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§ 1. IL] EEV. T. P. KIEKMAIf ON THE THEOEY OF THE POLYEDEA. 
3. Mixed symmetry, both zoned and zoneless axial ; 
4. Neuter symmetry, neither zoned nor zoneless axial ; 
5. No symmetry. 
The greater part of the polyedra are entirely asymmetric. 
In considering this symmetry, we have no regard to mere lengths or inclinations of 
edges. The symmetry is descriptive, not metrical. For example, if the cutting edge of 
a wedge he removed by any quadrilateral section, the solid acquires, for our purposes, 
all the symmetry of the cube ; and this remains, however the figure may be distorted. 
1. Zoned Symmetry. 
II. The polyedra which have a zoned symmetry, and only such, are 
a. Monozone polyedra ; 
1. m-zoned monaxine heteroids ; 
c. Zoned triaxines ; 
d. m-zoned monarchaxines, having one principal and m secondary axes ; 
e. Zoned polyarchaxines, having the axial systems of the regular polyedra. 
The terms will be explained below. 
Def. A zone is any closed line drawn or drawcthle on a polyedron, ivMch divides it into 
halves, either of which is the reflected image of the other, no regard being had to mere 
lengths of edges. 
The closed line may or may not be all in one plane, and the halves may or may not 
be metrically equal. We have a right to conceive, when it is convenient, that the solid 
is constructed with the greatest possible symmetry, in which case the halves will be 
exactly equal, and the zone will be a plane, having to them the geometrical relation 
which a mirror has to an 'object touching it and to its image. 
This relation we shall assume as always existing, however the polyedron may be dis- 
torted ; that is, we assume that any zoneless edge will meet, on any zonal plane, if it be 
produced, the edge which is its refiected image in respect of that zonal plane. 
For example, any section of a cube which passes through two opposite faces, and con- 
tains either two edges or none of the solid, is a zone. 
Every zone has a zonal signature, Z, which describes it by the number of its zoned 
features, i. e. its zonal faces, and its zonal summits, through which it passes, and its 
zonal and epizmal edges, but gives no account of the number of edges in the zoned faces 
or summits, nor of the order of the zoned features. 
Any edge contained, by the zone Z, is a zonal edge of Z. Any edge cut by the zone, is 
an epizonal edge of Z. 
It will create no confusion if we denote both the zone and its signature by the same 
name Z. 
We represent zonal and epizonal edges by the symbols 0 and 0 (zero faces), and we 
write the number of such edges as an index over the proper symbol. All such indices 
in a zonal signature are coefficients. 
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