On the "Infinite serie of extreme Delaunay polytope"Recently I stumbled across an article of Mathieu Dutour Sikirić from 2013: "Infinite serie of extreme Delaunay polytope", as can be viewed at

http://arxiv.org/pdf/math/0305196v1.pdf. It describes a quite interesting set of polytopes, even so the article is quite error prone. (Eg. neither the circumcenter nor the upper lacing edge length is calculated correctly - at least wrt. the therein provided coordinates. Even the symmetry assertion (for n=6) of the main theorem is wrong when using those coordinates, because the layer distance there is not conform to an circumscribing hypersphere (as for Delaunay cells needed), rather it is given fixed.)

The main idea of that article is to consider a series of some special type of lace towers polytopes with dimensionally analoguous layers, running through all applicable dimensions. There he considers especially the stack of a single point (top) atop a (pseudo) demihypercube (equatorial) atop a crosspolytope (bottom). - In Wendy Krieger's lace notation that fellow would generally be described as oxo3ooo3ooo *b3ooo...ooo3oox&#yt (where the lacing edge y quite generally could be considered as independent from the equal layerwise edge sizes x). Clearly the 4D representant then ought be compressed into oxo3oox3ooo&#yt and the 3D one then even could be described as oxx oox&#yt.

In terms of the layer distance d the lacing edge length y = y(d) can be given only depending on the dimension value n as: y = sqrt((n-1)/8 + d

^{2}), at least when using unit edge lengths x here. - In the crystallographic context of Delaunay cells it ought be essential now to provide a further formula d = d(n), which asserts the existance of an unique circumradius. - In the mathematical context of polytopists it might be more interesting however to provide a different such formula d = d*(n), which provides equal sized edges throughout, i.e. y = x. This simply can be achieved by using d = sqrt((9-n)/8). This latter choice then clearly provides the there proclaimed coincidence of the 6D (n=6) case with the well-known Schläfli-Gosset polytope 2

_{2,1} (or "jak" in Jonathan Bowers' terms).

We even could - again generally (but then n>3) - consider this fellow as the following lace city.

- Code: Select all
` P where:`

P = o3o3o *b3o...o3o

H h H = x3o3o *b3o...o3o

h = o3o3x *b3o...o3o

P C P C = o3o3o *b3o...o3x

Refering to that picture the coordinates can be given like this (again using unit edge length x and arbitrary layer distance d):

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

For the remainder I will consider equal edge lengths throughout now, i.e. assuming the above provided formula for d.

The

case n=3 is not too interesting here. 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 trigonal prism (trip). Whereas the top segment even becomes a mere (dimensionally degenerate) triangle. Moreover it is evident, that the derived "polyhedron" is concave.

The

case n=4 is a bit more intricate: The equatorial pseudo layer here is the tetrahedron (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.

The

case n=5 then becomes interesting. The equatorial pseudo layer here is the hexadecachoron (hex), while the bottom layer is the same solid, they simply are in different orientations. Thus the upper segment represents the segmentoteron hexpy. The lower one is just the hemipenteract (hin). The lacing elements of the upper segment clearly are all pentachora (pen). 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 tetrahedral (line)tegums (aka. tetrahedral dipyramids, tete).

Accordingly the total facet count here becomes (8+8) pens + (8+1) hexes + 8 tetes. This polyteron then clearly is CRF.

As already stated,

n=6 is uniform, the even higher-symmetrical jak. It clearly is a convex polyexon. The equatorial pseudo facet here is hin. That one dissects the lacing triacontaditera (tac), which strech through both segments, into pairs of (thus corealmic) hexpies.

The total facet count is 27 tacs + 72 hixes (hexatera).

I'll skip some further dimensions here. More interesting then is again the

case n=9. 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 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 decayotta (day) plus 16 hemihepteractic pyramids. Esp. hocto thus needs to have a circumradius of 1.

Today I now considered that first of those degenerate n=9 cases in more detail. By its description it is given as .xo3.oo3.oo *b3.oo3.oo3.oo3.oo3.ox&#zx, where the dots just refer to the here omitted "opposite" single point (top P - and therefore these freely could be neglected as well) and the trailing &#zx just states that the height here equals zero, while the lacing edges have the same size x as the "layers". (Note that mutual 8D circumradii of the 2 individual "layers" differ according to our choice of d=0.) By little of research one recognizes that these 16 ek vertices under that hull operation define just simply shallow pyramids attached to any of the 16 hesa facets of that other "layer", hocto.

Accordingly this here considered new polyzetton consists out of the 128 octaexa (oca) of hocto plus 16x the sloping facets of those hesa-pyramids, which are 14 hemihexeractic pyramids (haxpy) plus 64 ocas each. Thus giving rise to a total of 224 haxpies + 1152 ocas. - And, by construction, that polyzetton is still a biform CRF!

Btw., Dutour's restriction, of n to be even and >5, here nowhere had to be taken account for.

--- rk