The Golden Section: The “True” Ratio?

1978 ◽  
Vol 46 (3) ◽  
pp. 831-834 ◽  
Author(s):  
Jochen Piehl
Keyword(s):  
Golden Section ◽  
Golden Rectangle ◽  
Pair Comparisons ◽  
True Ratio ◽  
Aesthetic Preferences

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.

Preference for Proportions as a Function of Context

1997 ◽  
Vol 15 (2) ◽  
pp. 143-151 ◽  
Author(s):  
John Benjafield ◽  
Keith McFarlane
Keyword(s):  
Golden Section ◽  
Current Controversy ◽  
The Aesthetic ◽  
Order Of Presentation ◽  
Aesthetic Preferences

The debate over the aesthetic pleasingness of the golden section is still ongoing, over 100 years after Fechner's pioneering investigations. The present study attempts to advance the debate by investigating the role of context in determining which rectangular proportions are preferred. Participants were shown three different ranges of proportions in three different orders. The order of presentation of ranges influenced aesthetic preferences most when the first range presented contained relatively “thin” rectangles. However, when the first range presented contained relatively “thick” rectangles, or had the golden section as its mid-point, then the most preferred proportion was in the vicinity of the golden section. These results are discussed in relation to the current controversy concerning the aesthetic significance of the golden section.

scholarly journals Aesthetic preference of object position on pictures

Psihologija ◽  
2003 ◽  
Vol 36 (3) ◽  
pp. 313-330 ◽  
Author(s):  
Oliver Toskovic ◽  
Slobodan Markovic
Keyword(s):  
Aesthetic Experience ◽  
Golden Section ◽  
Aesthetic Preference ◽  
Golden Rectangle ◽  
Black Circle ◽  
Different Shapes ◽  
The Aesthetic ◽  
The Right ◽  
The Given ◽  
Optimal Effect

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).

From the Golden Rectangle and Fibonacci to Pedagogy and Problem Posing

1976 ◽  
Vol 69 (3) ◽  
pp. 180-188
Author(s):  
Stephen I. Brown
Keyword(s):  
Number Theory ◽  
Thirteenth Century ◽  
Golden Section ◽  
Golden Ratio ◽  
Mathematical Journal ◽  
Problem Posing ◽  
Total Height ◽  
Regular Pentagon ◽  
Golden Rectangle ◽  
Relative Placement

There is a wealth of valuable material on the golden section, and “fallout” from it as well, Indeed, the problem of constructing the golden rectangle with straightedge and a pair of compasses, and its relationship to constructing the regular pentagon with similar tools, is documented in Euclid's Elements (Euclid 1956; vol. 1, bk. 2, prop. 11; vol. 2, bk. 4. prop. 11). In the thirteenth century, the Italian mathematician Leonardo Fibonacci found connections between that aspect of geometry and number theory, and today there is a mathematical journal that is devoted exclusively to such issues. The golden section has appeared not only in mathematics but in architecture and art as well. The Parthenon in Greece, for example, has the proportions of the golden rectangle, and various portions of Michelangelo's David, from the joints of the fingers to the relative placement of the navel with respect to the total height, exemplify the golden ratio.

Golden Section’s Numerical Approximations in Terms of Graphics and Application

10.12737/5585 ◽  
2014 ◽  
Vol 2 (2) ◽  
pp. 15-20 ◽  
Author(s):  
Сафиулина ◽  
Yu. Safiulina ◽  
Шмурнов ◽  
V. Shmurnov
Keyword(s):  
Line Segment ◽  
Da Vinci ◽  
Golden Section ◽  
Leonardo Da Vinci ◽  
Problem Solution ◽  
Numerical Approximations ◽  
Circle Problem ◽  
Golden Rectangle ◽  
Gradual Development ◽  
Original Scheme

Golden proportion’s graphic plotting methods have been considered. A history related to gradual development of views on Golden section problem as «law of beauty» is traced. Numerical ratios most frequently used in the art for approximations related to division of a line segment in extreme and mean ratio have been provided. An original scheme for «Golden rectangle» construction based on application offered by Leonardo da Vinci for the quadrature of circle problem solution has been proposed.

scholarly journals Stability of aesthetic preference of object position on pictures

Psihologija ◽  
2004 ◽  
Vol 37 (4) ◽  
pp. 507-525
Author(s):  
Oliver Toskovic
Keyword(s):  
Golden Section ◽  
Aesthetic Preference ◽  
Object Position ◽  
Golden Rectangle ◽  
Different Shapes

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.

The Golden Section and the Shape of Objects

1997 ◽  
Vol 15 (2) ◽  
pp. 131-141 ◽  
Author(s):  
Frans Boselie
Keyword(s):  
Golden Section ◽  
Aesthetic Value ◽  
Aesthetic Quality ◽  
Golden Rectangle ◽  
Different Shapes ◽  
The Way

Though the claim that the golden section has special aesthetic value is a very general one, till now it has been tested only for a very small number of different shapes, most research being restricted to the golden rectangle. An experiment is reported in which the way golden ratios are implemented in sides of objects is varied, producing a variety of shapes. It was found that the golden section does not have a special aesthetic quality as compared to the ratio 1.8.

Application of the Golden Section and Golden Rectangle to Fountain Location in Landscape Design

2012 ◽  
Vol 174-177 ◽  
pp. 2554-2557
Author(s):  
Ling Xu ◽  
Chuan Gui Yang ◽  
Tong Chun Wei
Keyword(s):  
Golden Section ◽  
Golden Ratio ◽  
Landscape Design ◽  
Garden Design ◽  
Golden Rectangle

The paper mainly conducts the study on the application of The Golden Ratio in landscape design through the planning of Liuqing Park. Exploring firstly the popularity of The Golden Ratio in plants, animals, and human, and as well as the exploitability in art, architecture, and formal garden design, it makes creative designs to the determination of a fountain and the path annexed to the minor roads. Under the guide of The Golden Ratio, detailed drawing methods to locate the position and the paths have been given, and to be a good try to combine mathematics and landscape design.

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