. 150. OF COMBINATIONS. 191 



. 150. HEMI-PYRAMIDAL COMBINATIONS. 



A combination of the pyramidal system possesses 

 a Hemi-pyramidal Character, if one or more of the 

 simple forms contained in it, appear with only half 

 the number of their faces. 



The isosceles four-sided pyramid may be resolved into 

 halves, like the octahedron (. 129.). The result is a pair 

 of forms contained under four equal and similar isosceles tri- 

 angles, Figs. 61. 62., the first being produced from the isosce- 

 les four-sided pyramid, Fig. 8., by the enlargement of the faces 

 a and , the second by the enlargement of & and b. None 

 of their faces are parallel to each other. This form is ana- 

 logous to the tetrahedron. A combination, containing one 

 or several forms of this kind, is termed a hemi-pyramidal 

 combination of inclined faces. If we enlarge parallel faces, no 

 finite forms can be obtained ; and this seems to be the reason 

 why hemi-pyramidal forms thus possessing parallel faces 

 have not yet been found in nature. 



The scalene eight-sided pyramid, if resolved by enlarg- 

 ing its alternate faces, gives forms contained under eight 

 irregular trapezoidal faces, Figs. 63. 64., which on that ac- 

 count receive the denomination of four-sided Trapczohcdro. 

 These forms agree exactly with the six-sided trapezohedrons 

 obtained by the same mode of resolution from the di-pyra- 

 mid, . 146., except in the number of their faces. The two 

 forms thus produced from the eight-sided pyramid, are also 

 distinguished from each other by the difference of Right and 

 Left, as the forms of the rhombohedral system already 

 mentioned. Fig. 63. represents a right four-sided Trape- 

 zohedron, produced from the eight-sided pyramid, Fig. 12., 

 by the enlargement of the faces a, a, while Fig. 64. is the 

 left one which arises from the enlargement of 6 and ft. 



By enlarging parallel faces we obtain two forms absolute- 

 ly similar to isosceles four-sided pyramids, except in their 

 position, since the faces of such hemi-pyramidal forms are 

 always situated like the faces of that eight-sided pyramid, 



