11 / 22
Crystallography 2018
Structural Chemistry & Crystallography Communication
ISSN: 2470-9905
Page 63
June 04-05, 2018
London, UK
3
rd
Edition of International Conference on
Advanced Spectroscopy,
Crystallography and Applications
in Modern Chemistry
N
on-Euclidean crystallography seems to be a new direction in
modelling new phenomena as fullerenes, nanotubes, quasi
crystals. The existence of such materials gives us the feeling [1],
[2] that our experience space in small size can be non-Euclidean,
for instance hyperbolic (H3) in the sense of János Bolyai and
Nikolai I. Lobachevsky. Polyhedral models as for nanotube in
Figure, and the unified projective metric geometry with the newer
linear algebraic description provide us with these methods. The
mathematical tools have also been overviewed in our conference
papers [3], [4] with my colleagues, we are working on this topic.
Author’s hyperbolic football manifolds on some Archimedean
solids are described in [1], in particular the classical football
on {5, 6, 6} had already been published in 1988 (in a Dubrovnik
Proceedings) without any fullerene reference. Extremal ball
packing (with density 0.77147) and covering (with density
1.36893), realized at the football tiling, are better than those of
the Euclidean cases. These are our recent results in [5]. These
latter investigations led us also to a polyhedral scheme in Figure
from [6], as fundamental domain (asymmetric unit under a
symmetry group) for so-called cobweb (or tube) manifold Cw(6,
6, 6), where identifying (as with topological glue) the base faces
s-1 and s by 1/3 screw motion s, and repeating this process, we
get a tube structure. At some vertices four polyhedra Cw can
meet, imitating carbon (C) atoms with four bonds. We can extend
this construction for polyhedra Cw(2z, 2z, 2z) (3 ≤ z odd natural
number). So we get an infinite series of compact hyperbolic
manifolds (i.e. every point has a ball-like neighborhood) as new
topological structures. With these we also get new models for
possible nanotube structures realized in the hyperbolic space
(H3), maybe also in our experience space (Euclidean, E3) in small
size (?).
emolnar@math.bme.huNON-EUCLIDEAN CRYSTALLOGRAPHY
Emil Molnar
Budapest Univ. Technology Econ. Hungary
Struct Chem Crystallogr Commun 2018, Volume 4
DOI: 10.21767/2470-9905-C1-006