When Can You Fold a Map?

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; the simplest one-layer simple fold rotates a portion of paper about a crease in the paper by +-180 degrees. We first consider the analogous questions in one dimension lower -- bending a segment into a flat object -- which lead to interesting problems on strings. We develop efficient algorithms for the recognition of simply foldable 1D crease patterns, and reconstruction of a sequence of simple folds. Indeed, we prove that a 1D crease pattern is flat-foldable by any means precisely if it is by a sequence of one-layer simple folds. Next we explore simple foldability in two dimensions, and find a surprising contrast: ``map'' folding and variants are polynomial, but slight generalizations are NP-complete. Specifically, we develop a linear-time algorithm for deciding foldability of an orthogonal crease pattern on a rectangular piece of paper, and prove that it is (weakly) NP-complete to decide foldability of (1) an orthogonal crease pattern on a orthogonal piece of paper, (2) a crease pattern of axis-parallel and diagonal (45-degree) creases on a square piece of paper, and (3) crease patterns without a mountain/valley assignment.Comment: 24 pages, 19 figures. Version 3 includes several improvements thanks to referees, including formal definitions of simple folds, more figures, table summarizing results, new open problems, and additional reference

Constructing Buildings and Harmonic Maps

In a continuation of our previous work, we outline a theory which should lead to the construction of a universal pre-building and versal building with a $\phi$-harmonic map from a Riemann surface, in the case of two-dimensional buildings for the group $SL_3$. This will provide a generalization of the space of leaves of the foliation defined by a quadratic differential in the classical theory for $SL_2$. Our conjectural construction would determine the exponents for $SL_3$ WKB problems, and it can be put into practice on examples.Comment: 61 pages, 24 figures. Comments are welcom

Infinite All-Layers Simple Foldability

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 infinite line and must fold all layers of paper that intersect this line. This model is motivated by folding in manufacturing such as sheet-metal bending. We improve on [Arkin et al., 2004] by giving a deterministic $O(n)$-time algorithm to decide simple foldability of 1D crease patterns in the all-layers model. Then we extend this 1D result to 2D, showing that simple foldability in this model can be decided in linear time for unassigned axis-aligned orthogonal crease patterns on axis-aligned 2D orthogonal paper. On the other hand, we show that simple foldability is strongly NP-complete if a subset of the creases have a mountain-valley assignment, even for an axis-aligned rectangle of paper

Simpler learning of robotic manipulation of clothing by utilizing DIY smart textile technology

Deformable objects such as ropes, wires, and clothing are omnipresent in society and industry but are little researched in robotics research. This is due to the infinite amount of possible state configurations caused by the deformations of the deformable object. Engineered approaches try to cope with this by implementing highly complex operations in order to estimate the state of the deformable object. This complexity can be circumvented by utilizing learning-based approaches, such as reinforcement learning, which can deal with the intrinsic high-dimensional state space of deformable objects. However, the reward function in reinforcement learning needs to measure the state configuration of the highly deformable object. Vision-based reward functions are difficult to implement, given the high dimensionality of the state and complex dynamic behavior. In this work, we propose the consideration of concepts beyond vision and incorporate other modalities which can be extracted from deformable objects. By integrating tactile sensor cells into a textile piece, proprioceptive capabilities are gained that are valuable as they provide a reward function to a reinforcement learning agent. We demonstrate on a low-cost dual robotic arm setup that a physical agent can learn on a single CPU core to fold a rectangular patch of textile in the real world based on a learned reward function from tactile information

Development and Marketing of a Repurposed Textile Product for Homeless Individuals in Northwest Arkansas

Growing concerns over waste disposal methods have led to a greater focus on recycling efforts in the textile industry. Second only to the oil industry, the textile industry continues to be one of the most wasteful among leading businesses around the world and determining ways to repurpose fashion materials could be a reasonable solution to this growing problem (Dobilaite, V., Mileriene, G., Juciene, M., & Sacevičienė, 2017). In addition to alleviating disposal issues, repurposed materials could serve the humanitarian needs of local communities, and even more importantly, could specifically benefit homeless populations. The purpose of this project was to design, execute, analyze, critique and report on the development of a sleeping bag prototype using repurposed materials created to benefit homeless populations in Northwest Arkansas. The product logo for the sleeping bag, identified by the name, “HIP”, was established to represent the slogan, “Homelessness Is Personal.” Using the design methodology established, the HIP prototype could be distributed to individuals living in homeless communities in Northwest Arkansas. By using repurposed materials for the end product, waste reduction of textile products may result. Further, engagement of the community could be accomplished by using the design methodology to establish work groups for production of the sleeping bags. Ultimately, this design methodology was designed so that distribution to other communities outside of the Northwest Arkansas region could be a foreseeable future goal of the committee. Materials collected from Goodwill in Fayetteville, Arkansas were used in creating the prototype and production was completed at the University of Arkansas Apparel Merchandising and Product Development (AMPD) design and development labs. Projections indicate that the design, development, and marketing efforts surrounding the HIP prototype are potentially conducive to the creation of a lasting and sustainable project, which might continue to creatively engage students for many years to come

FOLDED MYSTERY

The body of work in my thesis project titled Folded Mystery are the metaphors for how we exchange knowledge, how perception widens our perspective, and how observation deepens our understanding of the reality in which we live. I seek works of art that activate once the viewer is involved. Folded Mystery is about challenging viewers\u27 perception of multi-perception embodiment through 2D and 3D drawings, sculptural paintings, and installations that focus on the interaction of geometric abstract forms, colors, reflective objects and layering grid-like materials in space