(551) 
- 
1,2 . 
of C,, corresponds to 0, TA these symbols, difference in orientation 
16 
being disregarded EE eN A Ee 
oO "eoar § ay ; , ETE IOA eN fe | Sea Ie 
eme isregardea, may je reduced: to 16° 16 16° 16’ 16 16 
We now consider each of these cases separately. 
4,8 
Case (= 75) In this space-filling the middle section (26, 42, 18) 
8 
with the fraction err oe orientation only, whilst the poly- 
? 
hedron (11,18, 9) with the fraction occurs in two different posi- 
tions passing into one another by a rotation of 180° about the axis 
MN with the period 3. In order to make this space-filling perfectly 
clear we project it successively on a plane normal to the common 
direction of the axes and on one of the axes. These projections can 
be found immediately, if we know how the composing polyhedra are 
to be put in contact with each other. Therefore we indicate first 
that two polyhedra of different form in facial contact with each 
other have always a deltoid in common, whilst this contact can be 
realized for two polyhedra (11, 18,$) by a lozenge only, for two 
polyhedra (26, 42,18) by a hexagon only; this is clear if we bear 
in mind that all the axes MN are parallel. 
The projection of the space-filling on to a plane normal to the 
axes MN may be regarded as the superposition to one another of 
two wellknown plane-fillings (fig. 7°), that of regular hexagons and 
that of equilateral triangles, the vertices of the polygons of the one 
being the centres of the polygons of the other and vice versa. If 
this space-filling is cut by a plane bisecting an axis MN normally 
the result is the plane-filling by triangles or that by hexagons accord- 
ing as that axis belongs to the form (11, 18, 9) or to the form 
(26, 42, 18). 
The projection of the space-filling on to an axis is given in fig. 84 
in two layers of which the upper one is related to the axes of the 
central section, the one below to the axes of the polyhedra (11, 18, 9). 
The axes MN of the first group with the length 2/2 fill the whole 
line, whilst the axes J/’N’ of the second group with the length 
4 
3 V2 leave parts of the line uncovered. On one axis of both 
groups the projections of the vertices of the polyhedron have been 
indicated, on MN the points A, B, C, D, on M'N’ the points 
