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Within a 2013 article M. Dutour investigates somehow "extremal" Delone cells of laminated D_{n} lattices (i.e. infinite stacks of such lattices which are themself lattices). The first cell type, he is describing, mainly is a generalization of jak (Gosset polypeton 2_{2,1}, i.e. x3o3o3o3o *c3o), the Delone cell of lattice E_{6}. That polytope lends for a lace tower description as pt || (pseudo) hin || tac. Using the according extended Dynkin diagram notation of W. Krieger, this latter description can also be given symbolically as oxo3ooo3ooo *b3ooo3oox&#xt.
Based on the above observation, he quite generally piles up the tower stack of a point P, a demihypercube H, and a crosspolytope C: P || H || C = oxo3ooo3ooo *b3ooo...ooo3oox&#yt with some lacing length y depending on the layer distance. He then relates them to the above laminated lattices and states the further restriction to n≥6 even (where n represents the dimension of embedding space, i.e. of the full laminate and not that of the individual layers). But that additional restriction is only related to his extremality consideration. E.g. it also guarantees the inversion symmetry of the medial layer.
In contrast we will consider these here mainly as mere polytopes. Then we need not to bother about his restriction. Further it might be mentioned that each of the layers can be considered as a tower itself, which thus provides a lace city display. Thereby we can interdimensionally relate the mere tower and the lace city components: P || C_{n} || P = C_{n+1} and also H_{n} || alternate H_{n} = H_{n+1}.
P where (n≥4): P = o3o3o *b3o...o3o H h H = x3o3o *b3o...o3o h = o3o3x *b3o...o3o P C P C = o3o3o *b3o...o3x
From this display one easily derives the coordinates. Using unit edges within the layers and some additional layer distance d one gets:
P (top layer): | (0; 0, 0, 0, ...; +d) |
H (equatorial left): | (-1/sqrt(8); -1/sqrt(8), 1/sqrt(8), 1/sqrt(8), ...; 0) & all even changes of sign in the medial block of coordinates |
h (equatorial right): | (1/sqrt(8); 1/sqrt(8), 1/sqrt(8), 1/sqrt(8), ...; 0) & all even changes of sign in the medial block of coordinates |
P (bottom left/right): | (±1/sqrt(2); 0, 0, 0, ...; -d) |
C (bottom central): | (0; 1/sqrt(2), 0, 0, ...; -d) & all permutations and changes of sign in the medial block of coordinates |
Here the first coordinate each represents the horizontal alignment, the last the vertical. The medial block of coordinates then is the not represented orthogonal space of this projection. Coordinates are chosen such that we have layerwise unit-edged polytopes. d thereby is the layer distance. These coordinates are valid from n≥3. Note that moreover for n>3 the center of both H resp. h is given by setting the medial block coordinates to 0. Thus these centerpoints then are exactly the midpoints of the single top P and the respective bottom P. Therefrom we further derive that these latteral sloped towers P || H || P then always fall into a single subspace, i.e. represent H-dipyramids. (For n=4 these ridges then represent – assuming all unit edges – 60°-rhombs, for n=5 tetes, and for n=6 these happen to become regular tacs; etc.)
Thus the length of the lacing edge of the upper segment has size y = sqrt[(n-1)/8 + d^{2}]. Obviously the same holds both for the distance to the nearer bottom P and the near vertex of the bottom central C. Therefore, using as mere polytopes the above mentioned restriction to even n is not necessary, and even the lower bound can be downsized to n≥3. – If we additionally would consider all unit edged polytopes only, then the layer distance d should vary in accordance to n due to the further restriction y = 1, providing d = sqrt[(9-n)/8].
n | P || H || C | Notes |
3 |
oxx oox&#xt | Here the medial block of above coordinates has just the count 1. Thus neither even changes of sign nor permutations are possible there. The bottom layer degenerates into a mere square, the equatorial one into a single unit edge. The bottom segment becomes a trip. Whereas the top segment even becomes a mere (dimensionally degenerate) triangle. Moreover it is evident, that the derived "polyhedron" is concave. |
4 |
oxo3oox3ooo&#xt |
The equatorial pseudo layer here is the tet. Thus the upper segment represents the
segmentochoron K-4.1: pen, the lower one is
K-4.5: rap. But as in this external blend the respective lacing
cells tet (upper segment) and oct (lower segment – those are the only ones,
which are fully connected to the equatorial layer) happen to become corealmic, they can be united.
But then their lacing faces in turn are coplanar, resulting in 60°-rhombs. Thus, even so this polychoron is convex, it still is not a CRF. |
5 |
oxo3ooo3ooo *b3oox&#xt |
The equatorial pseudo layer here is the hex, while the bottom layer is the same solid, they simply are in
different orientations. Thus the lower segment is just hin. The upper one represents the
segmentoteron hexpy.
The lacing elements of the upper segment clearly are all pens. The diteral angle between those and the base
of that pyramid equals arccos(1/sqrt(5)) = 63.434949°.
The lacing elements of the lower segment, which are fully connected to the equatorial layer, are hexes and
pens. The diteral angle between the hexes here is 90°, while that between the equatorial hex and the lacing
pens is arccos[-1/sqrt(5)] = 116.565051°. Thus the upper lacing pens and the directly connected lower pens are corealmic and connect into
tetes. Accordingly the total facet count here becomes (8+8) pens + (8+1) hexes + 8 tetes. This polyteron clearly is CRF. |
6 |
oxo3ooo3ooo *b3ooo3oox&#xt | This now is just the uniform and thus higher-symmetrical jak (Gosset polytope 2_{2,1}). It clearly is a convex polyexon. The equatorial pseudo facet here is hin. That one dissects the lacing tacs, which strech through both segments, into pairs of (thus corealmic) hexpies. The total facet count is 27 tacs + 72 hixes. |
7 |
oxo3ooo3ooo *b3ooo3ooo3oox&#xt |
Here likewise the upper segment is obviously just the segmentoexon haxpy.
