Solving Jigsaw Puzzles By the Graph Connection Laplacian

We propose a novel mathematical framework to address the problem of automatically solving large jigsaw puzzles. This problem assumes a large image, which is cut into equal square pieces that are arbitrarily rotated and shuffled, and asks to recover the original image given the transformed pieces. The main contribution of this work is a method for recovering the rotations of the pieces when both shuffles and rotations are unknown. A major challenge of this procedure is estimating the graph connection Laplacian without the knowledge of shuffles. We guarantee some robustness of the latter estimate to measurement errors. A careful combination of our proposed method for estimating rotations with any existing method for estimating shuffles results in a practical solution for the jigsaw puzzle problem. Numerical experiments demonstrate the competitive accuracy of this solution, its robustness to corruption and its computational advantage for large puzzles

Computational design of steady 3D dissection puzzles

Dissection puzzles require assembling a common set of pieces into multiple distinct forms. Existing works focus on creating 2D dissection puzzles that form primitive or naturalistic shapes. Unlike 2D dissection puzzles that could be supported on a tabletop surface, 3D dissection puzzles are preferable to be steady by themselves for each assembly form. In this work, we aim at computationally designing steady 3D dissection puzzles. We address this challenging problem with three key contributions. First, we take two voxelized shapes as inputs and dissect them into a common set of puzzle pieces, during which we allow slightly modifying the input shapes, preferably on their internal volume, to preserve the external appearance. Second, we formulate a formal model of generalized interlocking for connecting pieces into a steady assembly using both their geometric arrangements and friction. Third, we modify the geometry of each dissected puzzle piece based on the formal model such that each assembly form is steady accordingly. We demonstrate the effectiveness of our approach on a wide variety of shapes, compare it with the state-of-the-art on 2D and 3D examples, and fabricate some of our designed puzzles to validate their steadiness

Dr. KID: Direct Remeshing and K-set Isometric Decomposition for Scalable Physicalization of Organic Shapes

Dr. KID is an algorithm that uses isometric decomposition for the physicalization of potato-shaped organic models in a puzzle fashion. The algorithm begins with creating a simple, regular triangular surface mesh of organic shapes, followed by iterative k-means clustering and remeshing. For clustering, we need similarity between triangles (segments) which is defined as a distance function. The distance function maps each triangle's shape to a single point in the virtual 3D space. Thus, the distance between the triangles indicates their degree of dissimilarity. K-means clustering uses this distance and sorts of segments into k classes. After this, remeshing is applied to minimize the distance between triangles within the same cluster by making their shapes identical. Clustering and remeshing are repeated until the distance between triangles in the same cluster reaches an acceptable threshold. We adopt a curvature-aware strategy to determine the surface thickness and finalize puzzle pieces for 3D printing. Identical hinges and holes are created for assembling the puzzle components. For smoother outcomes, we use triangle subdivision along with curvature-aware clustering, generating curved triangular patches for 3D printing. Our algorithm was evaluated using various models, and the 3D-printed results were analyzed. Findings indicate that our algorithm performs reliably on target organic shapes with minimal loss of input geometry

The Development of Early Spatial Thinking

The different spatial experiences in the lives of young boys and girls may partly explain sex differences in spatial skills (Baenninger & Newcombe, 1995; Nazareth et al., 2013; Newcombe, Bandura & Taylor, 1983). While several studies have examined the influence of spatial activities on the development of spatial skills (e.g., Nazareth et al., 2013) there currently exists no widely used comprehensive measure to assess children’s concurrent participation in spatial activities and engagement with spatial toys. Study 1 of the current dissertation filled this gap in the field of spatial research through the creation of the Spatial Activity Questionnaire, a comprehensive survey designed to assess children’s involvement in spatial activities and engagement with spatial toys of diverse gender-typed content. The toys and activities 295 children were reported to have access to and engage with were explored to assess patterns of play with spatial and gender-stereotyped toys and activities. A sample of 76 children between 4 and 6 years of age and their primary caregivers participated in studies 2, 3, and 4 to explore the toys and activities young children have access to and play with (study 2), the link between play and mental rotation (study 3), and the relation between play, gender stereotypes, and mental rotation skills (study 4). Findings reveal great variability in the toys and activities children have access to and play with, with sex difference suggesting girls play with low-spatial and stereotypically feminine toys and activities more than boys while boys play with highly-spatial and stereotypically masculine toys and activities more than girls. Adding to the exiting literature suggesting the inconsistency of sex differences in early mental rotation skills, our results suggest no sex differences in children’s mental rotation ability. Furthermore, no relations were discovered between children’s play, gender stereotypes, and mental rotation ability. These findings point to the need to further explore the influence of play on when and how sex differences in mental rotation ability develop in order to promote fun and easy ways to support spatial learning in young boys and girls

Boxelization: folding 3D objects into boxes

We present a method for transforming a 3D object into a cube or a box using a continuous folding sequence. Our method produces a single, connected object that can be physically fabricated and folded from one shape to the other. We segment the object into voxels and search for a voxel-tree that can fold from the input shape to the target shape. This involves three major steps: finding a good voxelization, finding the tree structure that can form the input and target shapes' configurations, and finding a non-intersecting folding sequence. We demonstrate our results on several input 3D objects and also physically fabricate some using a 3D printer