Shapes, Design and Architecture by the Theory of Privileged Angles and Remarkable Proportions Mathieu Jean -Pierre

Shapes, Design and Architecture  by the Theory of Privileged Angles and Remarkable Proportions

Mathieu Jean -Pierre1

1Honorarus professor , 50, rue du Transvaal, 44300 . NANTES.Cedex 3. FRANCE-(,

 Chaiman of KAIROS-ID, 33 avenue du Maine, 75015, PARIS,

Summary of the chapter

The purpose of this chapter is to show that the theory of privileged angles and remarkable proportions and its application to the shape of commonly used objects and products allow to shed a geometrical and physical light, both on the nature and role of aesthetics in perception, but also on the categorisation’s and memorisation’s processes, the whole, starting from the consumer’s point of view. In a first part, we present this theorical approach in the field of experimental aesthetics, physics, neurophysiology and cognitive psychology, then in the field of its effectiveness, especially in art, as well figurative as abstract. In a second part, we apply it to marketing and design, and, precisely, to the angular analysis of advertising image and, very particularly, to the coupling of text and image, then, afterwards, to the analysis of the shape, and, literally speaking, of the architecture of two beverage’s bottles (extending so several of our already published analysis concerning shapes and architecture of wine’s bottles). To conclude, we shall evoke research’s tracks and managerial applications to sensoriel marketing in its links with design.

Key words: Experimental aesthetics, shape perception, role of privileged angles and « sacred » proportions, shape structure, design, consumer’s behaviour, sensorial marketing.


Innovation takes place in a process which results of exchanges of points of view and of decisions in terms of conception and positioning of a product on a market. But the result is not other than a gathering of shapes, lines, proportions, colors, mechanisms and technology which does not possess in itself a signification, excepted its exchange value, which subtends its compensation power before a social deficiency. Most often the innovation which succeeds on a market find expression through an optimisation and an harmony, between performance and aesthetics concerning what is perceived and lived by all actors around the product. It’s in this perspective that a theory of shapes should not be missed to explicit the contribution of privileged angular components, of certain remarkable proportions and of their filiations and complementarities, to aesthetics, harmony and efficiency.

In this chapter, we shall present a theory of shapes, its main foundations, its involvement in natural and artificial categories, and its method of application. We shall describe later the quantization of shapes and proportions of two products of beverages recently innovated, and then, the quantization of angular couplings in an advertisement image and the text and logo associated, for a service (Russian insurance company). Finally we shall conclude, while emphasizing the contribution brought to the management by the knowledge and the use of the theory and a practice of shapes, based on privileged angles and remarkable proportions, in the analysis and creation processes related to innovation.

Existence and theory of universal shapes : privileged angles and sacred proportions.

The visual characteristics of a product, but also the tactile sensations that come from its covering, its weight or its ergonomic facilities, the animal or vegetal models which have inspired its morphology and the artistic vibrations that it arouses, make it an object of desire and pleasure. So the objects are aimed to satisfy our senses, and the sensorial analysis i.e « the use of human organs as measurement instruments » (Mac Leod, 1996) seems to be quite appropriate to conceive and to test them. However, the study of the relations that users and designers have with their product brings forth the unavoidable notion of « aesthetics » and, for this, the criteria for a comparison between art and design have been established. As a matter of fact, for us, the identification and then the choice of a product are the result, first of all, of the perception of its original shape(Mathieu, 1994), as appearing aesthetic to the consumer who perceives it. For example, beauty according to Fechner « designates anything that has the property of arousing pleasure immediately » (Fechner, 1876) This phenomenon of natural pre-selection is, in our opinion, based on the perception of universal shapes, whose angular structures are created by Privileged Angles (Le Ray, 1980). These have the ability to radiate « the aesthetic experience » created by the designer and decoded by the consumer, where the influence of certain angles and of their shapes (or design) associated as a criteria of performance and aesthetics proves to be universal throughout time and space] ( Martinache, Le Ray and Levin,.1983).

Recent advances in physics and the observations made by a research team over a number of years have revealed that Privileged Angles make a particular contribution to  « beauty, harmony and effectiveness ». Some of these angles also fit the classic proportions known since Antiquity. The observations made first in physics and then in art, nature, architecture and many manmade products (Le Ray, 1980) highlight that “ the use of these angles is the key criterion for performance, and these angles are the trigger that fixes the image” (Mathieu and Le Ray, 1999-06). In marketing, (Gavard- Perret, 1993) has well shown the importance of the relation text/image and the results of its research enlighten the notions of congruence/redundancy beetween the image and the text of an advertising image. It’s thus particularly important from a sensorial point of view, and, specially, while beginning by consumer’s vision (the eye being an essential organ to put in touch the human organism with external world), to explicit the angular structures which subtend the shapes of images and products. In this view, we shall briefly outline the various areas of research in which these angles and other geometric shapes are sources of beauty, balance and harmony ; it may be hypothesized that these particular shapes tend to be memorable because of the feelings of serenity they induce in people.

