Rectangles and spirals

Every reader knows about the Golden Rectangle (see [1, pp. 85, 119], [2, 3]), and that it can be subdivided into a square and a smaller copy of itself, and that this process can be continued indefinitely, converging towards the intersection point of diagonals of any two successive rectangles in the sequence. The circumscribed logarithmic spiral passing through the vertices and converging to the same point is also familiar (see [3, 4]), and is analogous to the circumcircle of a regular polygon or a triangle. The approximate logarithmic spiral obtained by drawing a quarter-circle inside each of the squares is equally well known [3, p. 64]. Perhaps slightly less familiar is the inscribed spiral, which is tangential to a side of every rectangle, like the incircle of a triangle or a regular polygon. It does not (quite) coincide with the spiral passing through the point of subdivision of each side, as discussed in [3, pp. 73-77]. The Golden Rectangle, its subdivisions, and the circumscribed and inscribed spirals are illustrated in Figure 1.

APPLICABILITY OF GOLDEN RECTANGLE IN FAÇADE OF BUILT FORM

All of us want to promote pleasing architecture but is there a clear theory that explains what makes a design good. Why do some buildings look pleasing to us while the others not so much? Why do a few of them give us a sense of wellbeing whereas a few of them make us feel out of place. Architecture never has a single answer and is very subjective. It is evident that there could be a tangible relationship between the visual aspect of a building and the intangible that is the way we feel about it. This research tries to understand these tangible characteristics of a built form and tries to objectify aesthetics by analyzing it through the lens of scale and proportion. “Geometry existed before creation” - Johan Kepler “Nature seems to be written in the language of mathematics” - Galileo Galilei These statements form the basis of this research. It will start with the study of the relationship of aesthetics of façade and their proportions by overlaying the guiding principles attached to these attributes on several built forms. Once established as to how through history buildings have related to different simple and complex equations the research moves on to perceive a certain number of buildings in current scenario. These buildings may or may not have been built on some geometrical guiding principles. Later through survey the research will try and identify whether or not an aesthetically pleasing building has some relationship with scale. It will try and identify the exact equation that guided the form of the building. The relationship between proportion and aesthetics will be established by running experiments, overlaying and calculating proportion theories on to the façade of that particular built form.

Investigate, because of the golden rectangle

“Investigate, because of the golden rectangle” offers mathematics students and enthusiasts inspiration for mathematical play by way of a guided construction of the golden rectangle. The discussion is illustrated with numerous hand-drawn sketches. A golden rectangle is a rectangle whose side lengths are in the golden ratio, which is, where the Greek letter (pronounced “phi”) is approximately equal to. Readers learn that an indirect, even haphazard, approach in mathematical play may lead to unanticipated discoveries. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Shrinkage Points of Golden Rectangle, Fibonacci Spirals, and Golden Spirals

We investigated the golden rectangle and the related Fibonacci spiral and golden spiral. The coordinates of the shrinkage points of a golden rectangle were derived. Properties of shrinkage points were discussed. Based on these properties, we conduct a comparison study for the Fibonacci spiral and golden spiral. Their similarities and differences were looked into by examining their polar coordinate equations, polar radii, arm-radius angles, and curvatures. The golden spiral has constant arm-radius angle and continuous curvature, while the Fibonacci spiral has cyclic varying arm-radius angle and discontinuous curvature.

FINDING HARMONY IN CHAOS: THE ROLE OF THE GOLDEN RECTANGLE IN DECONSTRUCTIVE ARCHITECTURE

It is generally accepted that compositions in deconstructive architecture are irrational, fragmented, and do not follow proportional systems or principles of architecture, such as harmony, continuity, and unity. These compositions are understood as the result of compilations of random geometries that are often non-rectilinear, distorted, and displaced. In spite of this, deconstructive architecture is widely accepted and practiced in the last couple of decades. On the other hand, geometrical proportions have long been considered as a self-guided method of aesthetically proven designs. This paper examines the hypothesis that the golden rectangle as a proportional system is manifested, to a varying degree, in deconstructive architecture. Methodologically, the hypothesis was tested using two inter-related methods. First, Tension Points of three famous examples of deconstructivist architecture were identified using the Delphi method by a panel of experts. Second, a matrix of displaced golden rectangles was used to test the degree of correspondence between the tension points of the case studies and the golden rectangle. It was found that deconstructive architecture is not a type of “free-form” architecture; and that conventional proportional systems and aesthetics laws, such as the golden ratio, are partially manifested in its compositions and forms, thus confirming the hypothesis. This paper argues that since architects are trained to capture proportional systems and design according to certain organizational and proportional principles, this would inevitably be consciously or unconsciously reflected on their designs.