Related papers: The Pattern Complexity of the 2-Dimensional Paperf…
We give an exact formula for the number of distinct square patterns of a given size that occur in the Squiral tiling.
We explore the following problem: given a collection of creases on a piece of paper, each assigned a folding direction of mountain or valley, is there a flat folding by a sequence of simple folds? There are several models of simple folds;…
We give exact formulas for the number of distinct triangular patterns (or subtriangles) of a given size that occur in the Sierpi\'{n}ski Triangle.
Consider a curve $\Gamma$ in a domain $D$ in the plane $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding $P$ in the Euclidean space $\boldsymbol R^3$. The singular set $C$ of $P$ as a space curve is…
We find the exact formula for the number of distinct $n \times n$ square patterns which appear in a Robinson tiling made of one infinite order supertile.
For each integer n, an n-folding curve is obtained by folding n times a strip of paper in two, possibly up or down, and unfolding it with right angles. Generalizing the usual notion of infinite folding curve, we define complete folding…
Continuing results from JCDCGGG 2016 and 2017, we solve several new cases of the simple foldability problem -- deciding which crease patterns can be folded flat by a sequence of (some model of) simple folds. We give new efficient algorithms…
The paperfolding sequences form an uncountable class of infinite sequences over the alphabet $\{ -1, 1 \}$ that describe the sequence of folds arising from iterated folding of a piece of paper, followed by unfolding. In this note we observe…
Two-dimensional patterns are used in many research areas in computer science, ranging from image processing to specification and verification of complex software systems (via scenarios). The contribution of this paper is twofold. First, we…
We study the problem of deciding whether a crease pattern can be folded by simple folds (folding along one line at a time) under the infinite all-layers model introduced by [Akitaya et al., 2017], in which each simple fold is defined by an…
We compute the Parisi overlap distribution for paperfolding sequences. It turns out to be discrete, and to live on the dyadic rationals. Hence it is a pure point measure whose support is the full interval [-1; +1]. The space of paperfolding…
Complex patterns generated by the time evolution of a one-dimensional digitalized coupled map lattice are quantitatively analyzed. A method for discerning complexity among the different patterns is implemented. The quantitative results…
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…
Using methods of conformal field theory, we conjecture an exact form for the probability that n distinct clusters span a large rectangle or open cylinder of aspect ratio k, in the limit when k is large.
The 2x2 space-filling curve is a type of generalized space-filling curve characterized by a basic unit is in a "U-shape" that traverses a 2x2 grid. In this work, we propose a universal framework for constructing general 2x2 curves where…
We show that the abelian complexity function of the ordinary paperfolding word is a 2-regular sequence.
A two-dimensional configuration is a coloring of the infinite grid Z^2 with finitely many colors. For a finite subset D of Z^2, the D-patterns of a configuration are the colored patterns of shape D that appear in the configuration. The…
In this paper we consider developable surfaces which are isometric to planar domains and which are piecewise differentiable, exhibiting folds along curves. The paper revolves around the longstanding problem of existence of the so-called…
As a confined thin sheet crumples, it spontaneously segments into flat facets delimited by a network of ridges. Despite the apparent disorder of this process, statistical properties of crumpled sheets exhibit striking reproducibility.…
An interesting class of automatic sequences emerges from iterated paperfolding. The sequences generate curves in the plane with an almost periodic structure. We generalize the results obtained by Davis and Knuth on the self-avoiding and…