The purpose of this overview of the contributions of quantum physics, neurophysiology and cognitive psychology to the recognition of shapes and hence to the processes of perception, identification and remembrance is to gain a better understanding of the complex phenomena underlying the reactions of individuals to external stimuli.

The relation established between the microscopic level and the macroscopic scale, due to the presence of privileged angles, has led to observe, in the human creation, a strong tendency, more or less conscious, to want to use these angles, sources of astonishment, adhesion, and even enthusiasm. Other general shapes associated to those privileged angles favor equally at the level of perception by human beings, feelings of equilibrium, stability and quietness

The notion of Privileged Angles developped by physical sciences and between microscopic and macroscopic scales.

This notion of Privileged Angles is known at microscopic scale since 1925-1930 and is located firstly at the scale of simplest atom (hydrogen) in which the rotation of the electron around the nucleus occurs, in the manner of that of a spinning top, around an axis located on a cone whose generating lines make with an external magnetic field or with the axis of an external rotation, one of the Privileged Angles defined a little further. In 1972-1973, generalizing at a much greater scale this capital fact, Le Ray and his coworkers (Le Ray, Deroyon and all, 1972) (Deroyon, Deroyon and Le Ray, 1973) demonstrate the existence in superfluid liquid helium of helicoidal vortices (local axes of rotation in a fluid) obeying at macroscopic scale to the “macroscopic quantization of angular momentum”, and, particularly, making with their axes angles given by the formula below, and only these ones. A supplementary step (Deroyon, Deroyon and Le Ray, 1975)is got over in 1975-76, when is established the key role played in the stability conditions of vortices systems by two families of angles defined by the formula below :

(l and m being integer numbers)

 The cosine of an angle θ, quoted Cosθ, is defined, we recall, or indicate it, as the ratio of the side of the right angle adjacent to the concerned angle over the hypotenuse, second side of the concerned angle, in a right rectangle of which the concerned angle is one of the non-right angles.

These two families correspond to a simple relation between l and m and to successive values of these two numbers or of one of them :

  • for the first family (l=m=1,2,3,4,5,6,7,8,9,……..), these angles equal to 45°;35,3°;30°; 26,6°;24.1° ;22,2° ;20,7°, 19,4° ;18.4°; 17,5°; 16,8°….
  • For the second one (m=2 ;l ≥2), one is concerned with angles equal to 35,3° (this angle being common to both families) ;54,7° ;63 ;4° ;68,6° ;72° ;74.5° ;76,4° ;77,8° ;79° ;80° ;…

It is very important to remark the existence of several combinations of these Privileged Angles :45°=26,6°+18,4° ;63,4°=45°+18,4°=18,4°+26,6°+18,4°and,finally :54.7°=35,3°+19,4°, and of several complementarities of two of these angles : 45° with itself ;54,7° with 35,3° ;63,4°with 26,6°.


The Privileged Angles and the Shapes built by the Man or Nature

Starting from Aerodymamics, Le Ray and his coworkers have questionned themselves about the possible presence of these same Privileged Angles in some natural structures submitted to a skirting by water or air currents, particularly by natural wind. The neighbouring rectilinear or inflexive lines in the branches of trees or along the arrests of very fine sand dunes such as those of Sahara, have been confronted with the Privileged Angles.Thousands checkings have provided, as early as 1976-77, the proof of the adaptation of these natural shapes to the equilibrium angular conditions of the vortices (Proof equally given in the case of sails, airplane wings, cars, ship’s hulls and keels, great velocity train’s locomotives ; see particularly (Le Ray, 1980-85). In 1980 Le Ray (1980) publishes a first synthesis paper from the analysis of thousands of lines on hundreds of documents (works of art, information’s photos ; advertisement images,…) which shows the existence of the coupling between physically or psychologically important lines according to Privileged Angles.

Figures 1,2,3,4 illustrate in major part the Universality of the Privileged Angles phenomena in the flows and the shapes created by Man or Nature in view of a necessary adaptation to privileged relative positions (illustrated by Privileged Angles) of rotation’s axes, commonly called vortices, in the fluid (air or water, most often), these local axes having to be stable with respect to the solid shape to be skirted by the fluid, and having, moreover, in several cases, to be attached to an arrest or a strong curvature zone of the concerned shape..

