1,054 research outputs found
On the number of non-hexagons in a planar tiling
We give a simple proof of T. Stehling's result, that in any normal tiling of
the plane with convex polygons with number of sides not less than six, all
tiles except the finite number are hexagons.Comment: 2 pages, 2 figure
Thermomechanical effects in uniformly aligned dye-doped nematic liquid crystals
We show theoretically that thermomechanical effects in dye-doped nematic
liquid crystals when illuminated by laser beams, can become important and lead
to molecular reorientation at intensities substantially lower than that needed
for optical Fr\'eedericksz transition. We propose a 1D model that assumes
homogenous intensity distribution in the plane of the layer and is capable to
describe such a thermally induced threshold lowering. We consider a particular
geometry, with a linearly polarized light incident perpendicularly on a layer
of homeotropically aligned dye-doped nematics
Billiards in convex bodies with acute angles
In this paper we investigate the existence of closed billiard trajectories in
not necessarily smooth convex bodies. In particular, we show that if a body
has the property that the tangent cone of every
non-smooth point is acute (in a certain sense) then there is
a closed billiard trajectory in .Comment: 8 pages, 2 figure
Long geodesics on convex surfaces
We review the theory of intrinsic geometry of convex surfaces in the
Euclidean space and prove the following theorem: if the surface of a convex
body K contains arbitrary long closed simple geodesics, then K is an isosceles
tetrahedron.Comment: 8 pages, 10 figure
On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape
We study the lengths of curves passing through a fixed number of points on
the boundary of a convex shape in the plane. We show that for any convex shape
, there exist four points on the boundary of such that the length of any
curve passing through these points is at least half of the perimeter of . It
is also shown that the same statement does not remain valid with the additional
constraint that the points are extreme points of . Moreover, the factor
cannot be achieved with any fixed number of extreme points. We
conclude the paper with few other inequalities related to the perimeter of a
convex shape.Comment: 7 pages, 8 figure
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