Using an apparently simple problem, ''Design a cylindrical can that will hold a liter of milk,'' this paper demonstrates how engineering design may facilitate the teaching of the following ideas to secondary students: linear and non-linear relationships; basic geometry of circles, rectangles, and cylinders; unit measures of area and volume;…

This paper describes the procedures of reconstructing ancient architecture using solid modeling with geometric analysis, especially the Golden Ratio analysis. In the past the recovery and reconstruction of ruins required bringing together fragments of evidence and vast amount of measurements from archaeological site. Although researchers and…

The general purpose of this article is to shed some light on the understanding of real numbers, particularly with regard to two issues: the real number as the limit of a sequence of rational numbers and the development of irrational numbers as a continued fraction. By generalizing the expression of the golden ratio in the form of the limit of two…

Many assertions about the occurrence of the golden ratio phi in art, architecture, and nature have been shown to be false, unsupported, or misleading. For instance, we show that the spirals found in sea shells, in particular the "Nautilus pompilius," are not in the shape of the golden ratio, as is often claimed. Some of the most interesting…

Cubic symmetry is used to build the other four Platonic solids and some formalism from classical geometry is introduced. Initially, the approach is via geometric construction, e.g., the "golden ratio" is necessary to construct an icosahedron with pentagonal faces. Then conventional elementary vector algebra is used to extract quantitative…

Typically, the mathematical properties concerning the golden ratio are stated correctly, but much of what is presented with respect to the golden ratio in art, architecture, literature, and aesthetics is false or seriously misleading. Discussed here are some of the most commonly repeated misconceptions promulgated, particularly within mathematics…

In 1854 the German scientist Zeising claimed that the ratio of a person's height to the height of their navel is in the same ratio as the Golden Ratio ([phi] = 1.62). There is so much hype about the Golden Ratio that it is worth reading an article at http://www.maa.org/devlin/devlin_06_04.html. It explains why some of the more far-fetched ideas…

The Golden Ratio is sometimes called the "Golden Section" or the "Divine Proportion", in which three points: A, B, and C, divide a line in this proportion if AC/AB = AB/BC. "Donald in Mathmagicland" includes a section about the Golden Ratio and the ratios within a five-pointed star or pentagram. This article presents two computing exercises that…

This article demonstrates how within an educational context, supported by the notion of hidden mathematics curriculum and enhanced by the use of technology, new mathematical knowledge can be discovered. More specifically, proceeding from the well-known representation of Fibonacci numbers through a second-order difference equation, this article…

A generalization of the golden ratio is made, called the silver ratio. Some examples where the golden ratio appears are provided so that the silver ratio appears. (MNS)

Diagrams that aid in relating the golden ratio to pi are discussed, with the theorem and its proof. (MNS)

Using investigations in teaching mathematics has for many years become an established feature of most curricula around the world. Investigations can be a vehicle for enabling children to experience the genuine excitement that comes from mathematical discovery. The true spirit of inquiry and investigation lies in the mind-set that continually asks…

This article explores the multiple representations (verbal, algebraic, graphical, and numerical) that can be used to study the golden ratio. Emphasis is placed on using technology (both calculators and computers) to investigate the algebraic, graphical, and numerical representations. (JAF)

Describes computation of a continued radical to approximate the golden ratio and presents two well-known geometric interpretations of it. Uses guided-discovery to investigate different repeated radicals to see what values they approximate, the golden-rectangle interpretation of these continued radicals, and the golden-section interpretation. (KHR)

Discusses the golden ratio and how to make golden triangles using recursive Logo programs. Presents some outputs of the Penrose tile patterns. (YP)