Figure 1 :Vortices above a Delta Wing whose Leading Edges make between them a Privileged Angle.

Figure 1 shows how, above a Delta Wing placed in a relative wind with respect of which this wing is strongly inclined (inclination, or incidence, that would be observed on a profile view, and which is, evidently, aimed to provide a lifting force simply called lift), spread out vortices called hyperlifting vortices, around which the flow wraps oneself, while increasing its velocity and decreasing the pressure prevalent above the wing. If smoke is emitted in the immediate vicinity of the wing front top, of which are issued the vortices, it divides and follows one or the other of these vortices [here rectilinear as are also the « leading edges » (front or upstream edges) of this wing] since, along the axis of each of these vortices, the fluid’s rotation is zero, while it becomes very intense all around them. The smoke realizes thus here a visualization of a couple of vortices, showing in the event that a wing whose leading edges make between them the Privileged Angles θ11=45° creates vortices placed at θ33=30° one with respect to the other (Figure 1). A wing open at θ22=35,3° creates, for its part, vortices 24,1° angularly apart, just as a wing of aperture angle called « Apex Angle » θ33=30° generates an Intervortices Angle θ77=20,7° and that, finally, one attends again to the « Filiation » θ44=26,6°à θ99=18,4°. Two other filiations are also essential : θ42=63,4°à θ11=45°, and θ32=54,7°à θ22=35,3.

So, have been established (Le Ray, Deroyon and all, 1985) « filiation rules » which, as we shall see it later, are proved to play also a key role in the structure of monuments, objects and text-image coupling

Figure 2 : Concorde and its angular couplings

Figure 2 shows the Leading Edges of Concorde with their angle 30° between their front or upstream parts on each wing, those of 24,1° between each rectilinear part and the following inflexive part on each wing, and, finally, this of 54,7° between the inflexive part of each leading edge and back edge called « Trailing Edge ». These angles have largely contributed to the stability and to the very weak noise level of the flow around the airplane (noise distinct from that generated by the jet engines) during the 34 flight’s years of the most beautiful bird realized until then by man.

As for Figures 3 and 4, they illustrate the presence of privileged angles in natural structures submitted continuously to wind action, or having, essentially, to create, with respect to it, a relative wind so efficient as possible (case of the wing of the flying bird) :

  • Dunes of very fine sand in the desert of Sahara (Figure 3) where the grains are so fine that the arrests place with an extreme accuracy their successive inflexions qualified by the writer Roger FRISON-ROCHE of « pure and quasi abstract wire which ties together their undulations» in such a way that « these moving hills are so right in proportion that while seeing them one could not be able to give them a dimension and that one must think that, while creating them, Nature has divinely respected the Golden Number » (Frison-Roche, 1954).

Figure 3 : Dunes of very fine sand, in the desert of Sahara, near Douiret (Southern Tunisia)

  • Successive elements of bird wing’s leading edge (Figure 4) : the Golden Angles θ42=63,4° gives an impressive majesty to the argented seagull’s flight, evoking this statement of Gaston BACHELARD (1943) « the motion of flight gives, at once, in a lightning abstraction, a perfect, achieved, total dynamic image»

Figure 4 : Argented Seagull in flight

 Angles d’or et autres formes géométriques

The Privileged Angles seem, conjugated with other geometrical shapes (Golden Rectangle, Dynamic Rectangles, ..) to constitute a frame particularly favorable to feelings of quietness and equilibrium, as well as to dynamism of artistic or (and) commercial object, design or communication and, even, to adhesion to the aims of these ones (Le Ray, 1980) (Le Ray and Martinache, 1984) (Mathieu and Le Ray 2001-06) Numerous antique art’s monuments and some contemporary product packagings are witnessing the use of these geometrical shapes to arouse these feelings (Le Ray, 1980) (Mathieu and Le Ray, 2004-06).

Figure 5 : Generation of the Dynamic Rectangles from the Square and Relation between the Difference of two consecutive Dynamic Rectangles and the Dynamic Rectangle preceding this Difference..

As early as the immediately post-pythagorician era, these rectangles and these numbers were regarded as beauty and harmony standards. Actually, the privileged angles of first family, are also the angles between great side and diagonal of the « ROOT RECTANGLES  so named because the ratios of their sides are equal to the square roots of the successive integer numbers (√1=1 ;√2=1,414 ;√3=1,732 ;√4=2 ;√5=2,236 ;….), and called by PLATON « DYNAMIC RECTANGLES ». These rectangles are obtained, each from the previous one, when one pulls down a diagonal of this latter on a great side and its prolongation. One calls order of Dynamic Rectangle its running number in the series which has been defined, the Dynamic Rectangle of order 1 being the square. For instance, the square, of order 1, having evidently for characteristic angle 45°, the Dynamic Rectangle of order 3 corresponds to an angle 30°, and that of order 6 to that of 22,2°, in all cases between great side (or simply one side in the case of the square) and diagonal (Figure 5).

