1,563 research outputs found
Developing a Mathematical Model for Bobbin Lace
Bobbin lace is a fibre art form in which intricate and delicate patterns are
created by braiding together many threads. An overview of how bobbin lace is
made is presented and illustrated with a simple, traditional bookmark design.
Research on the topology of textiles and braid theory form a base for the
current work and is briefly summarized. We define a new mathematical model that
supports the enumeration and generation of bobbin lace patterns using an
intelligent combinatorial search. Results of this new approach are presented
and, by comparison to existing bobbin lace patterns, it is demonstrated that
this model reveals new patterns that have never been seen before. Finally, we
apply our new patterns to an original bookmark design and propose future areas
for exploration.Comment: 20 pages, 18 figures, intended audience includes Artists as well as
Computer Scientists and Mathematician
A computational approach to Conway's thrackle conjecture
A drawing of a graph in the plane is called a thrackle if every pair of edges
meets precisely once, either at a common vertex or at a proper crossing. Let
t(n) denote the maximum number of edges that a thrackle of n vertices can have.
According to a 40 years old conjecture of Conway, t(n)=n for every n>2. For any
eps>0, we give an algorithm terminating in e^{O((1/eps^2)ln(1/eps))} steps to
decide whether t(n)2. Using this approach, we improve the
best known upper bound, t(n)<=3/2(n-1), due to Cairns and Nikolayevsky, to
167/117n<1.428n.Comment: 16 pages, 7 figure
Boundary operator algebras for free uniform tree lattices
Let be a finite connected graph, each of whose vertices has degree at
least three. The fundamental group of is a free group and acts on
the universal covering tree and on its boundary ,
endowed with a natural topology and Borel measure. The crossed product
-algebra depends only on the rank of
and is a Cuntz-Krieger algebra whose structure is explicitly
determined. The crossed product von Neumann algebra does not possess this
rigidity. If is homogeneous of degree then the von Neumann algebra
is the hyperfinite factor of type
where if is bipartite, and
otherwise
Short-range and long-range order: a transition in block-gluing behavior in Hom shifts
Hom shifts form a class of multidimensional shifts of finite type (SFT) and
consist of colorings of the grid Z2 where adjacent colours must be neighbors in
a fixed finite undirected simple graph G. This class includes several important
statistical physics models such as the hard square model. The gluing gap
measures how far any two square patterns of size n can be glued, which can be
seen as a measure of the range of order, and affects the possibility to compute
the entropy (or free energy per site) of a shift. This motivates a study of the
possible behaviors of the gluing gap. The class of Hom shifts has the interest
that mixing type properties can be formulated in terms of algebraic graph
theory, which has received a lot of attention recently. Improving some former
work of N. Chandgotia and B. Marcus, we prove that the gluing gap either
depends linearly on n or is dominated by log(n). We also find a Hom shift with
gap {\Theta}(log(n)), infirming a conjecture formulated by R. Pavlov and M.
Schraudner. The physical interest of these results is to better understand the
transition from short-range to long-range order (respectively sublogarithmic
and linear gluing gap), which is reflected in whether some parameter, the
square cover, is finite or infinite
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