12 research outputs found
Generalized Erdos Numbers for network analysis
In this paper we consider the concept of `closeness' between nodes in a
weighted network that can be defined topologically even in the absence of a
metric. The Generalized Erd\H{o}s Numbers (GENs) satisfy a number of desirable
properties as a measure of topological closeness when nodes share a finite
resource between nodes as they are real-valued and non-local, and can be used
to create an asymmetric matrix of connectivities. We show that they can be used
to define a personalized measure of the importance of nodes in a network with a
natural interpretation that leads to a new global measure of centrality and is
highly correlated with Page Rank. The relative asymmetry of the GENs (due to
their non-metric definition) is linked also to the asymmetry in the mean first
passage time between nodes in a random walk, and we use a linearized form of
the GENs to develop a continuum model for `closeness' in spatial networks. As
an example of their practicality, we deploy them to characterize the structure
of static networks and show how it relates to dynamics on networks in such
situations as the spread of an epidemic
Geometric Mechanics of Curved Crease Origami
Folding a sheet of paper along a curve can lead to structures seen in
decorative art and utilitarian packing boxes. Here we present a theory for the
simplest such structure: an annular circular strip that is folded along a
central circular curve to form a three-dimensional buckled structure driven by
geometrical frustration. We quantify this shape in terms of the radius of the
circle, the dihedral angle of the fold and the mechanical properties of the
sheet of paper and the fold itself. When the sheet is isometrically deformed
everywhere except along the fold itself, stiff folds result in creases with
constant curvature and oscillatory torsion. However, relatively softer folds
inherit the broken symmetry of the buckled shape with oscillatory curvature and
torsion. Our asymptotic analysis of the isometrically deformed state is
corroborated by numerical simulations which allow us to generalize our analysis
to study multiply folded structures
An additive framework for kirigami design
We present an additive approach for the inverse design of kirigami-based
mechanical metamaterials by focusing on the design of the negative spaces
instead of the kirigami tiles. By considering each negative space as a four-bar
linkage, we discover a simple recursive relationship between adjacent linkages,
yielding an efficient method for creating kirigami patterns. This shift in
perspective allows us to solve the kirigami design problem using elementary
linear algebra, with compatibility, reconfigurability and rigid-deployability
encoded into an iterative procedure involving simple matrix multiplications.
The resulting linear design strategy circumvents the solution of non-convex
global optimization problems and allows us to control the degrees of freedom in
the deployment angle field, linkage offsets and boundary conditions. We
demonstrate this by creating a large variety of rigid-deployable, compact
reconfigurable kirigami patterns. We then realize our kirigami designs
physically using two new simple but effective fabrication strategies with very
different materials. All together, our additive approaches pave a new way for
mechanical metamaterial design and fabrication based on paper-based
(ori/kiri-gami) art forms