184 
EEV. T. P. KIEKMAN ON AUTOPOLAE POLTEDEA. 
rank and collocation with respect to at least one a-ace a in it, of its faces and remaining 
summits. This summit a maybe considered the pole of that face A. A^>-edraly)-acron, 
whose summits are the poles of its j) faces, is an autopolar polyedron. 
To every edge AB, between the a-gon A and /3-gon B, in such a figure, corresponds 
an edge ah^ between the a-ace a and the /3-ace and vice versd. Two such edges I call 
a gamic jpair^ or a pair of gamics, either being the gamic of the other. If these edges 
the edge af, and so on, until we arrive at a face containing the edge ae ; we have, for example, 
'K~ealcd 
J.=haf.. 
M=/«y . . 
N =gah . . . 
P — Jiai . . 
Q =-iae . . . 
and we thence derive the vertex-symbol 
a=KLMNPQ, (2.) 
where the contiguous duads KL, LM, MN, NP, PQ, QK represent in order the edges through the vertex a. 
The remarks before made with respect to the face-symbols apply to the vertex-symbols. A non-contiguous 
duad such as KN of the vertex-symbol a cannot be a duad contiguous, or non-contiguous of any other 
vertex-symbol, but each contiguous duad such as KL of the vertex-symbol a must be a contiguous duad of 
one and only one other vertex-symbol h. And the symbols of two vertices cannot contain more than one 
contiguous duad the same in each symbol. 
“ Any edge of the polyedron admits of a double representation ; it is the junction of two vertices, or the 
intersection of two faces. Thus ah and KL will represent the same edge, or we may write 
o6=KL (3.) 
“ It is to be remarked, that in this system each to each equation o5=KLi, there corresponds one and only 
one equation of the form ae=KQ, i.e. to an edge considered as drawn from a given vertex in a given face 
there corresponds one and only one other edge from the same vertex in the same face. 
“ It has been shown how the system of face-symbols (1) leads to the system of vertex-symbols (2) and 
the system of edge symbols (3) ; and generally, any one of the three systems leads to the other two ; and 
the three systems conjointly, or each system by itself, is a complete representation of the polyedron. As 
an example, take the hexaedron : the three systems are, — 
'K.=-ahcd (1) 
L —ahfe 
lsl=hfge 
N =-gcdh 
P =d}iea 
(^ = hefg 
a5=KL 
Jc = KM 
c<7=KN 
de=KP 
a=LPK (2) 
S=LMK 
c = MNK 
<^=NPK 
e=LPQ 
/=LMQ 
y=MNQ 
A=NPQ, 
ae = PL 
5/=:LM 
cy = MN 
K h 
^=LQ (3) 
/y=MQ 
a7e=NQ, 
7ia = PQ 
“ Consider, now, two polyedi'a having the same number of vertices and also the same number of faces. 
