo *b3oo3ox3oo&#x (haxarag = hax || rag) xo3oo3oo *b3oo3oo3ox&#x (gahax = gee || hax) ox3xo3oo *b3ox3oo3oo&#x (broxasirhax = brox || sirhax) ox3xo3oo *b3oo3ox3oo&#x (broxa sophax = brox || sophax) ox3xo3oo *b3oo3oo3ox&#x (broxasochax = brox || sochax) ox3oo3oo *b3xo3ox3oo&#x (bragasophax = brag || sophax) ox3oo3oo *b3xo3oo3ox&#x (bragasochax = brag || sochax) ox3oo3oo *b3oo3xo3ox&#x (ragasochax = rag || sochax) xo3ox3oo *b3xo3ox3oo&#x (sirhaxa siborg = sirhax || siborg) xo3ox3oo *b3xo3oo3ox&#x (spogasirhax = spog || sirhax) xo3ox3oo *b3oo3xo3ox&#x (sophaxaspog = sophax || spog) xo3oo3oo *b3ox3xo3ox&#x (sophaxasrog = sophax || srog) xo3ox3oo *b3xo3ox3xo&#x (siborga crohax = siborg || crohax) | |
h = 0 (degenerate) |
oo3oo3oo *b3oo3ox3oo&#x (ragpy = point || rag) oo3xo3oo *b3oo3ox3oo&#x (ragabrox = rag || brox) oo3oo3oo *b3xo3oo3ox&#x (gabrag = gee || brag) ox3oo3ox *b3xo3oo3oo&#x (bragarax = brag || rax) ox3oo3xo *b3oo3oo3ox&#x (haxalsochax = hax || gyro sochax) oo3ox3oo *b3xo3oo3ox&#x (bragaspog = brag || spog) oo3oo3oo *b3ox3xo3ox&#x (ragasrog = rag || srog) xo3ox3xo *b3oo3oo3xo&#x (broxascag = brox || scag) xo3oo3ox *b3xo3ox3oo&#x (sophaxalsirhax = sophax || gyro sirhax) xo3oo3ox *b3oo3xo3ox&#x (sochaxalsophax = sochax || gyro sophax) oo3xo3oo *b3ox3xo3ox&#x (srogasiborg = srog || siborg) xo3ox3xo *b3xo3ox3oo&#x (siborga saborx = siborg || saborx) xo3ox3xo *b3oo3xo3ox&#x (spoga sobpoxog = spog || sobpoxog) | |
(others are only possible within according
| ||
| 8D |
h = 1 |
oo3oo3oo *b3oo3oo3oo3oo&#x (line = point || point) |
h = 1/sqrt(2) = 0.707107 |
oo3oo3oo *b3oo3oo3oo3ox&#x (zeepy = point || zee) xo3oo3ox *b3oo3oo3oo3oo&#x (hocto = hesa || gyro hesa) oo3xo3oo *b3ox3oo3oo3oo&#x (sezabersa = sez || bersa) oo3oo3oo *b3xo3ox3oo3oo&#x (barzasez = barz || sez) oo3oo3oo *b3oo3xo3ox3oo&#x (rezabarz = rez || barz) oo3oo3oo *b3oo3oo3xo3ox&#x (zarez = zee || rez) ox3xo3ox *b3oo3oo3oo3oo&#x (bersa arasa = bersa || rasa) | |
h = 1/sqrt(8) = 0.353553 |
