2,611 research outputs found
Rep-tiling for triangles
AbstractIn this paper we prove that one can only tile a triangle with tiles all congruent to each other and similar to the original triangle when k2, l2 + k2, or 3k2 tiles are used. The result is based on the geometry of packing and a result of I. Niven's on rational trigonometric values. In addition we describe how to tile most triangles
Archimedean-like colloidal tilings on substrates with decagonal and tetradecagonal symmetry
Two-dimensional colloidal suspensions subject to laser interference patterns
with decagonal symmetry can form an Archimedean-like tiling phase where rows of
squares and triangles order aperiodically along one direction [J. Mikhael et
al., Nature 454, 501 (2008)]. In experiments as well as in Monte-Carlo and
Brownian dynamics simulations, we identify a similar phase when the laser field
possesses tetradecagonal symmetry. We characterize the structure of both
Archimedean-like tilings in detail and point out how the tilings differ from
each other. Furthermore, we also estimate specific particle densities where the
Archimedean-like tiling phases occur. Finally, using Brownian dynamics
simulations we demonstrate how phasonic distortions of the decagonal laser
field influence the Archimedean-like tiling. In particular, the domain size of
the tiling can be enlarged by phasonic drifts and constant gradients in the
phasonic displacement. We demonstrate that the latter occurs when the
interfering laser beams are not adjusted properly
Quasi Regular Polyhedra and Their Duals with Coxeter Symmetries Represented by Quaternions I
In two series of papers we construct quasi regular polyhedra and their duals
which are similar to the Catalan solids. The group elements as well as the
vertices of the polyhedra are represented in terms of quaternions. In the
present paper we discuss the quasi regular polygons (isogonal and isotoxal
polygons) using 2D Coxeter diagrams. In particular, we discuss the isogonal
hexagons, octagons and decagons derived from 2D Coxeter diagrams and obtain
aperiodic tilings of the plane with the isogonal polygons along with the
regular polygons. We point out that one type of aperiodic tiling of the plane
with regular and isogonal hexagons may represent a state of graphene where one
carbon atom is bound to three neighboring carbons with two single bonds and one
double bond. We also show how the plane can be tiled with two tiles; one of
them is the isotoxal polygon, dual of the isogonal polygon. A general method is
employed for the constructions of the quasi regular prisms and their duals in
3D dimensions with the use of 3D Coxeter diagrams.Comment: 22 pages, 16 figure
- …