ISOMETRIC SYSTEM, 



19 



occur the forms shown in figs. 8 to 11. 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 



12. 





13. 





^s^\ 



\ H ^ 







j 



^ 



H 



7? 





H 



i 









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 apices of the 

 solid angles in the octahedron. 



Again, between a cube and a dodecahedron there occur forms 

 like figs. 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, cube 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. 