Figure 6 : Square, Golden Rectangles and Golden Number.

Moreover, it is very easy to establish mathematically (As, from an other hand, Figure 5 shows it, in the case of the angle θ55=24,1°) that the difference (case of Figure 5) or the sum of two consecutive Dynamic Rectangles has for angle between its diagonals, the angle between diagonal and great side of the smallest of these two Dynamic Rectangles. As for the Privileged Angles of second family, they are, for the 35,3°, common with the first family, for the 54,7° and 63,4°, complementary with the 35,3° and 26,6° of the first family, the 63,4° being, from an other hand, the angle between the diagonals of the famous Golden Rectangle (Figure 6) whose ratio of sides Ф=(1+√5)/2=1.618 is the equally famous Golden Number. In what follows, we shall systematicaly call this Angle θ42=63,4° the “Golden Angle”.

We recall that the Golden Rectangle is the rectangle such that, if one removes from it a square, the remaining rectangle, which is also a Golden Rectangle, is similar to the initial rectangle, answering so to an « Invariance’s Thirst of the Central Nervous System » (Paillard, 1974).

The contribution of neurophysiology

As early as 1980, Le Ray suggested that it might exist a connection between the presence of Privileged Angles as key factors in the impact, fixing and memorizing of images, and the presence in the brain of structures and mechanisms that could explain the central nervous system’s “strong need for invariancy”. The possible existence of “genetically pre-wired detectors of shapes” was already bound up with the functional specialization of the neurones in the occipital cortex (areas 17 and 18), which had been studied by the Harvard neurophysiologists Hubel and Wiesel since the 1960s (Hubel and Wiesel 1963-79). These two researchers produced about 20 publications from 1962 onwards, and had their efforts rewarded by a Nobel Prize for Medicine and Physiology in 1981.

The informative areas

One of their crucial discoveries relates to the existence of preferred orientations on the part of neurones in the visual cortex, or to be more precise, that each of these neurones has its own preferred orientation. Experiments were made with cats and monkeys in which a tiny electrode was inserted in the visual cortex at an angle to the exterior envelopp. The animals’ eyes were immobilized beforehand, and they were then confronted with stimuli such as black lines against a white background or white lines against a black background, or by the boundary between a black and a white area. Each of these cases corresponds to what psychophysiologists term “informative areas”. Hubel and Wiesel published more than a dozen articles with diagrams showing the orientations preferred by the neurones in the visual cortex, the preference being expressed by the obtention in each region of a maximum electrical potentiel when the stimulus passes through a “preferred orientation”.

The structure of neurones in the visual cortex and special angles

 Martinache, Le Ray and Levin (1983) enlarged these diagrams, and compared the preferred orientations detected in the successive neuronal columns with the preferred orientations found in adjacent or nearby columns. The result of these comparisons can be summarized as follows : “every preferred orientation makes angles with at least one other orientation (though more often with two or more) that are the same as one of the Privileged Angles”. Given these results, it appears increasingly likely that the visual cortex contains neurone structures that make it easier to perceive Privileged Angles.

Angularly remarkable dipolar character of natural and brain processes

A dipole is, most generally speaking, a couple of a « source » and a « sink » (emitter and attractor of a flow in hydrodynamics or aerodynamics, positive charge and negative charge in electrostatics) placed very close one from the other and defining with the support of the segment joining them, the axis of the dipole so constituted. This structure in « source and sink » is identical by its effects, i.e by its velocity field, to that constituted by a loop of very concentrated local rotation of a fluid on itself, called vortex ring, running the fluid at the ring’s interior and bringing it again at outside along closed stream-lines. Now, the common shape of the fluid stream-lines created by a hydrodynamic or aerodynamic dipole and of electrical field or magnetic field lines created by an electrostatic or electrical (constituted by a small loap of electrical current) dipole presents remarkable properties of correspondance between the angles characterizing the position of a point with respect to the dipole and to its axis, and the angle characterizing the position of the velocity of the fluid or the electrical or magnetic field at this point with respect to the line joining the dipole to the considered point, line which is the second side of the preceding angle. These observations take all their sense if one brings them nearer of the magnitude, direction and sense, of the magnetic fields issuing from the various cortex of the human brain, measurement which has allowed to reconstitute a map of the internal magnetic fields in the cortex itself, as well as that of the electrical currents creating them. The so displayed electrical current loaps create actually magnetic fields distributions whose lines of field are very close those created by an ideal dipole constituted by a small circular loop of electrical current. As well the electrical dipolar point of view in brain as the vortex point of view in natural phenomena explain then the omnipresence of privileged angles.

