The Orbifold Notation for Two-Dimensional Groups

@article{Conway2002TheON,
  title={The Orbifold Notation for Two-Dimensional Groups},
  author={J. Conway and D. Huson},
  journal={Structural Chemistry},
  year={2002},
  volume={13},
  pages={247-257}
}
This paper gives a detailed introduction to the orbifold notation for two-dimensional (2-D) symmetry groups. It discusses the correspondence between properties of orbifolds and symmetries in the original surface. The problem of determining a group in situ is addressed. Elementary proofs of the classification of the Euclidean and spherical 2-D symmetry groups are presented. 
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References

SHOWING 1-9 OF 9 REFERENCES
Generators and relations for discrete groups
1. Cyclic, Dicyclic and Metacyclic Groups.- 2. Systematic Enumeration of Cosets.- 3. Graphs, Maps and Cayley Diagrams.- 4. Abstract Crystallography.- 5. Hyperbolic Tessellations and FundamentalExpand
The geometry and topology of three-manifolds
Visions of Symmetry: Notebooks, Periodic Drawings and Related Work of M. C. Escher by Doris Schattschneider (review)
Visions of symmetry contains Esch r's complete set of symmetry drawings reproduced in full colour and two Escher notebooks (which contain his theory of symmetry). In all, there are more than 350Expand
Contrib
  • Geometry Algebra
  • 2001
InGroups, Combinatorics and Geometry
  • (London Mathematical Society Lecture Note Series
  • 1992
Notebooks, Periodic Drawings and Related Work of M
  • C. Escher
  • 1990
Geometry and Topology of Three-Manifolds (Princeton University: Princeton
  • New Jersey,
  • 1980
Canad
  • J . Math .
  • 1967