432 
132. Hemihedral or Half-symmetrical Forms of the Cubic System. 
In the holohedral or perfectly symmetrical forms of the 
cubical system, the solid form of the crystal is bounded by 
the lines where any one plane or face is intersected by the 
adjacent planes or faces. There are, however, symmetrical 
forms where half the number of the holohedral faces are 
omitted, the planes of the remaining faces forming a solid by 
the intersection of the adjacent planes. 
These, called hemikedral or half- symmetrical faced forms, 
are of two kinds, — the inclined , in which no one face is parallel 
to the other; and the parallel, in which the faces are parallel 
in pairs. 
133. The inclined hemihedral forms are the tetrahedron 
(figs. 15 and 16, Plate III.), the twelve-faced trapezohedron 
(figs. 17 and 18), the four-faced tetrahedron (figs. 19 and 20), 
and the six-faced tetrahedron (figs. 21 and 22); these being 
the hemihedral forms respectively derived from the octahedron , 
three-faced octahedron, twenty -four -faced trapezohedron , and 
six-faced octahedron, half of whose faces are produced to meet 
each other. 
There are two hemihedral forms with parallel faces, — the 
twelve-faced pentagon, derived from the four-faced cube (figs. 
23 and 24), and the irregular twenty -four -faced trapezohedron , 
derived from the six-faced octahedron . 
The cube and rhombic dodecahedron do not produce hemi- 
hedral forms, according to the laws of symmetry by which the 
preceding are formed. 
134. The tetrahedron (figs. 15 and 16, Plate III.) is formed 
by taking half the faces of the octahedron (fig. 7, Plate I.), in 
the following order, — CfJ^C 3 , C Y C f 5 0 4 , C 2 C 5 C 6 , and C 4 0 3 0 6 , and 
producing these planes to intersect in the lines 0 4 0 2 , 0 2 0 5) 
0 2 0 7 , 0 4 0 5 , 0 4 0 7 , and 0 7 0 5 . Referring to (fig. 14, Plate II.), 
we see that these edges are diagonals of the square faces of 
the cube in which the octahedron is inscribed, one edge for 
each face of the cube. 
The tetrahedron is therefore geometrically inscribed in the 
same cube in which the octahedron, from which it is derived, 
is also inscribed. (Pig. 16, Plate III.) shows the face of the 
octahedron shaded on the corresponding face of the tetrahedron. 
Since 0 2 0 4 , 0 2 0 5 , and 0 4 0 5 are diagonals of equal squares, 
each face of the tetrahedron is an equilateral triangle, 0 2 0 4 0 5 
(fig. 33, Plate IV.). If we bisect the three sides of this 
equilateral triangle in the points C lt C 2 , and C 3 , and join these 
points, the equilateral triangle C 1 C 2 Cz will be a face of the 
octahedron. 
