25 research outputs found
Prototiles and Tilings from Voronoi and Delone cells of the Root Lattice A_n
We exploit the fact that two-dimensional facets of the Voronoi and Delone
cells of the root lattice A_n in n-dimensional space are the identical
rhombuses and equilateral triangles respectively.The prototiles obtained from
orthogonal projections of the Voronoi and Delaunay (Delone) cells of the root
lattice of the Coxeter-Weyl group W(a)_n are classified. Orthogonal projections
lead to various rhombuses and several triangles respectively some of which have
been extensively discussed in the literature in different contexts. For
example, rhombuses of the Voronoi cell of the root lattice A_4 projects onto
only two prototiles: thick and thin rhombuses of the Penrose tilings. Similarly
the Delone cells tiling the same root lattice projects onto two isosceles
Robinson triangles which also lead to Penrose tilings with kites and darts. We
point out that the Coxeter element of order h=n+1 and the dihedral subgroup of
order 2n plays a crucial role for h-fold symmetric aperiodic tilings of the
Coxeter plane. After setting the general scheme we give examples leading to
tilings with 4-fold, 5-fold, 6-fold,7-fold, 8-fold and 12-fold symmetries with
rhombic and triangular tilings of the plane which are useful in modelling the
quasicrystallography with 5-fold, 8-fold and 12-fold symmetries. The face
centered cubic (f.c.c.) lattice described by the root lattice A_(3)whose
Wigner-Seitz cell is the rhombic dodecahedron projects, as expected, onto a
square lattice with an h=4 fold symmetry.Comment: 22 pages, 17 figure
Chiral Polyhedra Derived From Coxeter Diagrams and Quaternions
There are two chiral Archimedean polyhedra, the snub cube and snub
dodecahedron together with their duals the Catalan solids, pentagonal
icositetrahedron and pentagonal hexacontahedron. In this paper we construct the
chiral polyhedra and their dual solids in a systematic way. We use the proper
rotational subgroups of the Coxeter groups ,
, and to derive the orbits representing the solids of
interest. They lead to the polyhedra tetrahedron, icosahedron, snub cube, and
snub dodecahedron respectively. We prove that the tetrahedron and icosahedron
can be transformed to their mirror images by the proper rotational octahedral
group so they are not classified in the class of chiral
polyhedra. It is noted that the snub cube and the snub dodecahedron can be
derived from the vectors, which are non-linear combinations of the simple
roots, by the actions of the proper rotation groups and
respectively. Their duals are constructed as the unions of
three orbits of the groups of concern. We also construct the polyhedra,
quasiregular in general, by combining chiral polyhedra with their mirror
images. As a by product we obtain the pyritohedral group as the subgroup the
Coxeter group and discuss the constructions of pyritohedrons. We
employ a method which describes the Coxeter groups and their orbits in terms of
quaternions.Comment: 22 pages, 19 figure