184 
THE PROBLEM OF POLYHEDRA. 
[259 
ab = KL, ae — PL, ef=LQ, (3) 
be = KM, bf = LM, fg = MQ, 
cd = KN, cg = MN, gh = NQ, 
de = KP, dh = NP, he = PQ. 
a 
K 
b 
e 
L 
71 
K P Q M K 
h 9 
N 
cl 
c 
Consider, now, two polyhedra having the same number of vertices and also the 
same number of faces. And let the vertices and faces of the first polyhedron taken in 
any order be represented by 
abcde...KLM..., 
and the vertices and faces of the second polyhedron taken in a certain order be 
represented by 
then, forming the substitution symbol 
a'b'c'd'e'. ..K'L'M'. ..abede. ..KLM..., 
which denotes that a' is to be written for a, b' for b...K' for K, &c., if operating 
with this upon the symbol system of the first polyhedron, we obtain the symbol system 
of the second polyhedron, the second polyhedron will be syntypic with the first. It 
should be noticed, that there may be several modes of arrangement of the vertices 
and faces of the second polyhedron, which will render it syntypic according to the 
foregoing definition with the first polyhedron, i.e. the second polyhedron may be syntypic 
in several different ways with the first polyhedron. This is, in fact, the same as saying 
that a polyhedron may be syntypic with itself in several different ways. Suppose, next, 
that the number of vertices of the second polyhedron is equal to the number of faces 
of the first polyhedron, and the number of faces of the second polyhedron is equal to 
the number of vertices of the first polyhedron; and let the vertices and faces of the 
first polyhedron in any order be represented by 
abede ... KLM..., 
and the faces and vertices of the second polyhedron in a certain order be represented by 
A'B’C'UE' ...k’l'm!....