442 
at o v o 2 , &c., o 8 , is cut off through the lines ^ 2 , S 2 $ 5 , and 
£ ^ &c., forming a solid bounded by twenty equilateral 
triangles, — eight being parallel to the faces of the octahedron 
inscribed in the dodecahedron, and the remaining twelve faces 
of the pentagonal dodecahedron. 
Ozonam, in his Mathematical Becreations, remarks that 
cc The ancient geometricians made a great many geometrical 
speculations respecting these bodies ; and they form almost the 
whole subject of the last books of Euclid's Elements. They 
were suggested to the ancients by their believing that these 
bodies were endowed with mysterious properties, on which the 
explanation of the most secret phenomena of nature depended. 
142. The irregular twenty -four- faced trapezohedron is a half- 
symmetrical form with parallel faces derived from the six-faced 
octahedron. It is called the irregular twenty -four -faced 
trapezohedron because its trapezoidal faces have only two 
equal edges, and to distinguish it from the twenty-four-faced 
trapezohedron, which is a holohedral form and has the four 
edges of its trapezoidal faces equal in pairs. 
It is bounded by twenty-four . irregular trapeziums (figs. 25 
and 26, Platen.). § . 
It is also called the hemi-octahis-hexahedron , the trapezoidal 
ico si-tetrahedron, the dyakis dodecahedron, the diploid, and 
the diplopyritoid. 
It is formed from the six-faced octahedron by taking three 
out of the six faces which meet in o v o 2 , &c., o 8 (fig. 31, 
Plate I.), and producing them to meet each other and form a 
solid bounded by twenty-four irregular trapeziums. 
Thus (fig. 3, Plate I.) the twenty-four faces C f 1 o 1 d 1 , 0 2 o 1 d 5 , 
(LoA, C 2 oJ 8 , 0 1 o 4 d 1 , C 3 o 4 d 4 , &c., are produced to meet in the 
points Sj, S 2 , &c., S 12 (fig. 25, Plate III.)* to form the po situ e 
irregular twenty-four-faced trapezohedron. 
The remaining twenty-four-faces if produced will form the 
negative trapezohedron. 
To obtain a face of the irregular twenty- four-faced trapezo- 
hedron geometrically from that of the six-faced octahedron 
from which it is derived. — Describe (fig. 35, Plate IV.), as 
previously constructed for finding a face of the six-faced 
octahedron, § 68 and § 137. Join G 2 N cutting produced 
in 8 r Let G 2 o x d 5 (fig. 38, Plate IV.) be a face of the six-faced 
octahedron. Produce G 2 d 5 to S 5 , and make G 2 d-§ 5 , fig. 38, 
= 0 <L$ ( fi g- 35 ) • Join on base °sPv describe the triangle 
0*8.01, fi avin g 0 &= 0£ 1 fig- 35 > and °A = 0 A fig- 38 - „ _ 
0&0& will be a face of the irregular twenty-four-faced 
trapezohedron, and twenty-four such faces will form a net for 
the same, which can be inscribed in a cube whose faces are 
equal to the square 0 1 0 5 0 8 0 i (fig. 27, Plate IV.) 5 