Contribution of cognitive psychology

Even though they are not referred explicitly, the privileged angles and the remarkable angular combinations, appear to us indirectly in the theories of  cognitive psychology, particularly in shape recognition and in the attentionnal processes where memory plays a crucial role. At the outset, studies on shape reccognition were part opart of the investigation of more general phenomena related to perception, such as Associational and Gestalt (Guillame, 1937), but, very quickly, two theories were developped each based on different concept of identification :

  • The first concept emphasizes matching among prototypes : issuing from the Gestalt approach (Wertheimer and Köhler, 1923), it argues that identification is based on the perception of a whole. At a second step, Rosch (1973-76), introduces the idea that the comparison between an exemplar and a prototype can lead to an assessment of similarity or proximity : certain exemplars would then be judged to be closer to the prototype, more typical, than others.
  • The second concept, issuing from the « Associational » approach, is based on a description of the image’s component features in terms of dashes with the identification models (Neisser, 1964) or componential (Smith and all, 1974).


The various contributions from quantum physics, neurophysiology and cognitive psychology have rarely been brought to bear on the fields of marketing and design, despite the fact that they are closely concerned, since images are the key to success in selling products to a wide range of consumers.

The spatial quantization of an object.

In what follows, we shall present some analyses concerning two objects, and, more precisely, the new bottle of mineral water Perrier, as well as that of Coca-Cola. Beforehand, we refer the reader to some publications of the authors or of one or two of them, and, particularly, to the first of these papers, which is the most exhaustive, with 16 graduated figures in the domains going from hydrodynamics and aerodynamics until pictural art, passing through natural shapes (Le Ray, 1980), as well as to one of the most recent of them, equally very exhaustive (Mathieu and Le Ray, 2004-06).

Any object, before any stylistic study (Christofol and all, 1996) as well new as innovated, presented to a potential consumer, must posses an attracting shape, and, as we have shown it (Mathieu and Le Ray, 2001-06) in the case of wine bottles, this attractive character is related, it also, to the presence of privileged angles and remarkable proportions, as well as of resonant combinations, of these ones, in the various parts of the object, and between these parts. In this view and in order to complete the precited works in the case of bottles, we have selected the new bottle Perrier of a shape identical to that of type « FLUO », as well as its equivalent on the market, the new bottle Coca-Cola, which, both, make now reference on the soft drinks market. As we shall note it in the following analysis, the two bottles turn out to harbour very dynamical and   structuring fundamental directions.

In the Perrier bottle, left-right and right-left crossings of non-corresponding sides of upper and lower divergence angles of, respectively, 35, 3° and 54,7° , determine angles 45°=(35,3°+54,7°)/2 and 45°=(54,7°+35,3°)/2 (figure 1.7) which contribute to secure the shape’s unity, while the divergence angle, towards above, this time,

Figure 7 : Bottle Perrier 50cl, type « FLUO »

, of 24, 1° (figure 19.8) transforms the simple filiation 54,7°à35,3° in double filiation 54,7°à35,3°à24 ,1°.

Figure 8 : Bottle Perrier 50cl, type « Fluo »

The Coca-Cola bottle, a priori more classical and more austere, perhaps, less spontaneausly dynamic, presents, however, a more worked out resonant structuration, from which, literally, comes out a feeling of authority. Figure10 (with the angles de 45°, 54,7° and 35,3°) is particularly subtle, complete, and balanced..

Figure 9 : Bottle Coca-Cola 50cl

Figure 9 shows only the initial divergence-convergence of 45° (to be compared probably with that of the bottles « Burgundy Tradition ») that one remarks firstly. This divergence-convergence is evidently to take into account in  figure 11, already evoked, to enlighten effects of perpendicularly and of resonance with elements of bottle’s main part.

figure 10 : Bottle Coca-Cola 50cl

Figure 10, with the angles 63,4° and 26,6° and the effects of perpendicularity which are associated with them, shows phenomena related with two « Golden Trapezes » (with the angles 63,4° between their diagonals), both of vertical axis, one elongated and the other flattened. The various angular couplings between diagonals indicate a relevance of the location of the « waist » (narrowest part) of the bottle, in the more strict frame of general architecture of this bottle, as defined by next figure 11. The appearance of the 63,4° and 26,6° completes almost totally (at the exception of the 30°) the use in the bottle geometry of the most significant privileged angles.

