Book Review: How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics

If you think about it, mathematics is really just one big analogy. For one example, the very concept of the number three is an drawing an analogy between a pile with three rocks, a collection of three books, and a plate with three carrots on it. For another, the idea of a group is drawing an analogy between adding real numbers, multiplying matrices, and many other mathematical structures. So much of what we do as mathematicians involves abstracting concrete things, and what is abstraction other than a big analogy? [excerpt

Network protocol scalability via a topological Kadanoff transformation

A natural hierarchical framework for network topology abstraction is presented based on an analogy with the Kadanoff transformation and renormalisation group in theoretical physics. Some properties of the renormalisation group bear similarities to the scalability properties of network routing protocols (interactions). Central to our abstraction are two intimately connected and complementary path diversity units: simple cycles, and cycle adjacencies. A recursive network abstraction procedure is presented, together with an associated generic recursive routing protocol family that offers many desirable features.Comment: 4 pages, 5 figures, PhysComNet 2008 workshop submissio

Building Student’s Mathematical Connection Ability in Abstract Algebra: The Combination of Analogy-Contruction-Abstraction Stages

The objective of the study was to describe the effect of six types of mathematical connections (representation connections, structural connections, procedural connections, implication connections, generalization connections, and hierarchy connections) on abstract algebraic materials through four stages, i.e., abstraction, analogy-abstraction, construction-analogy, and construction. The study employed qualitative descriptive approaches, including tests, questionnaires, and interviews. The subjects of the study were chosen based on the responses to a questionnaire regarding the employed stages. Then, two subjects who could converse and were willing to be interviewed were chosen from each stage. Data collection techniques were conducted through four stages, i.e., 1) identifying the stages used; 2) identifying the ability of six types of student mathematical connections through predictive indicators; 3) describing the capabilities of the six types of connections through interviews; and 4) conducting source triangulation and method triangulation. The results indicated that the subjects who utilized the construction stage tended to be able to construct six types of mathematical connection links in a set, as well as standard and non-standard binary operations. The subjects who utilized the construction-analogy stage likely to be able to build three forms of representation connections, structural connections, and procedural connections in a set of standard binary operations. In characterizing the symbol of a set element and the binary operation of the standard form inside the closed property of the standard form, the subjects who used the analogy-abstraction stage have the same tendency as subjects who use the abstraction-construction stage

Network protocol scalability via a topological Kadanoff transformation

A natural hierarchical framework for network topology abstraction is presented based on an analogy with the Kadanoff transformation and renormalisation group in theoretical physics. Some properties of the renormalisation group bear similarities to the scalability properties of network routing protocols (interactions). Central to our abstraction are two intimately connected and complementary path diversity units: simple cycles, and cycle adjacencies. A recursive network abstraction procedure is presented, together with an associated generic recursive routing protocol family that offers many desirable features.Comment: 4 pages, 5 figures, PhysComNet 2008 workshop submissio

Use of elastic stability analysis to explain the stress-dependent nature of soil strength

The peak and critical state strengths of sands are linearly related to the stress level, just as the frictional resistance to sliding along an interface is related to the normal force. The analogy with frictional sliding has led to the use of a ‘friction angle’ to describe the relationship between strength and stress for soils. The term ‘friction angle’ implies that the underlying mechanism is frictional resistance at the particle contacts. However, experiments and discrete element simulations indicate that the material friction angle is not simply related to the friction angle at the particle contacts. Experiments and particle-scale simulations of model sands have also revealed the presence of strong force chains, aligned with the major principal stress. Buckling of these strong force chains has been proposed as an alternative to the frictional-sliding failure mechanism. Here, using an idealized abstraction of a strong force chain, the resistance is shown to be linearly proportional to the magnitude of the lateral forces supporting the force chain. Considering a triaxial stress state, and drawing an analogy between the lateral forces and the confining pressure in a triaxial test, a linear relationship between stress level and strength is seen to emerge from the failure-by-buckling hypothesis

Analogy Mining for Specific Design Needs

Finding analogical inspirations in distant domains is a powerful way of solving problems. However, as the number of inspirations that could be matched and the dimensions on which that matching could occur grow, it becomes challenging for designers to find inspirations relevant to their needs. Furthermore, designers are often interested in exploring specific aspects of a product-- for example, one designer might be interested in improving the brewing capability of an outdoor coffee maker, while another might wish to optimize for portability. In this paper we introduce a novel system for targeting analogical search for specific needs. Specifically, we contribute a novel analogical search engine for expressing and abstracting specific design needs that returns more distant yet relevant inspirations than alternate approaches