Some remarks on triangulating a d-cube
Spacefilling knots.
Spectral Pairs in Cartesian Coordinates.
Spiral Tilings with S-curves and C-curves Using Combinatorics to Augment Tradition
Substitutions, abstract number systems and the space filling property
In this paper we study multi-dimensional words generated by fixed points of substitutions by projecting the integer points on the corresponding broken halfline. We show for a large class of substitutions that the resulting word is the restriction of a linear function modulo and that it can be decided whether the resulting word is space filling or not. The proof uses lattices and the abstract number system associated with the substitution.
Symmetry breakings of the cube tiling and the spatial chess board by - symbols.
Système dynamique à spectre discret et pavage périodique associé à une substitution
On donne une condition combinatoire effective suffisante pour que le sytème dynamique associé à une substitution de type Pisot ait un spectre purement discret. Dans le cas unimodulaire, cette condition est nécessaire dès que la substitution n'a qu'un cobord trivial ; elle est vérifiée si et seulement si le fractal de Rauzy associé à la substitution engendre un pavage auto-similaire et périodique. On en déduit des conditions de connexité des fractals de Rauzy.
The Countability of a Tiling Family and the Periodicity of a Tiling.
The local theorem for monotypic tilings.
The lonely vertex problem.
The optimal ball and horoball packings of the Coxeter tilings in the hyperbolic 3-space.
The optimal ball and horoball packings to the Coxeter honeycombs in the hyperbolic -space.
The possibility of extending factorization results to infinite Abelian groups.
The projective interpretation of the eight 3-dimensional homogeneous geometries.
The size of an annihilator in a factorization.
Tiling with L's and squares.
Tilings in topological spaces.
Tropical convexity.
Unimodular Pisot substitutions and their associated tiles
Let be a unimodular Pisot substitution over a letter alphabet and let be the associated Rauzy fractals. In the present paper we want to investigate the boundaries () of these fractals. To this matter we define a certain graph, the so-called contact graph of . If satisfies a combinatorial condition called the super coincidence condition the contact graph can be used to set up a self-affine graph directed system whose attractors are certain pieces of the boundaries . From this graph...
When can you tile a box with translates of two given rectangular bricks?