ox3oo3oo *b3oo3oo3oo3oo&#x (hesapy = point || hesa) xo3ox3oo *b3oo3oo3oo3oo&#x (hesa abersa = hesa || bersa) xo3oo3oo *b3ox3oo3oo3oo&#x (hesa asez = hesa || sez) xo3oo3oo *b3oo3ox3oo3oo&#x (hesa abarz = hesa || barz) xo3oo3oo *b3oo3oo3ox3oo&#x (hesa arez = hesa || rez) xo3oo3oo *b3oo3oo3oo3ox&#x (zahesa = zee || hesa) ox3xo3oo *b3ox3oo3oo3oo&#x (bersa asirhesa = bersa || sirhesa) ox3xo3oo *b3oo3ox3oo3oo&#x (bersa asphosa = bersa || sphosa) ox3xo3oo *b3oo3oo3ox3oo&#x (bersa asochesa = bersa || sochesa) ox3xo3oo *b3oo3oo3oo3ox&#x (bersa asuthesa = bersa || suthesa) xo3ox3oo *b3xo3ox3oo3oo&#x (sirhesa astrasaz = sirhesa || strasaz) xo3ox3oo *b3xo3oo3ox3oo&#x (sirhesa asibpaz = sirhesa || sibpaz) xo3ox3oo *b3xo3oo3oo3ox&#x (scazasirhesa = scaz || sirhesa) xo3ox3oo *b3oo3xo3ox3oo&#x (sphosa asibpaz = sphosa || sibpaz) xo3ox3oo *b3oo3xo3oo3ox&#x (sphosa ascaz = sphosa || scaz) xo3ox3oo *b3oo3oo3xo3ox&#x (sochesa ascaz = sochesa || scaz) xo3oo3oo *b3ox3xo3ox3oo&#x (sphosa asebraz = sphosa || sebraz) xo3oo3oo *b3ox3xo3oo3ox&#x (spazasphosa = spaz || sphosa) xo3oo3oo *b3ox3oo3xo3ox&#x (sochesa aspaz = sochesa || spaz) xo3oo3oo *b3oo3ox3xo3ox&#x (sochesa asarz = sochesa || sarz) ox3xo3oo *b3ox3xo3ox3oo&#x (strasazacrohesa = strasaz || crohesa) ox3xo3oo *b3ox3xo3oo3ox&#x (turhesa astrasaz = turhesa || strasaz) ox3xo3oo *b3ox3oo3xo3ox&#x (sibpaza turhesa = sibpaz || turhesa) ox3xo3oo *b3oo3ox3xo3ox&#x (sibpaza tuphesa = sibpaz || tuphesa) ox3oo3oo *b3xo3ox3xo3ox&#x (sebraza tuphesa = sebraz || tuphesa) xo3ox3oo *b3xo3ox3xo3ox&#x (crohesa acraze = crohesa || craze) | |
h = 0 (degenerate) |
oo3oo3oo *b3oo3oo3ox3oo&#x oo3xo3oo *b3oo3ox3oo3oo&#x oo3oo3oo *b3xo3oo3ox3oo&#x oo3oo3oo *b3oo3xo3oo3ox&#x oo3ox3oo *b3xo3oo3oo3ox&#x oo3oo3oo *b3ox3xo3oo3ox&#x oo3oo3oo *b3oo3ox3xo3ox&#x xo3ox3xo *b3oo3oo3oo3xo&#x oo3xo3oo *b3ox3xo3ox3oo&#x oo3xo3oo *b3ox3oo3xo3ox&#x oo3oo3oo *b3xo3ox3xo3ox&#x ox3xo3ox *b3ox3xo3oo3oo&#x ox3xo3ox *b3oo3ox3xo3oo&#x ox3xo3ox *b3oo3oo3ox3xo&#x | |
(others are only possible within according
| ||
From this small listing it becomes apparent that for the height of any lace prism of overall axially demihypercubic symmetry with ox at one of its tiny legs and with oo (or likewise with xx) at the other leg will follow h2(d) = (9-d)/8. I.e. according series remain finite and would stop with d = 9. – The most prominent member of these stopping cases for sure is fy = 42,1 = xo3oo3oo *b3oo3oo3oo3ox3oo&#zx = hocto || rek.
