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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.

In researches of basic theory of TCM, the Luoshu has direct relationship with the Jiugong bafeng theory. In this paper based on analysis of mathematical relations among the Fibonacci sequence, the Fibonacci rectangle and golden rectangle, according to geometric properties of the golden rectangle as well as numeric distribution rules implied in Luoshu, relationship between the Fibonacci sequence including its modular series by 10 and Luoshu is studied, and inherent and consistent mathematical relationship between them is found in spatial orientation, digital running sequence, numeric grouping and generating rules, etc. The study results of this paper can help to build up close mathematical association between Luoshu and the Fibonacci sequence, which can lay key theoretical foundation for the ultimate breakthrough for scientific mysteries containing in Luoshu and the Jiugong bafeng theory in TCM.

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.

As part of a larger research aimed at developing design aids for architects, this paper presents the “automated” generation of the “best connected” rectangular floor plans, satisfying given topological and dimensional constraints. It has been seen that architects, knowingly or unknowingly, have often used either the golden rectangle or the Fibonacci rectangle in their works throughout history. But it was hard to find any specific reason for such use, other than aesthetic. In 2015, Shekhawat showed that they are among the best connected rectangular arrangements (dimensionless rectangular floor plans) and that this may well be another reason for their frequent use in architectural design. In this work, an alternative algorithm is presented which generates n − 3 best connected rectangular arrangements, being n the number of rooms. Then, this concept is further extended for constructing the best connected dimensioned rectangular floor plans. The goal is to provide an optimal solution for the rectangular space allocation problem, while satisfying given topological and dimensional requirements.

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.

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.

The aim of this work is is to give answers to question is aesthetic preference of object position on pictures stabile, or is there a change of aesthetic preference with increase of number of objects and with the change of picture orientation (horizontal-vertical). In conducted experiments subjects had a task to put one, two or three circles on three different shapes of backgrounds (square, golden rectangle, rectangle), in such way that given configuration is the most beautiful one in their own opinion. In some experiments backgrounds were observed horizontaly, and in other verticaly. When the backgrounds were horizontal, aesthetic preference of golden section position did not change with increase of the number of circles. When the backgrounds were vertical golden section position was prefered one in cases with one and two circles, while in the experiment with three circles aesthetic preference of golden section position decreased. In most situations circles were ordered on backgrounds in such way to balance each other. Distance between two circles on same shapes of backgrounds, on repeated situations, is relativly constant in both orientations of backgrounds.

In this study three hypothesis were evaluated. The first claims that the golden section position is an ideal position of an object on a picture and that this position does not depend on picture shape, or on the number of objects on it. According to the second hypothesis, the aesthetically optimal effect is achieved when the focus is on the right side of the picture ( for asymmetrically composed pictures). According to the third hypothesis, there is an influence of previous stimulation on aesthetic experience; that is, because of the monotony, the aesthetic preference of observers will change. An experiment was done, with two sections. In the first section, subjects were asked to put a little black circle, on three different shapes of cards (square, golden rectangle and rectangle), in a such way that the given configuration is the most beautiful one in their own opinion. The second section of the experiment was almost identical to the first one, with the exception that the subjects were asked to put two circles on each of the cards. Each one of the three hypothesis was confirmed by the results of this experiment. The preferred position of the circle is the same as the position of the golden section and it does not change with the change of card shape and number of objects. There is a clear preference of the upper-right corner of cards. The preferred position of an object is changed with repetition of the same stimulation (the same shape of cards and the same number of circles).

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.

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.

Aesthetics in user interfaces addresses font definitions, typesetting conventions, color combinations, graphics design considerations, high resolution for viewscreens, and the shapes of windows. Computer viewscreens are evolving into pictorial media, communicating information with visual immediacy. The more interesting interfaces make use of multiple windows, menus, icons, and other visual effects to waken and sustain user interest and effectiveness. A pretty window is a viewscreen window with the dimensions of a golden rectangle, a rectangle whose width and height form the golden ratio of Euclid. Psychologists believe golden rectangles are aesthetically more pleasing than arbitrary rectangles, and subjects tend to select them in preference to other rectangles in tests.

Conformal transformations are given which are suitable for the discussion of the space outside a finned rectangle. These include figures such as rectangle, cross, three-sided box, form like the letter H, “windmill”, and the transformation could be applied to any rectangle with four fins of any lengths at its corners, each of them in either of two directions. The transformations convert the outer space into the inside of a rectangle, which can then be transformed back into the outside of a circle, with the same point at infinity as the original-space. By a generMization of a special case of this transformation another is given, whereby an irregular star, composed of any number of rays of any lengths and at any angles branching out from a centre, can be transformed into a circle. This includes, as a simple example, a “dihedral” of two equal arms at any angle.

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.