 Figure 11 : Bottle Coca-Cola 50cl

The section of the bottle’s main part, from the upper bulge of this one, at the basis of its divergent part, until to the largest section of the lower part (section characterized by a seal, forseable on the photograph, but also clearly visible and palpable on the object itself) appears as a rectangle whose diagonals make between them the angle q32=54,7°, which corresponds to a ratio great side over small side, thus, here, height over width (diameter of the cylinder envelopping the bottle) equal: x = (Ö3+1)/Ö2 = Ö2/(Ö3-1)= 1,932. This ratio can be decomposed in x = Ö2+(Ö3-1)/Ö2, i.e x = Ö2+1/x (A), relation very analogous to : f = 1+1/f (B), defining the Golden Number and the associated Golden Rectangles, whose difference is a square, and whose angles between their diagonals are equal to q42 = 63,4° . Here the role played in (B) by the unity is played in (A) by Ö2 =1,414. The relation (A) shows clearly that the rectangle elongated vertically, whose angle  between diagonals is 54,7° is the  succession :

  • of a rectangle of relative height Ö2 which is, evidently, a dynamic rectangle of order 2, characterized by its angle between great side and diagonal q22 = 35,3°, and its angle, complementary of the first one, between small side and diagonal, equal to q32 = 54,7°.
  • And of a rectangle of relative height 1/x=(Ö3-1)/Ö2=Ö2/(Ö3+1)=0.518, similar, evidently, to the great rectangle of proportion x, but, it, « flattened » (i.e with horizontal great sides), while the first is « elongated », (i.e vertical great sides). This new rectangle, similar to the whole rectangle, possesses, itself also, diagonals at 54,7° one with the other, each of the diagonals being perpendicular to a diagonal of the similar vertical rectangle (as one can see on the figure 2.1) and making the Angle q22 = 35,3° (not written on Figure, but rather easy to identify in right triangles involving, on an other hand the angle q32 = 54,7°=90°-35,3°)with the other diagonal of the great vertical rectangle.


Finally, in its turn, the dynamic rectangle of proportion Ö2 (with, we recall it, its angles q22 = 35,3° et q32 = 54,7° between diagonals and , respectively, its great and small sides)  can be decomposed as the sum of a square (with, evidently, its angles 45° between sides and diagonals) and the difference of two consecutive dynamic rectangles of proportions Ö2 et Ö1=1. Now, one knows, from the general theorems concerning the sum and the difference of two consecutive dynamic rectangles, that the angle between the diagonals of the rectangle difference is, then, equal, it also, to q11 = 45°.

The angular analysis of the bottle (Figure 11) lets actually appear all successive elements of various decompositions which have been just detailled. The sides and diagonals of each component of the general combination are, for most of them, remarkably associated to morphologic or graphic elements of  the bottle or of its label.

The square below extends from the maximum section quoted in the beginning of this analysis until a first excrescence of the outline just below the lower limit of the label. Comes then, above, the rectangle whose diagonals make between them 45°. Its lower side is evidently the preceding square’s upper side, i.e the bulge or excrescence which has just been cited.

The upper side of this rectangle, which is, evidently, the lower side of the following rectangle in the decomposition, is common with the  lower level of the  acronym Coca-Cola, inscribed on the label, putting, prominently, this label and the very famous acronym..

This emphasis is particularly striking, since the diagonals at 45°, one with respect to the other, are, each of them, on one hand, perpendicular to one of the sides of the angle, of 45°, itself also, which encompasses the upper bottle’s divergence, and, on the other hand, at 45°with the other side of this divergence angle. One is, here, in the presence of a fundamental whole of elements of stability, but, also, of resonance, and of aerial and dynamic characteristics, that one can compare, as we already have done it, in the case of some wine bottles, with the aesthetic and aerodynamic qualities of a delta wing, and, particularly of a « swallow tail » delta wing, involwing such privileged angles, and, more specially, angles of 45° (Le Ray, Deroyon and all, 1985)(Mathieu and Le Ray, 2001-06).