Therefore there remains just 1 single additional infinite series of fundamental lace prisms, which truely is describable in this type of symmetry. In fact, any other already would have been listed within the axially hypercubical symmetry above (according to the nodewise equivalence A3B3A *b3... = ...3B3A4o). That single additional one thus is
Their fundamental representant each happens to be nothing but the axial description of the according next-dimensional demihypercube itself.
| E.g. | 4D |
hex |
= tet || dual tet |
| 5D |
hin |
= hex || gyro hex | |
| 6D |
hax |
= hin || gyro hin | |
| 7D |
hesa |
= hax || gyro hax | |
| 8D |
hocto |
= hesa || gyro hesa | |
| 9D |
henne |
= hocto || gyro hocto | |
| etc. | |||
| 3D |
h = 1 |
oo5oo&#x (line = point || point) |
h = sqrt[(5+sqrt(5))/10] = 0.850651 |
xo5ox&#x (pap = {5} || gyro {5}) | |
h = sqrt[(5-sqrt(5))/10] = 0.525731 |
ox5oo&#x (peppy = point || {5}) | |
| 4D |
h = 1 |
oo3oo5oo&#x (line = point || point) |
h = (1+sqrt(5))/4 = 0.809017 |
oo3xo5ox&#x (doaid = doe || id) | |
h = 1/2 |
xo3oo5ox&#x (ikadoe = ike || doe) ox3xo5ox&#x (idasrid = id || srid) | |
h = (sqrt(5)-1)/4 = 0.309017 |
ox3oo5oo&#x (ikepy = point || ike) xo3ox5oo&#x (ikaid = ike || id) | |
(others are only possible within according
| ||
| 5D |
h = 1 |
oo3oo3oo5oo&#x (line = point || point) |
h = 0 (degenerate) |
oo3xo3oo5ox&#x (roxahi = rox || hi) oo3oo3xo5ox&#x (hiarahi = hi || rahi) ox3oo3xo5ox&#x (rahia sidpixhi = rahi || sidpixhi) xo3ox3xo5ox&#x (srixasrahi = srix || srahi) | |
(others are only possible within according
| ||
| 7D |
h = 1 |
oo3oo3oo3oo3oo *c3oo&#x (line = point || point) |
h = 1/sqrt(3) = 0.577350 |
ox3oo3oo3oo3oo *c3oo&#x (jakpy = point || jak) xo3ox3oo3oo3oo *c3oo&#x (jakarojak = jak || rojak) xo3oo3oo3oo3ox *c3oo&#x (jaka = jak || inv jak) xo3oo3oo3oo3oo *c3ox&#x (jakamo = jak || mo) oo3xo3ox3oo3oo *c3oo&#x (rojakaram = rojak || ram) oo3xo3oo3ox3oo *c3oo&#x (rojaka = rojak || inv rojak) oo3xo3oo3oo3oo *c3ox&#x (moarojak = mo || rojak) ox3xo3oo3oo3ox *c3oo&#x (rojakatrim = rojak || trim) ox3xo3oo3oo3oo *c3ox&#x (rojaka hejak = rojak || hejak) ox3oo3xo3ox3oo *c3oo&#x (ramashopjak = ram || shopjak) ox3oo3xo3oo3oo *c3ox&#x (hejaka ram = hejak || ram) oo3ox3xo3oo3oo *c3ox&#x (ramaharjak = ram || harjak) xo3ox3xo3ox3oo *c3oo&#x (sirjakabarm = sirjak || barm) xo3ox3xo3oo3oo *c3ox&#x (sirjaka harjak = sirjak || harjak) xo3ox3oo3xo3ox *c3oo&#x (shopjaka = shopjak || inv shopjak) | |
h = 0 (degenerate) |
oo3oo3oo3oo3oo *c3ox&#x xo3oo3oo3ox3oo *c3oo&#x oo3oo3xo3oo3oo *c3ox&#x ox3xo3oo3ox3oo *c3oo&#x ox3oo3xo3oo3ox *c3oo&#x oo3ox3xo3ox3oo *c3oo&#x xo3ox3xo3oo3ox *c3oo&#x xo3oo3ox3oo3xo *c3xo&#x ox3xo3ox3xo3ox *c3oo&#x ox3xo3ox3xo3oo *c3xo&#x ox3xo3ox3oo3xo *c3xo&#x ox3xo3ox3xo3ox *c3xo&#x | |
(others are only possible within according
| ||
| 8D |
h = 1 |
oo3oo3oo3oo *c3oo3oo3oo&#x (line = point || point) |
h = 1/2 |
oo3oo3oo3oo *c3oo3oo3ox&#x (naqpy = point || naq) xo3oo3oo3ox *c3oo3oo3oo&#x (laqalin = laq || lin) xo3oo3oo3oo *c3oo3oo3ox&#x (naqalaq = laq || naq) oo3xo3oo3ox *c3oo3oo3oo&#x (linarolaq = lin || rolaq) oo3xo3oo3oo *c3ox3oo3oo&#x (rolaqabranq = rolaq || branq) oo3oo3xo3oo *c3ox3oo3oo&#x (branqalanq = branq || lanq) oo3oo3oo3xo *c3oo3ox3oo&#x (linaranq = lin || ranq) oo3oo3oo3oo *c3xo3ox3oo&#x (ranqabranq = ranq || branq) oo3oo3oo3oo *c3oo3xo3ox&#x (naqaranq = naq || ranq) ox3xo3oo3ox *c3oo3oo3oo&#x (rolaqashilq = rolaq || shilq) ox3xo3oo3oo *c3oo3oo3ox&#x (stanqarolaq = stanq || rolaq) ox3oo3xo3ox *c3oo3oo3oo&#x (shilqalanq = shilq || lanq) ox3oo3xo3oo *c3ox3oo3oo&#x (lanqashiplaq = lanq || shiplaq) ox3oo3oo3oo *c3xo3ox3oo&#x (branqascalq = branq || scalq) ox3oo3oo3oo *c3oo3xo3ox&#x (ranqastanq = ranq || stanq) oo3ox3xo3ox *c3oo3oo3oo&#x (lanqahirlaq = lanq || hirlaq) oo3ox3xo3oo *c3oo3oo3ox&#x (shocanqalanq = shocanq || lanq) oo3oo3ox3xo *c3oo3xo3oo&#x (lanqaharnaq = lanq || harnaq) oo3oo3oo3ox *c3xo3oo3ox&#x (branqashanq = branq || shanq) xo3ox3xo3ox *c3oo3oo3oo&#x (hirlaqa sirlaq = hirlaq || sirlaq) xo3ox3xo3oo *c3ox3oo3oo&#x (sirlaqa berlin = sirlaq || berlin) xo3ox3oo3oo *c3xo3ox3oo&#x xo3ox3oo3oo *c3oo3xo3ox&#x oo3xo3ox3oo *c3xo3ox3oo&#x oo3xo3ox3oo *c3oo3xo3ox&#x oo3oo3xo3ox *c3ox3oo3xo&#x oo3oo3oo3xo *c3ox3xo3ox&#x oo3oo3ox3xo *c3xo3ox3xo&#x | |
