
ISOMETRIC SYSTEM. 19 
£ va x > 
occur the forms shown in figs. 8 to 1l. Fig. 12 is a cube; 
fig. 8 represents the cube with a plane on each angle, equally 
inclined to each cubic face; 9, the same, with the planes on the 
angles more enlarged and the ‘cubic faces reduced in size; and 

then 10, the octahedron, with the cubic faces quite small ; 
and fig. 11, the octahedron, the cubic faces having disappeared 
altogether. This transformation is easily performed by the — 
student with cubes cut out of chalk, clay, or a potato. It shows 
the fact that the cubic axes (fig. 12) connect the age of the 
solid angles in the octahedron. 
Again, between a cube and a dodecahedron there occur forms 
like fies. 13 and 14; fig. 12 being a cube, fig. 13 the same, with 
planes. truncating the “edges, each plane being equally inclined 
to the adjacent cubic faces, and fig. 14 another, with these 
planes on the edges large and the cubic faces small; and then, 
when the cubic faces disappear by farther enlargement of the 
planes on the edges, the form is a dodecahedron, fig. 15. The 
student should prove this transformation by trial with chalk or 
some other material, and so for other cases mentioned beyond. 
The surface of such models in chalk may be made hard by a 
coat of mucilage or varnish. 
Again, between a cube and a trapezohedron there are the 
forms 17 and 18; 16 being the cube, 17, cnbe with three planes 
placed symmetrically on each angle; 18, the same with the 
cubic faces greatly reduced (but also with small octahedral faces), 
and 19, the trapezohedron, the cubic faces having disappeared. 
