Mathematical Understandings Of A Rubik\u27s Cube

Many people are familiar with the 3x3x3 Rubik’s Cube as a puzzle or a toy. But, what most people do not realize is that the cube is a great physical visual of a group. The goal of this paper is to discuss the Rubik’s Cube as a group and dive into a specific subgroup of the cube. Through this discussion, we will also explore homomorphisms in the slice group. This paper will also give insight on permutations, commutators, and conjugates in terms of the cube, as well as “God’s number”

Towards an Effective Theory of Reformulation

This paper describes an investigation into the structure of representations of sets of actions, utilizing semigroup theory. The goals of this project are twofold: to shed light on the relationship between tasks and representations, leading to a classification of tasks according to the representations they admit; and to develop techniques for automatically transforming representations so as to improve problem-solving performance. A method is demonstrated for automatically generating serial algorithms for representations whose actions form a finite group. This method is then extended to representations whose actions form a finite inverse semigroup

Proceedings of the Workshop on Change of Representation and Problem Reformulation

The proceedings of the third Workshop on Change of representation and Problem Reformulation is presented. In contrast to the first two workshops, this workshop was focused on analytic or knowledge-based approaches, as opposed to statistical or empirical approaches called 'constructive induction'. The organizing committee believes that there is a potential for combining analytic and inductive approaches at a future date. However, it became apparent at the previous two workshops that the communities pursuing these different approaches are currently interested in largely non-overlapping issues. The constructive induction community has been holding its own workshops, principally in conjunction with the machine learning conference. While this workshop is more focused on analytic approaches, the organizing committee has made an effort to include more application domains. We have greatly expanded from the origins in the machine learning community. Participants in this workshop come from the full spectrum of AI application domains including planning, qualitative physics, software engineering, knowledge representation, and machine learning

Zero knowledge with Rubik's cubes and non-abelian groups

International audienceThe factorization problem in non-abelian groups is still an open and a difficult problem [12]. The hardness of the problem is illustrated by the moves of the Rubik's cube.We will define a public key identification scheme based on this problem, in the case of the Rubik's cube, when the number of moves is fixed to a given value. Our scheme consists of an interactive protocol which is zero-knowledge argument of knowledge under the assumption of the existence of a commitment scheme. We will see that our scheme works with any non-abelian groups with a set of authorized moves that has a specific property. Then we will generalize the scheme for larger Rubik's cubes and for any groups. © Springer International Publishing 2013

Interpreting parametric-biomimicry design from cad тo bim software: digital modelling based on a sketch of nandi flame

This research represents an application of two digital modelling softwares, first digital modelling software, chosen as representative of Computer-Aided Design – CAD modelling tool was Fusion 360. The representative of Building Information Modelling (BIM) as second digital modelling software was ArchiCAD. The aim of the research was to translate the same parametric-biomimicry design methodology used in CAD process modelling into BIM environment. African species Spathodea campanulata P. Beauv, whose common name in Kenya is Nandi flame, has been selected for the purpose of this digital modelling processes. As one of the most spectacular flowering plants, Nandi flame is indigenous to the tropical dry forests in Kenya. The decorative flower of this species was the basic model, more precisely the botanical sketches of the flower. This sketches were implemented into digital modelling softwares and used for parametric modelling. The results of this processes were represented as urban models or installations (landscape-architectural elements) in open space. This approach of digitally generating conceptual solutions from nature elements has capability to boost the formulation of new creative inventions in the different fields. The unique geometric patterns found in the flower of Spathodea campanulata P. Beauv served as a good example of how we may transform these ideas into actual design installations– using CAD or BIM software tools. This research has been carried out with the aim to find the position of BIM tools in parametric biomimicry design