h = 0 (degenerate) |
ox3oo3oo3oo *c3oo3oo3oo&#x xo3ox3oo3oo *c3oo3oo3oo&#x xo3oo3oo3oo *c3oo3ox3oo&#x oo3xo3ox3oo *c3oo3oo3oo&#x oo3xo3oo3oo *c3oo3ox3oo&#x oo3oo3oo3xo *c3ox3oo3oo&#x oo3oo3oo3xo *c3oo3oo3ox&#x ox3xo3oo3oo *c3oo3ox3oo&#x ox3oo3xo3oo *c3oo3ox3oo&#x ox3oo3oo3oo *c3xo3oo3ox&#x oo3ox3xo3oo *c3oo3ox3oo&#x oo3oo3ox3xo *c3xo3oo3oo&#x oo3oo3ox3xo *c3oo3oo3xo&#x oo3oo3oo3ox *c3xo3ox3oo&#x oo3oo3oo3ox *c3oo3xo3ox&#x xo3ox3xo3oo *c3oo3ox3oo&#x xo3ox3oo3oo *c3xo3oo3ox&#x xo3oo3ox3xo *c3oo3oo3xo&#x xo3oo3oo3oo *c3ox3xo3ox&#x oo3xo3ox3oo *c3xo3oo3ox&#x oo3oo3xo3ox *c3ox3xo3oo&#x oo3oo3xo3ox *c3oo3ox3xo&#x ox3xo3ox3xo *c3oo3oo3xo&#x ox3xo3ox3oo *c3xo3oo3ox&#x ox3xo3oo3oo *c3ox3xo3ox&#x ox3oo3xo3ox *c3ox3xo3oo&#x ox3oo3xo3ox *c3oo3ox3xo&#x ox3oo3xo3oo *c3ox3xo3ox&#x oo3ox3xo3oo *c3ox3xo3ox&#x xo3ox3xo3ox *c3ox3xo3oo&#x xo3ox3xo3ox *c3oo3ox3xo&#x xo3ox3xo3oo *c3ox3xo3ox&#x | |
(others are only possible within according
| ||
| 9D |
h = 1 |
oo3oo3oo3oo *c3oo3oo3oo3oo&#x (line = point || point) |
h = 0 (degenerate) |
oo3oo3oo3oo *c3oo3oo3oo3ox&#x xo3oo3oo3ox *c3oo3oo3oo3oo&#x xo3oo3oo3oo *c3oo3oo3ox3oo&#x xo3oo3oo3oo *c3oo3oo3oo3ox&#x oo3xo3oo3ox *c3oo3oo3oo3oo&#x oo3xo3oo3oo *c3ox3oo3oo3oo&#x oo3xo3oo3oo *c3oo3ox3oo3oo&#x oo3oo3xo3oo *c3ox3oo3oo3oo&#x oo3oo3oo3xo *c3oo3ox3oo3oo&#x oo3oo3oo3xo *c3oo3oo3ox3oo&#x oo3oo3oo3oo *c3xo3ox3oo3oo&#x oo3oo3oo3oo *c3oo3xo3ox3oo&#x oo3oo3oo3oo *c3oo3oo3xo3ox&#x ox3xo3oo3ox *c3oo3oo3oo3oo&#x ox3xo3oo3oo *c3oo3oo3ox3oo&#x ox3xo3oo3oo *c3oo3oo3oo3ox&#x ox3oo3xo3ox *c3oo3oo3oo3oo&#x ox3oo3xo3oo *c3ox3oo3oo3oo&#x ox3oo3xo3oo *c3oo3ox3oo3oo&#x ox3oo3oo3oo *c3xo3ox3oo3oo&#x ox3oo3oo3oo *c3xo3oo3ox3oo&#x ox3oo3oo3oo *c3oo3xo3ox3oo&#x ox3oo3oo3oo *c3oo3xo3oo3ox&#x ox3oo3oo3oo *c3oo3oo3xo3ox&#x oo3ox3xo3ox *c3oo3oo3oo3oo&#x oo3ox3xo3oo *c3oo3oo3ox3oo&#x oo3ox3xo3oo *c3oo3oo3oo3ox&#x oo3ox3oo3oo *c3xo3oo3oo3ox&#x oo3oo3ox3xo *c3oo3xo3oo3oo&#x oo3oo3ox3xo *c3oo3oo3xo3oo&#x oo3oo3oo3ox *c3xo3oo3ox3oo&#x oo3oo3oo3ox *c3xo3oo3oo3ox&#x