Search

Geometric properties of finite Blaschke products have been intensively studied by many different aspects. In this paper, our aim is to study geometric properties of finite Blaschke products related to the golden ratio $\alpha =\frac{1+\sqrt{5}}{2}$. Mainly, we focus on the relationships between the zeros of canonical finite Blaschke products of lower degree and the golden ratio. We show that the geometric notions such as "golden triangle, "golden ellipse" and "golden rectangle" are closely related to the geometry of finite Blaschke products.

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.

“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.

The Fibonacci sequence of numbers is known since the ancient past. It is considered as a proportion which can be frequently appreciated in nature. The associated ratio, called as the golden ratio is manifested in the works of art, nature, galaxies, monuments etc. Many famous mathematicians, philosophers, and artists have used this ratio in their work. Humans distinguish this proportion as pleasing, as it enlightens beauty when applied in living or non living entities. The ideal proportion is directly related to golden proportion. It ranges from 1 to 1.618. Moreover, the proportion can be applied in facial aesthetics also. Studies have showed that the beautiful faces have facial measurements close to golden ratio. Thus, the treatment following the standard will help in obtaining optimal facial aesthetics. The review is written with an aim to highlight the historical perspectives, its varied applications in the past and present; and further scope of research in future.

This paper presents an interesting deduction of the Golden Spiral equation in a suitable polar coordinate system. For this purpose, the concepts of Golden Ratio and Golden Rectangle, and a significant result for the calculation of powers of the Golden Ratio ϕ using terms of the Fibonacci sequence are mentioned. Finally, various geometrical considerations that help us deduce the sought equation are presented.

It can be found quite often in the literature that many well-known architects have employed either the golden rectangle or the Fibonacci rectangle in their works. On contrary, it is rare to find any specific reason for using them so often. Recently, Shekhawat (2015) proved that the golden rectangle and the Fibonacci rectangle are one of the best connected rectangular arrangements and this may be one of the reasons for their high presence in architectural designs. In this work we present an algorithm that generates n - 4 best connected rectangular arrangements so that the proposed solutions can be further used by architects for their designs.

The golden ratio has universal application and provides maximum performance in an artistic composition. This article discusses how knowledge about golden ratio can be used in jewelry design. The golden rectangle and golden pentagon were used as grids to generate jewelry with harmonic proportions.

A polyomino is called odd if it can tile a rectangle using an odd number of copies. We give a very general family of odd polyominoes. Specifically, consider an L-shaped polyomino, i.e., a rectangle that has a rectangular piece removed from one corner. For some of these polyominoes, two copies tile a rectangle, called a basic rectangle. We prove that such a polyomino is odd if its basic rectangle has relatively prime side lengths. This general family encompasses several previously known families of odd polyominoes, as well as many individual examples. We prove a stronger result for a narrower family of polyominoes. Let L n denote the polyomino formed by removing a 1 × (n−2) corner from a 2 × (n−1) rectangle. We show that when n is odd, L n tiles all rectangles both of whose sides are at least 8n 3, and whose area is a multiple of n. If we only allow L n to be rotated, but not reflected, then the same is true, provided that both sides of the rectangle are at least 16n 4. We also give several isolated examples of odd polyominoes.

The two most striking characteristics of the sequence of primes are that there are many of them but that their density is rather slim.

Suppose that a rectangle with a longitudinally symmetric slit is given. Then we want to find another rectangle with a square removed and a conformal mapping between them so that the ordered set of four vertices is mapped on the other ordered set of four vertices. We also determine the modulus of the image rectangle and the size of the square removed.

which is called the golden ratio [5] or its reciprocal [4], depending on the author. Almost any discussion oftthe golden rectangle will include the image in figure 1, which is constructed by starting with a golden rectangle, partitioning it into a square and a smaller golden rectangle, and then repeating the process indefinitely. We will refer to any such set as a golden set. Authors occasionally include the golden set in discussions of self-similarity [5, 6], because it is the union of a square and a smaller copy of itself, scaled by a factor of y. Some popularizations of the golden ratio [4, 7] also include examples of truly self-similar, planar fractals to illustrate properties of y. Now the golden set is not truly self-similar, because it is not a union of smaller copies of itself, but it suggests a problem: Can we construct a self-similar fractal made of golden sets?

The golden quadratic x 2 − x − 1 = 0, when re-expressed as (x)(1) = 1/(x − 1), x = 1.618, can be interpreted as the algebraic expression of division in extreme and mean ratio (DEMR) of a line of length x = 1.618 into a longer section of length 1 and a smaller of length (x − 1). It can, however, also be interpreted as the formulation of the area of a golden rectangle of sides x = 1.618 and 1, and as the system of equations constituted by y = x, and y = 1/(x − 1). Based on the well-known connection existing between the first two of these interpretations, the authors address the problem of finding out the thread connecting the golden rectangle with the system of equations referred to above. The results obtained indicate first that this system, like the golden rectangle, also carries in its geometry the essential traits of DEMR; and, second, that it implicitly subsumes the simpler rectangular geometry of its alternative interpretation. The process of developing these connections brought forward a heretofore apparently unreported golden trapezoid of sides Φ, 1, ϕ and \(\sqrt{2}\).

Geometer Mark Reynolds discusses four approximatons for finding the Golden Section of the circumference of a circle from the root-2 rectangle.

This paper gives algorithms to dissect a rectangle sequentially into polygons that can be reassembled to a rectangle, which is similar to the initial rectangle, and [Formula: see text]-similar equilateral triangles or regular hexagons and also pentagons or heptagons or octagons with symmetry.

We show how the unitary, genetic S-code description of the family of Kepler conic sections, not only enlightens the genesis of the so called Dandelin spheres but also naturally unfolds in the Kepler scenery the famous golden ratio, the golden rectangle and the Fibonacci sequence.

Can we produce a ratio ψ to rival the golden ratio ϕ? In this article I shall construct a variant of the classical golden rectangle by choosing a different geometrical property. The rectangle described has more advanced mathematical properties than its golden relation, though it is not extravagant. In this sense it could be described as a supergolden rectangle.

The asymptotic approach of a certain sequence of rectangles (rectangular parallelepipeds/hyperparallelepipeds) to a golden rectangle (rectangular paral‐lelepiped/hyperparallelepiped) is considered.

Some enrichment ideas suitable for the advanced algebra class or the mathematics club

Seven rectangles with different ratios of the lengths of their sides but of approximately equal areas were presented to 120 subjects to assess their aesthetic preferences for each rectangle. The method of pair comparisons was used for presentation of the rectangles. Subjects tended to prefer the golden rectangle. Also rectangles in the vicinity of the golden rectangle were preferred to those deviating markedly from it.

From two different series of dimensions were generated six ranges of nine rectangles each. Each range was presented to a different group of 15 men and 15 women, and a novel measure was used to assess aesthetic preference for each rectangle. The ratios between the lengths of the sides of the rectangles were the same in both series, but in one series the size of the rectangles covaried with the ratio between length and width, as in Godkewitsch's 1974 study, and in the other series the rectangles were of equal area. The subjects tended to prefer large to small rectangles in the former series and rectangles in the vicinity of the 'golden section' in the latter.