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

On Dissecting Polygons into Rectangles

What is the smallest number of pieces that you can cut an n-sided regular polygon into so that the pieces can be rearranged to form a rectangle? Call it r(n). The rectangle may have any proportions you wish, as long as it is a rectangle. The rules are the same as for the classical problem where the rearranged pieces must form a square. Let s(n) denote the minimum number of pieces for that problem. For both problems the pieces may be turned over and the cuts must be simple curves. The conjectured values of s(n), 3 <= n <= 12, are 4, 1, 6, 5, 7, 5, 9, 7, 10, 6. However, only s(4)=1 is known for certain. The problem of finding r(n) has received less attention. In this paper we give constructions showing that r(n) for 3 <= n <= 12 is at most 2, 1, 4, 3, 5, 4, 7, 4, 9, 5, improving on the bounds for s(n) in every case except n=4. For the 10-gon our construction uses three fewer pieces than the bound for s(10). Only r(3) and r(4) are known for certain. We also briefly discuss q(n), the minimum number of pieces needed to dissect a regular n-gon into a monotile.Comment: 26 pages, one table, 41 figures, 14 reference

Emotion and the Seduction of the Senses, Baroque to Neo-Baroque

Emotion and the Seduction of the Senses, Baroque to Neo-Baroque examines the relationship between the cultural productions of the baroque in the seventeenth century and the neo-baroque in our contemporary world. It asks the question: Is the baroque a recurring phenomenon that has returned in aspects of contemporary global culture, or is it something specific to the early modern period? It argues one of the common and central features of both styles is their appeal to emotion. This volume illuminates how, rather than providing rationally ordered visual realms, both the baroque and the neo-baroque construct complex performative spaces whose spectacle seeks to embrace, immerse, and seduce the senses and solicit the emotions of the beholder.https://scholarworks.wmich.edu/mip_smemc/1006/thumbnail.jp

MATHEMATICS VERSUS THE ARTS: A COMPARATIVE LOOK AT STUDENTS' ATTITUDES AND BELIEFS

Merged with duplicate record 10026.1/1937 on 15.02.2017 by CS (TIS)The words that students use to paint a picture of mathematics are very different from those which they use to describe their experiences in art and music. In the views of students, mathematics is pointless and repetitive while the arts are creative, relaxing and an expression of themselves. This thesis reports on the findings of a two part research project designed to investigate the attitudes of high school students when learning mathematics, art and music. The focus of this study was a comparative look at their confidence and enjoyment in learning these subjects. A questionnaire was designed and developed for use in a study of students in the United States and England (n = 1226). The intent of this questionnaire, which contained seven Likert-type questions and one open-ended response, was an in-depth look at the existence of confidence and enjoyment in learning mathematics, art and music. The results indicated that, the highest frequency of students in the mathematics group were confident in their ability in mathematics but did not enjoy learning it. This study also found, however, that there were very low percentages of students that were confident in art and music but did not enjoy learning them. Additionally there was a high frequency of students who had no confidence in art but did enjoy learning it compared to a low frequency of students in mathematics who were not confident but enjoyed learning it. To further explore these findings, repertory grid interviews were conducted on a selection of questionnaire participants from the United States (n = 42). Honey's method of content analysis was used to analyse the data. Among the differences found between students' confidence and enjoyment in learning mathematics compared to the arts were their perceptions of the routine nature of their daily lessons in mathematics versus their active, creative, personally engaging experiences while learning art and music

Considerations in designing a cybernetic simple 'learning' model; and an overview of the problem of modelling learning

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Learning is viewed as a central feature of living systems and must be manifested in any artifact that claims to exhibit general intelligence. The central aims of the thesis are twofold: (1) - To review and critically assess the empirical and theoretical aspects of learning as have been addressed in a multitude of disciplines, with the aim of extracting fundamental features and elements. (2) - To develop a more systematic approach to the cybernetic modelling of learning than has been achieved hitherto. In pursuit of aim (1) above the following discussions are included: Historical and Philosophical backgrounds; Natural learning, both physiological and psychological aspects; Hierarchies of learning identified in the evolutionary, functional and developmental senses; An extensive section on the general problem of modelling of learning and the formal tools, is included as a link between aims (1) and (2). Following this a systematic and historically oriented study of cybernetic and other related approaches to the problem of modelling of learning is presented. This then leads to the development of a state-of-the-art general purpose experimental cybernetic learning model. The programming and use of this model is also fully described, including an elaborate scheme for the manifestation of simple learning