id: 05983866
dt: j
an: 2012e.00656
au: Montesinos Amilibia, José María
ti: Crystallographic groups and topology of Escher.
so: Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 104, No. 1, 27-47 (2010).
py: 2010
pu: Real Academia de Ciencias Exactas, Físicas y Naturales, Madrid
la: ES
cc: H45
ut: orbifolds; crystallographic groups; space groups; wallpaper groups; frieze
groups; point groups; Euler characteristic
ci:
li:
ab: The paper (in Spanish) presents an alternative approach to listing
crystallographic groups in dimensions up to 3, as well as basic ideas
of its generalization to listing hyperbolic crystallographic groups.
The approach is based on the topological concept of orbifolds and their
Euler characteristics. Although the concepts are more advanced than the
ones needed to list crystallographic groups on basis of geometrical, or
even classical group-theoretic, arguments, the deduction based on the
concept of orbifolds is in itself quite simple and straightforward. The
essential terms are explained in not too technical a manner, and basic
theorems given, some with the corresponding proofs, but the overall
approach is not too technical and adapted to non-specialists. Based on
these general properties the deduction of the 17 wallpaper groups is
described in detail, and with slightly less detail the 7 frieze groups
and 32 crystallographic classes (corresponding to point groups, which
in turn correspond to crystallographic spherical groups) are presented.
All the ideas are well illustrated by examples from Escher
(tessellations of Euclidean and hyperbolic plane), that make this paper
a particularly interesting and enjoyable one for a reader with basic
knowledge of abstract mathematical concepts like groups and manifolds,
and can serve as a motivational introductory/review paper for a reader
interested in this relatively new topic of (crystallographic)
orbifolds.
rv: Franka Miriam Bruckler (Zagreb)