Dutour's Delone cells – and related CRF polytopes

First infinite series of Delone cells, applicable for n≥6 even

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 E₆. 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:

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 + d²]. 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].

oxx oox&#xt

oxo3oox3ooo&#xt

oxo3ooo3ooo *b3oox&#xt

oxo3ooo3ooo *b3ooo3oox&#xt

oxo3ooo3ooo *b3ooo3ooo3oox&#xt

oxo3ooo3ooo *b3ooo3ooo3ooo3oox&#xt

.xo3.oo3.oo *b3.oo3.oo3.oo3.oo3.ox&#zx
ox.3oo.3oo. *b3oo.3oo.3oo.3oo.3oo.&#zx

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 d² + 2dm + m² = d² - 2dm + m² + 1/2 or equivalentely m = 1/(8d). And thus for the circumradius itself r = d+m = (8d² + 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 + m²). Thus we get from r = M ver(H), using the above derived value for m, the equation (8d² + 1)²/(8d)² = (n-1)/8 + 1/(8d)² or equivalently 64d⁴ + 16d² + 1 = 8(n-1)d² + 1. Whenever d > 0 this runs down to 8d² = 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 = (8d² + 1)/(8d) = (n-2)/sqrt(8(n-3)). Or we could also provide the circumcenter M = M(d) = M(d(n)) explicitly 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 + d²) = 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.

Second infinite series of Delone cells, applicable for n≥7 odd

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 E₇.

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.