Related papers: Dilute Hitomezashi on the Isometric Grid
Hitomezashi, a form of traditional Japanese embroidery, gives rise to intricate arrangements of axis-parallel unit-length stitches in the plane. Pete studied these patterns in the context of percolation theory, and the first two authors…
Two mathematical aspects of the centuries-old Japanese sashiko stitching form hitomezashi are discussed: the encoding of designs using words from a binary alphabet, and duality. Traditional hitomezashi designs are analysed using these two…
Hitomezashi patterns, which originate from traditional Japanese embroidery, are intricate arrangements of unit-length line segments called stitches. The stitches connect to form hitomezashi strands and hitomezashi loops, which divide the…
Both Celtic knotwork and strips of hitomezashi stitching can be interpreted as being two-sided friezes wherein the patterns on the sides are interleaved. We prove which of the thirty-one two-sided friezes can, and cannot, be realized in…
The field of rigid origami concerns the folding of stiff, inelastic plates of material along crease lines that act like hinges and form a straight-line planar graph, called the crease pattern of the origami. Crease pattern vertices in the…
We study tilings of the plane composed of two repeating tiles of different assigned areas relative to an arbitrary periodic lattice. We classify isoperimetric configurations (i.e., configurations with minimal length of the interfaces) both…
We consider the edge- and vertex-isoperimetric probem on finite and infinite hexagonal grids: For a subset W of the hexagonal grid of given cardinality, we give a lower bound for the number of edges between W and its complement, and lower…
Thin surfaces are ubiquitous in nature, from leaves to cell membranes, and in technology, from paper to corrugated containers. Structural thinness imbues them with flexibility, the ability to easily bend under light loads, even as their…
We study the three-dimensional equilibrium shape of a shell formed by a deployed accordion-like origami, made from an elastic sheet decorated by a series of parallel creases crossed by a central longitudinal crease. Surprisingly, while the…
A flat sheet programmed with a planar pattern of spontaneous shape change will morph into a curved surface. Such metric mechanics is seen in growing biological sheets, and may be engineered in actuating soft matter sheets such as…
In this paper we consider the problem of approximating Euclidean distances by the infinite integer grid graph. Although the topology of the graph is fixed, we have control over the edge-weight assignment $w:E\to \mathbb{R}_{\ge 0}$, and…
Kirigami, the art of paper cutting, has become a paradigm for mechanical metamaterials in recent years. The basic building blocks of any kirigami structures are repetitive deployable patterns that derive inspiration from geometric art forms…
The main result of this note is that, for each $n\in \{1,2,3,\ldots\}$, there exists a Hodge metric on the $n$-th Hirzebruch surface whose positive holomorphic sectional curvature is $\frac{1}{(1+2n)^2}$-pinched. The type of metric under…
A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope \pm1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods,…
In this paper, we will show methods to interpret some rigid origami with higher degree vertices as the limit case of structures with degree-4 supplementary angle vertices. The interpretation is based on separating each crease into two…
Arrays of interacting magnetic nanostructures were introduced recently as a powerful approach to investigate experimentally the exotic many-body physics of frustrated spin models. Following a similar strategy based on…
The material characterization of ultra-thin solid sheets, including two-dimensional materials like graphene, is often performed through indentation tests on a flake suspended over a hole in a substrate. While this `suspended indentation' is…
We consider discrete spin models on arbitrary planar graphs and lattices with frustrated interactions. We first analyze the Ising model with frustrated plaquettes. We use an algebraic approach to derive the result that an Ising model with…
In this work, we study the Hermite interpolation on $n$-dimensional non-equally spaced, rectilinear grids over a field $\Bbbk $ of characteristic zero, given the values of the function at each point of the grid and the partial derivatives…
Kirigami, the art of paper cutting, has been widely used in the modern design of mechanical metamaterials. In recent years, many kirigami-based metamaterials have been designed based on different planar tiling patterns and applied to…