الملخص الإنجليزي
Voronoi and Delone cells of the root lattices of An and Dn are constructed.
Voronoi cell of the root lattice is constructed as the dual of the root polytope
which turns out to be the union of Delone cells. It is shown that the Delone cells
centered at the origin of the root lattice An are the polytopes obtained from the
weights ω1, ω2,. . . , ωn and the Delone cells of the root lattice Dn are the poly topes obtained from the weights ω1, ωn−1 and ωn. We calculated the volume of
the Voronoi cells of An as √
n + 1 via their (n − 1)−dimensional facets which are
generalized rhombohedra. The volumes of Voronoi cells of Dn lattices are always
2 and calculated using their (n − 1)−dimensional facets which are dipyramids
with a base of (n − 2)−dimensional cube. Projection of the Voronoi and De lone cells of An lattices in the Coxeter plane gives rhombic and triangular tilings
with (n+1)−fold symmetry respectively. Some of these aperiodic tilings represent
quasicrystallographic structures, for example rhombuses of the Voronoi cell of the
root lattice A4 project onto thick and thin Penrose rhombuses while the Delone
cells project onto two isosceles Robinson triangles. Platonic and Archimedean
solids possessing icosahedral symmetry H3 have been obtained by projections of
the sets of lattice vectors of D6 determined by a pair of integers (m1, m2) with
m1 + m2 = even. Vertices of the Danzer's tetrahedra ABCK are obtained as
the fundamental weights of H3. Tiling procedure for both ABCK tetrahedral
and < ABCK > octahedral tilings in 3D space with icosahedral symmetry H3,
and those related transformations in 6D space with D6 symmetry are specified
by determining the rotations and translations in 3D space. The tetrahedron K
constitutes the fundamental region of the icosahedral group H3 as being the cell of its Voronoi cell (rhombic triacontrahedron). The six Mosseri−Sadoc tiles have
been obtained from the projection of the 3D facets of Delone cells of D6 lattice
which in turn are assembled into four composite tiles that can be inflated to
any power of τ . The icosahedron, dodecahedron and icosidodecahedron whose
vertices are obtained from the fundamental weights of the icosahedral group are
dissected in terms of six tetrahedra. Dodecahedra with edge lengths of 1 and τ
naturally occur already in the second and third order of the inflations. Patches
of dodecahedral structures are obtained displaying local icosahedral symmetry.