Finally, this lower level of the acronym Coca-Cola is, not only, the upper side of the rectangle with 45° between its diagonals, and of that, vertically elongated, with 35,3° and 54,7° between its diagonals and, respectively, great vertical sides, and horizontal small sides, but it is, also, the lower side of the rectangle upperly limited by the bottle’s bulge, rectangle whose diagonals make between them the same angle 54,7° as that existing between the diagonals of the great rectangle, elongated vertically, envelopping the bottle’s main part, from the maximal section of the rounded lower part unitil the upper bulge, at the boarder of the conical (or divergent-convergent) part of the bottle. It results moreover, as one has it seen in the general study of these similar rectangles of great and small sides perpendicular between them, that each diagonal of one of these two rectangles makes with a diagonal of the other, either a right angle, or a privileged angle 35,3°=90°-54,7°.One imagines easily, yet here, the contribution that such simple and strong phenomena can bring to the universal brand of the product.



Taj Mahal, built in Agra between 1631 and 1648 by order of Mughal emperor Shah Jahan in Memory of his favourite wife, is the jewel of art in India and one of universally admired masterpieces of the world’s heritage. Indian marketing organizations have achieved success in building their brands by associating their product with Taj Mahal. This is basically because consumers associate the finest in India with Taj Mahal. Taj Mahal tea, India’s largest selling premium tea brand is one of the most successful brands in India. The brand was launched in the year 1966 and has developed a rich image over years. As we shall see in further analysis, Indian Ministry of Tourism releases several advertisements titled ‘Incredible India’ all around the world and Taj Mahal occupies the most important position in these advertisements (Mathieu, Gulawani and Le Ray, 2004). There are other well known examples of success associated with use of Taj Mahal picture in advertisement and, even, in brand itself. For instance, the Taj Hotels chain is one of the most renowned group of Hotels in India, and its success, even though due in great part to the quality of its buildings, sites and services, is also largely related to the universal symbol to which the chain is associated.

One of the main contemporary trends in modern societies appears to be toward a greater emphasis on aesthetics in daily life and hence also in consumption (Maffesoli, 1990). The possible applications are consequently extremely varied, and indeed it could be argued that everything with an aesthetic content is affected.

Beauty and harmony are necessary for the products themselves, and in the way they are presented in technical or commercial documentation, in advertising, in sales outlets and on store shelves.

An object or a product cannot be positioned without some knowledge of shapes using Privileged Angles and reflecting forms combining cognitive, neurobiological and physical aspects applied by someone with overall responsibility, possibly either the designer or the manager. In marketing, for example, this approach would enable marketing experts to use the special shapes so as to achieve greater beauty and harmony both in the design of the products themselves and in the whole range of marketing tools (for instance, the notion of Privileged Angles could be used in preparing a mock-up so as to improve the design, make it more homogeneous and reduce visual noise). This would add a competitive edge.The potential field of application is so vast that we shall deliberately focus here only on the promotion of a touristic and cultural site and, also, on the role that this site, highly representative, plays to emphasize the beauty of a region, and,  moreover of a whole country. In what follows, we propose to apply the model of the Privileged Angles, at a first time to the analysis of advertisement images of Taj Mahal.

But, before entering directly in the analysis of some Taj Mahal’s advertisements (and /or of some uses of Taj Mahal’s image to promote the region of Agra and of other cities in its relative vicinity, like for instance, Jaipur or Delhi or, even, to promote the Whole India), it seems important to start from a geometrical and aesthetic study of the Taj Mahal itself, independently of the frame in which its image will be afterwards included and used.

Figure 12: Transverse Sections of Taj Mahal’s Mausoleum and Minarets

Figure 12 represents a transverse section of the main central part of the the Mausoleum, and, also of two of its four minarets. But, in both cases, these minarets being either the front ones or the back ones, are not in the same plane as the transverse section of the rest of the Mausoleum evoked above. It results from this fact, that, for instance, the angles 35,3° and 54,7° seen on Figure 4 are the angles made with verticals (particularly with the Mausoleum Axis) by the line joining the dome spire’s top to the middle of the segment bringing together one front minaret’s to an other, located either before or behind this former one. From the presence of these complementary and very fundamental angles existing in a dynamic rectangle of order 2, and of two other simple features related to the Golden Angle 63,4° and to the equally previously defined and evoked Golden Number, it is relatively easy to deduce (Mathieu, Gulawani and Le Ray, 2004) some properties of the main spatial (or three dimensional) surfaces which constitute the most significative, and, even, striking, envelopps of the Mausoleum.

These capital envelopps appear to be defined, and can be commented, as follows:

Firstly, the volume, whose five summits are the dome spire’s top and the four minaret’s basis centers is an equilateral square-based pyramid, with its eight arrests equal, its four lateral faces being, for their parts, four equilateral triangles.These very simple properties  induce particularly that the equilateral triangles constituting the faces of the pyramid introduced above possess, evidently, medians which make with the sides angles of 30°=60°/2, i.e Privileged Angles of order 3, which, thus, exist, eight times, between a line joining the dome spire’s top to a minaret’s basis center and the line consecutive to the previous one, joining the same top with the middle of an interminaret square’s side.