And the lower one is gahax. Again the lacing hinpies
of either segment will blend together into hints. The total facet count here reads 12 hints + (32+160+32) hops + 60 hexascs + (32+1) gees. |
8 |
oxo3ooo3ooo *b3ooo3ooo3ooo3oox&#xt | The total facet count here reads 14 haxts + (64+560+64) ocas + 84 hinscs + 280 hexetes + (64+1) zees. |
9 (degenerate) |
.xo3.oo3.oo *b3.oo3.oo3.oo3.oo3.ox&#zx ox.3oo.3oo. *b3oo.3oo.3oo.3oo.3oo.&#zx |
The above equation for d freely provides that the 8D unit-edged demicube (hocto) and the 8D unit-edged crosspolytope
(ek) can be symmetrically superimposed as a compound and their hull then still will be a
unit-edged 8D polytope (hauhocto) with 128+16=144 vertices. And on the other hand, the 8D unit-edged demicube (hocto) itself is decomposable into all unit-edged centri-pyramids underneath all of its facets, i.e. into 128 days plus 16 hesa-pyramids. Esp. hocto needs to have a circumradius of 1. |
Just for completeness – and more as an aside tribute to Dutour's original aim – the corresponding crystallographic d = d(n) here gets derived as well. That one asks generally to have y = y(d) different from x. Instead the corresponding restriction is given by the existance of a well-defined circumradius, running through all vertices. (This then also is the definition of being a Delone cell.)
By consideration of the 3 P's of the lace city display, it becomes evident that the according circumcenter M of these 3 points ought be some (0; 0, 0, 0, ...; -m) and then r = MP_{top} = d+m as well as r = MP_{bottom} = sqrt((d-m)^{2} + 1/2) in either of the 2 cases. Thus one derives d^{2} + 2dm + m^{2} = d^{2} - 2dm + m^{2} + 1/2 or equivalentely m = 1/(8d). And thus for the circumradius itself r = d+m = (8d^{2} + 1)/(8d).
As P || C || P in the bottom layer defines a uniform crosspolytope, it becomes evident that our circumsphere also connects to all the vertices of C already. Whereas, in order to connect as well to the vertices of H (and then also of h) of the medial demihypercube H || h, we further have to evaluate the distance M ver(H) = sqrt((n-1)/8 + m^{2}). Thus we get from r = M ver(H), using the above derived value for m, the equation (8d^{2} + 1)^{2}/(8d)^{2} = (n-1)/8 + 1/(8d)^{2} or equivalently 64d^{4} + 16d^{2} + 1 = 8(n-1)d^{2} + 1. Whenever d > 0 this runs down to 8d^{2} = n-3 or the final result d = sqrt((n-3)/8).
We even could evaluate now the corresponding (crystallographic) radius formula from r = r(d) = r(d(n)) as r = (8d^{2} + 1)/(8d) = (n-2)/sqrt(8(n-3)). Or we could also provide the circumcenter M = M(d) = M(d(n)) explicitely as M = (0; 0, 0, 0, ...; -1/sqrt(8(n-3))). Further we can evaluate also the lacing edge size y = y(d) = y(d(n)) to y = sqrt((n-1)/8 + d^{2}) = sqrt((n-1)/8 + (n-3)/8) = sqrt(n-2)/2. (As throughout, all provided absolute values were scaled according to layerwise unit edges.)
It is obvious, that the single n, which solves both equations to the same value d (wrt. all unit edged at the one and wrt. an existing circumsphere on the other hand) ought have the same distance from 9 and 3. Thence it is n=6, in agreement to our previous knowledge that the n=6 fellow here is the well-known uniform figure jak = 2_{2,1}.
He also consideres a further infinite series of polytopes, which he constructs again as a lace tower according to P || D || inv D || P, where P again is a mere point, while D is the respective Dutour polytope of the first series of one dimension less. Accordingly here he then states that n≥7 odd.
Just as in the first series, this definition here too is guided by n=7, where this definition describes nothing but the uniform polyexon naq (Gosset polytope 3_{2,1}, i.e. o3o3o3o *c3o3o3x), the Delone cell of lattice E_{7}.
Using the lace tower description of D from the former section, we can provide here an lace city display too:
P where (n≥4): P H C P = o3o3o *b3o...o3o H = x3o3o *b3o...o3o C h P h = o3o3x *b3o...o3o C = o3o3o *b3o...o3x P
Within this display the extremal segments show that this provides interesting polytopes only, when all lacing edges y between the layers of any of these segments are of the same size. But as those are pyramids with a Dutour polytope of the first series as its base, this asks for a circumsphere of the base. And this in turn then implies that the crystallographical spacings are to be used! Accordingly we get 3 general edge classes: x, the ones within each lace city spot i.e. used within the respective perp space, the (now subdimensional) ones y = sqrt((n-1)-2)/2 = sqrt(n-3)/2 of the former section as lacings within each layer, and furthermore the (now uinque) lacings of the here applied stacking. But because of the mirror symmetry of the above lace city of these Dutour polytopes of the second kind along the H - h hyperplane, it is obvious that the new segmental lacings in here all could be chosen the same y as within the layers. Therefore, outside of n=7 these polytopes generally need 2 different edge lengths and therefore never become CRFs again.
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