>). 
(S53 ) 
on M’’N’’ the projections A’’, b’’,..., D’’ of vertices; here the 
pairs and triples (2, 7’), (FM), (@, A’, A’’), GH; B’, BY), (N,C’ 
coincide. 
Ch ; 6 I ne > . } Be : 49. 
an Te in this case of the unique polyhedron (14, 24, 12) 
occurring in two opposite orientations, bounded by an equatorial belt 
of six isosceles trapezia and two polar regions of triples of lozenges, 
the contact of the polyhedra takes place either by an isosceles 
trapezium or by a lozenge. 
The projection on a plane normal to the axes consists in the 
plane-filling of regular hexagons, each of these hexagons divided in 
the same way into three lozenges. 
The projection on an axis (fig 8°) consists of two layers; all axes 
MN, M'N’ have the same length. By the projections A, B, C, D 
of the vertiees each axis is divided into five equal parts; the last 
two segments CD, DN of MN cover the first two segments M'A’, 
A’ B’ of M'N’. 
If we consider the portion of the space-filling situated between the 
two planes normal to the plane of the diagram (fig. 8) according to 
a and 6 and if we imagine that the halves of the polyhedra (14, 24, 12) 
lying between these planes are hollow, we have before us a figure 
in space, imitating very nearly the shape of the honey-comb of the 
beehive. Indeed a space-filling, the polyhedra of which really are 
double beecells has been described by A. ANDreErNI'). But the space- 
filling derived here from fourdimensional space does not characterize 
itself by the known minimum property of the beecell; it is rather 
closely connected with the space-filling by granatohedra. If we divide 
a granatohedron in two equal halves by a plane normal to a diagonal 
and rotate one of the halves an angle of 180° about that diagonal, 
we generate a polyhedron also limited by six isosceles trapezia and 
six lozenges, but the trapezia have another shape. 
Space-fillings normal to OF. Expressed in the fractions belonging 
to the sections of C, we have to deal here with the two combinations 
0 okt yal 10 ti. 3 i nic! 
HIS RT)? men which may be reduced to 
4 6 3 5 
2’ ia) Md (ag? 7) 
1) In Anpremi’s memoir “Sulle reti di poliedri regolari e semiregolari e sulle 
corrispondenti reti correlative’ (Memorie della Società italiana delle Scienze, 
series 3, volume 14, p. 75—129, 1905) the three dodecahedra filling space are 
treated in the paragraphs 76, 77, 87. For the rest my study has nothing of any 
importance in common with his work, which is nicely illustrated by stereoscopic 
views of threedimensional space-fillings. 
