440 
is bounded by twelve equal and similar pentagons. These 
pentagons are, except in one species of the pentagonal dode- 
cahedron, irregular (figs. 23 and 24, Plate III.) ; four edges 
or sides of the pentagon being equal, and the fifth unequal. 
When the five edges are equal, the pentagonal dodecahedron is 
called the regular pentagonal dodecahedron, and is one of the 
five Platonic bodies. # 
It is also called the hemi-hexa-tetrahedron and pyritoid. 
It is formed from the four -faced cube by taking three out 
of the six faces (fig. 2, Plate I.) which meet in the points o v 
o &c., o 8 ; taking the faces alternately and producing them 
to form by their intersections a solid by twelve pentagonal 
faces 
Thus the faces 0 1 o 1 o 4 , G-p^o^, G 2 o 1 o 5 , 0 2 o 4 o 8 , G 2 o Y o 2 , 0 3 o 5 o 6 , 
0 4 o 2 o 6) C 4h o„ 0 , 0 , 0 ,, G,o, 0i , 0,o o, and 0,o,o 7 are produced to 
form the positive pentagonal dodecahedron ; the twelve 
remaining faces to form the negative pentagonal dodecahedron. 
The faces so produced meet in twenty-four equal edges o^, 
0l $ 2 , &c. (figs. 23 and 24, Plate III.); and six other edges, but 
unequal to the former S 4 S 9 , S 2 S 4 , &c. 
To obtain a face of the pentagonal dodecahedron geo- 
metrically from that of the four-faced cube from which it is 
derived (fig. 37, Plate IV.), being described as m § 53. 
Produce G 1 d 1 to meet D 1 G 2 in S r 
Describe 0 1 o 1 o 4 as in § 54, a face of the four-feced cube 
(fig. 34, Plate IV.). Bisect o 4 o 4 in d v Produce DA to b 19 
making 0^=0^ (fig. 37). Join and o&. Through 
G 1 draw \Cf> 2 parallel to o 4 o 4 . ^ 
Then (fig. 34) take Cf) 2 and Gf^ each equal G 2 d ± (fig. 3/). 
Join o 4 S 4 and o 4 S 2 . , _ . , 
Then S 4 S 2 oA °4 is a face of the pentagonal dodecahedron 
derived from the four-faced cube whose face is C 1 o 4 o 1 . 
Twelve such pentagonal faces form a net for the pentagonal 
dodecahedron which can be inscribed in the cube whose faces 
are equal to the square 0 1 0 4 0 8 0 5 (fig. 27, Plate IV.). 
The faces of the four-faced cube are shaded on those ot the 
pentagonal dodecahedron (fig. 24, Plate IV.). 
The following pentagonal dodecahedrons, having faces ol 
crystals parallel to tnem, have been observed in natuie . 
i-[l \ oo] ; °° ^-i. Naum an n ; 
2 
in pyrite. 
5 4 0 Miller ; -J- 5 4 Brooke, 
[1 | co] ; ‘2-2J ; 7 T 4 3 0 ; , in pyrite. 
u 
[1 | oo] ; 'e.Q-i ; 7 T 3 2 0 ; in pyrite. 
2 
