58 research outputs found
Polyominoes with nearly convex columns: An undirected model
Column-convex polyominoes were introduced in 1950's by Temperley, a
mathematical physicist working on "lattice gases". By now, column-convex
polyominoes are a popular and well-understood model. There exist several
generalizations of column-convex polyominoes; an example is a model called
multi-directed animals. In this paper, we introduce a new sequence of supersets
of column-convex polyominoes. Our model (we call it level m column-subconvex
polyominoes) is defined in a simple way. We focus on the case when cells are
hexagons and we compute the area generating functions for the levels one and
two. Both of those generating functions are complicated q-series, whereas the
area generating function of column-convex polyominoes is a rational function.
The growth constants of level one and level two column-subconvex polyominoes
are 4.319139 and 4.509480, respectively. For comparison, the growth constants
of column-convex polyominoes, multi-directed animals and all polyominoes are
3.863131, 4.587894 and 5.183148, respectively.Comment: 26 pages, 14 figure
Self-Healing Tile Sets
Biology provides the synthetic chemist with a tantalizing and frustrating challenge:
to create complex objects, defined from the molecular scale up to meters,
that construct themselves from elementary components, and perhaps
even reproduce themselves. This is the challenge of bottom-up fabrication.
The most compelling answer to this challenge was formulated in the early
1980s by Ned Seeman, who realized that the information carried by DNA
strands provides a means to program molecular self-assembly, with potential
applications including DNA scaffolds for crystallography [19] or for molecular
electronic circuits [15]. This insight opened the doors to engineering with the
rich set of phenomena available in nucleic acid chemistry [20]
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