Abstract
A description of the 11 well-known uninodal planar nets is given by Cayley color graphs or alternative Cayley color graphs of plane groups. By applying methods from topological graph theory, the nets are derived from the bouquet B n with rotations mostly as voltages. It thus appears that translation, as a symmetry operation in these nets, is no more fundamental than rotations.
Highlights
What is the origin of symmetry in crystal structures? The interplay between topology and symmetry in crystal structures has already been considered in [1] and [2]
As a important consequence, translational symmetry may follow as a combination of proper or improper rotational symmetry operations
Rotational symmetry operations may result from translational symmetry combined with topological restrictions, as previously discussed in [2]; that is, a full space-group may arise out of a small set of local topology and symmetry conditions
Summary
What is the origin of symmetry in crystal structures? The interplay between topology and symmetry in crystal structures has already been considered in [1] and [2]. The concept of symmetry-labeled quotient graphs was introduced by Eon [6] Such graphs were obtained as quotient graphs of the periodic net by a non-trivial subgroup H of their space-group G (i.e., T < H < G, where T is the full translation subgroup of the net), with edges assigned symmetry operations that generate H. These objects are truly combinatorial in nature, in the sense that the derived periodic net is isomorphic with. The paper ends with some general observations concerning the description of uninodal planar nets
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