Figure 13: Golden Angles 63,4° and  Cheops Pyramid Structure in Taj Mahal

 (Global Effect) .

Secondly, and, probably, even more important, the main part of the monument, itself, possesses many other striking features, as for instance, the fact that, excluding now the height of dome’s spire, the Mausoleum’s height, then reduced, is equal, with an accuracy of one or two per thousand, to a relative value (the unit being the half-side of the basis interminaret square) 1,272=√ф, ф=1,618 being the above-defined Golden Number. This property of a new pyramid, whose five summits are the dome’s summit (spire  not included, this time) and the four minaret’s basis centers, which is the same as that of the GREAT PYRAMID of CHEOPS in EGYPT, leads to the fact that  (figure 13) the four lateral summit angles (that is to say the angles between two consecutive lateral arrests), each with origin at the summit of the pyramid [taken at the summit of the dome, spire not included]) of this new pyramid, equal, this time, the angle present between the diagonals of the Golden Rectangle, i.e the Golden and Privileged Angle 63,4°.

Figure 14: Main Remarkable Proportions, Square Bases, Golden Number  and  Golden Cone as Taj Mahal’s Envelope.

Thirdly (figure 6), the  fact that the Mausoleum’s dome is exactly included in a cone, whose summit is, once again, the dome spire’s top, and whose aperture angle is the Golden Angle 63,4°, allowing us to call this cone a « Golden Cone », is probably in very strong resonance with the appearance , a little lower, on the same central monument’s axis, of the summit of the second mentioned pyramid, with its four succesive summit angles equal, them also, to the same Golden Angle 63,4°.We can add now that the ratio of the distance between consecutive minarets axes over the width of the main central monument equal exactly the Golden Number ф=(1+√5)/2=1.618. This property, compared with the existence of the “Golden Cone” previously evoked, leads to the consequence that the « Golden Cone » envelopping the dome, starting from spire’s top, with an aperture 63,4°, envelopps, thus, also, the Mausoleum’s basis envelopp, passing by the four corners of  this above mentioned square. Such a fact brings probably a decisive contribution to the feeling of equilibrium and eternity which is the main characteristic of the Taj Mahal.

Finally, many other angular properties of both Mausoleum’s and of its minarets configuration, taking particularly into account the Mausoleum’s lateral spires, are also of primordial importance, such as the Privileged Angles 30°, 24,1°, 20,7° and 16,8°. Actually, not only, these angles are present between great sides and diagonals of Dynamic Rectangles respective order 3, 5, 7, and 11 (i.e with ratio of  width over height √3, √5, √7 and √11), but, if we remenber the major role already analysed of the angles 63,4°, 54,7° and 45° (present particularly in all squares, and, also, between the lateral arrests and the axis of the equilateral pyramid previously evoked) and 35,3°, we remark that the two main triple filiations of Privileged Angles: 63,4°à45°à30°à20,7° and 54,7°à35,3°à24,1°à16,8°, commented above in aerodynamics and hydrodynamics, play, here, also, a major role in the Taj Mahal’s cascade of Harmony

19.5 Conclusion


We have presented a state of the art of the supply to light of the privileged angles and remarkable proportions, which is, quite naturally in keeping with the general pattern of the « Gestalt », while bringing an additionnal elaborate explanatory contribution, an operational power and an application of this later, to the quantization of the new and innovated objects, as well as of the most striking images and text-image couplings in advertising and, more generally, communication domains. At this term, the present chapter suggests, particularly, three paths of reflexion and application in order to take systematically into account the analysed phenomena, in view of a more efficient and playful innovation:

– First at all to update works capable to grasp better the influence of privileged angular components on perception, categorization, memorisation of shapes, and, thus, of objects, images, links with text and image, allowing so deeper awareness of researchs undertaken in marketing design, innovation and conception, especially in the frame of positioning strategies.

– To contribute to demonstrate the interest of angular analyses exhibiting spatial quantization of shapes and images, in the conception and evaluation of design and of its management, finally and moreover, in the study of objects and images stylistic, as well as of their setting, in relation, particularly, with the creation, analyse, maintain of interest and attractiveness, and revival, of the objects, products, services, and of the communication upon the innovative sight of these latter..

Finally to suggest the interest of a training, an experience, or, at least, of very large artistic and stylistic knowledges for ideas men and women, producers and actors of innovation.


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