Rhombic Axes. 
30. Describe a square B 1 G ] AG 2 (fig. 28*, Plate IV.*) having 
its equal sides one-half the side or edge of the circumscribing 
cube. Join the diagonals G X G 2 and B X A meeting in cl v Pro- 
duce B 1 G 1 to and C 2 A to B 5 , making G 1 0 1 and AB 5 each 
=AB r Join C l V 5 and O l A meeting in o v Draw O x d 5 
perpendicular to AB 5 . Then since GftJO^A is a rectangular 
parallelogram, it follows AO, is bisected in o,, o,d, = W x B, and 
Ad 6 =$AB 5 . 
Then referring to (fig. 12, Plate II.), — the square GJD-fi^A 
represents on a plane surface (fig. 28*), and the parallelogram 
C 1 AD b 0 1 the same figures shown in perspective in (fig. 12, 
Plate II.) ; the former being one -fourth of a section of the 
cube drawn through the points B^B^B^B^ and the latter one- 
fourth of the section drawn through 0 3 0 1 0 5 0 7 . 
G x o v 0l d 5 , &c., representing the lines similarly 
marked in the perspective figure of the rhombic dodecahedron 
inscribed in the cube. 
31. .Now fig. 30, Plate IV. Draw 0 2 0 3 = G-fi 2 (fig. 28*), on 
both sides G 2 G 3 as base, describe two isosceles triangles having 
their equal sides, such as G 2 o 1 = G 1 o 1 (fig. 28*) ; join the 
diagonals G 2 G 3 and o x o 5 meeting in d 5 . G 2 o 5 G 3 o x will represent 
on a plane surface a face of the rhombic dodecahedron, which 
can be inscribed in a cube whose edge is double G 2 B L or O x B 5 
(fig. 27*). 
32. (Fig. 28*, Plate IV.*) Dp^is perpendicular to G^G^ 
and also B 5 d 5 is perpendicular to o x d 5 . Hence, referring to 
(fig. 12, Plate II.), D x d x is perpendicular to G-^d-fi 2 , and B 5 d 5 is 
perpendicular to o x d v Hence, by symmetry and similarity of 
construction, D 5 d 5 is perpendicular to OjOg, and G 2 G 3 meeting 
in d 5 ; and therefore B 5 d 5 is perpendicular to the face o 1 G 2 o 5 G 3 
of the rhombic dodecahedron, and passes through d~. its centre 
of gravity. 
33. Hence by symmetry and similarity of construction 
comparing (fig. 12, Plate IV.) with (fig. 5, Plate I.), every axis 
DiD n , B 2 B 12 , D 3 Dp, &c., D 6 D 8 , joining the opposite centres of 
the edges of the circumscribing cube, are each perpendicular 
to, and pass through the centres of gravity of opposite and 
parallel faces of the inscribed rhombic dodecahedron. Thus 
n i D ii is perpendicular to G^ofy^ and D 4 o 6 (7 6 o 7 , B 2 B 12 is 
perpendicular to G 1 o 1 G 3 o 2 and G 5 o 8 G 6 o 7 , &c. From this property 
these axes are called the rhombic axes. 
34. Again referring to (fig. 28*, Plate IV.*), we see that 
Ao 1 — \A0 1 and Ad 1 z=^AD 1 , Hence by similarity and sym- 
metry of construction (fig. 12, Plate II.) we see that the rhombic- 
dodecahedron, inscribed in the cube, touches the centre of 