oo3oo3oo3ox *c3oo3xo3oo3ox&#x xo3ox3xo3ox *c3oo3oo3oo3oo&#x xo3ox3xo3oo *c3ox3oo3oo3oo&#x xo3ox3xo3oo *c3oo3ox3oo3oo&#x xo3ox3oo3oo *c3xo3ox3oo3oo&#x xo3ox3oo3oo *c3xo3oo3ox3oo&#x xo3ox3oo3oo *c3oo3xo3ox3oo&#x xo3ox3oo3oo *c3oo3xo3oo3ox&#x xo3ox3oo3oo *c3oo3oo3xo3ox&#x xo3oo3ox3xo *c3oo3oo3oo3xo&#x xo3oo3ox3oo *c3xo3oo3oo3ox&#x xo3oo3oo3oo *c3ox3xo3oo3ox&#x xo3oo3oo3oo *c3oo3ox3xo3ox&#x oo3xo3ox3oo *c3xo3ox3oo3oo&#x oo3xo3ox3oo *c3xo3oo3ox3oo&#x oo3xo3ox3oo *c3oo3xo3ox3oo&#x oo3xo3ox3oo *c3oo3xo3oo3ox&#x oo3xo3ox3oo *c3oo3oo3xo3ox&#x oo3xo3oo3oo *c3ox3xo3ox3oo&#x oo3xo3oo3oo *c3ox3oo3xo3ox&#x oo3oo3xo3ox *c3ox3oo3xo3oo&#x oo3oo3xo3ox *c3ox3oo3oo3xo&#x oo3oo3xo3ox *c3oo3ox3oo3xo&#x oo3oo3oo3xo *c3ox3xo3ox3oo&#x oo3oo3oo3xo *c3ox3xo3oo3ox&#x oo3oo3oo3xo *c3ox3oo3xo3ox&#x oo3oo3oo3xo *c3oo3ox3xo3ox&#x ox3xo3ox3xo *c3oo3oo3oo3xo&#x ox3xo3ox3oo *c3xo3oo3oo3ox&#x ox3xo3oo3oo *c3ox3xo3oo3ox&#x ox3xo3oo3oo *c3oo3ox3xo3ox&#x ox3oo3xo3ox *c3ox3xo3oo3oo&#x ox3oo3xo3ox *c3oo3ox3xo3oo&#x ox3oo3xo3ox *c3oo3oo3ox3xo&#x ox3oo3xo3oo *c3ox3xo3ox3oo&#x ox3oo3xo3oo *c3ox3oo3xo3ox&#x ox3oo3oo3oo *c3xo3ox3xo3ox&#x oo3ox3xo3oo *c3ox3xo3oo3ox&#x oo3ox3xo3oo *c3oo3ox3xo3ox&#x oo3oo3ox3xo *c3xo3ox3xo3oo&#x oo3oo3ox3xo *c3xo3ox3oo3xo&#x oo3oo3ox3xo *c3xo3oo3ox3xo&#x oo3oo3ox3xo *c3oo3xo3ox3xo&#x oo3oo3oo3ox *c3xo3ox3xo3ox&#x xo3ox3xo3ox *c3ox3xo3oo3oo&#x xo3ox3xo3ox *c3oo3ox3xo3oo&#x xo3ox3xo3ox *c3oo3oo3ox3xo&#x xo3ox3xo3oo *c3ox3xo3ox3oo&#x xo3ox3xo3oo *c3ox3oo3xo3ox&#x xo3ox3oo3oo *c3xo3ox3xo3ox&#x oo3xo3ox3oo *c3xo3ox3xo3ox&#x oo3oo3xo3ox *c3ox3xo3ox3xo&#x | |
(others are only possible within according
| ||
| 5D |
h = 1 |
oo3oo4oo3oo&#x (line = point || point) |
h = sqrt[4 sqrt(2)-5] = 0.810465 |
oo3xo4ox3oo&#x (ricoa = rico || gyro rico) | |
h = sqrt[sqrt(2)-1] = 0.643594 |
xo3oo4oo3ox&#x (icoap = ico || dual ico) ox3xo4oo3ox&#x (ricoaspic = rico || spic) xo3ox4xo3ox&#x (sricoa = srico || gyro srico) | |
h = 0 (degenerate) |
ox3oo4oo3oo&#x (icopy = point || ico) xo3ox4oo3oo&#x (icoarico = ico || rico) | |
(others are only possible within according
| ||
© 2004-2024 | top of page |