54. Observations of induced brightness

To a new visibility


When Hermann von Helmholtz in 1911 proved that a white plane on a black base looks bigger than a black plane on a white base (fig. 459), he opened an unexpected way to a new visibility: the optical illusion is, in the regulated environment of black and white, an indication for a world which is beyond the opposition of black and white. The illusion gives an insight in a higher dimension of division-thinking.


Fig. 459 – This test of Hermann von Helmholtz proved that a white plane on a black base looks bigger than a black plane on a white base. The representation was first published in 1911 and greatly influenced the new way of seeing, pioneered by artists such as Piet Mondriaan, Theo van Doesburg and Victor Vasarely at the beginning of the twentieth century.

The mechanism of this illusion must be clearly understood to see the connection with the quadralectic way of seeing. By looking at the illustration in a swift movement of the eye some four parameters are incorporated at once: a great white plane with a small black square and a large black plane with a small white square.

These four elements join in the act of observation. This moment is of the utmost importance in the present investigation, because the nature of the observation – the cause – is a unity (one look at the picture), but its actual effect is an optical illusion born in multiplicity. It is worthwhile to follow this process in minute detail and establish the lines of communication between the observer and the observed.

The act of looking (I) is drawn to a particular subject, in this case a black-and-white picture (III). In quadralectic terms (as an interpretation afterwards) it consists of a ‘jump’ from the First to the Third Quadrant: from invisible invisibility (absolute neutral looking with no point of reference) to visible visibility (the conscious attention to a physical point of reference). This action could be compared with a reflex: the natural, biological mechanism of action (the looking) and reaction (the seeing).

The reflex is only the beginning of the story, at least in the human environment. The (human) observer has to face the consequences of the first division. An observation is, in essence, a direct action. Even so, its processing is a complicated matter. The clarity between ‘yes and no’ and ‘either-or’ has to be questioned if life is lifted above the level of pure survival.

The key to life lies in the consciousness of a division. Alternatively, to put it in a spatial setting: in the cognitive area where the dynamism of the initial ‘jump’ is placed in a multitude (of thoughts). The movement is back to the process of looking again, but now armed with a knowledge of a physical existence. A contrast between the initial move from the invisible to the visible and a return again from the visible to the invisible comes into being. A comparison between the cause and the effect leads to the particular stage of causation.

Causation is the consciousness of dynamism. It opens an exciting world of possibilities, ideas, theories and conjectures. The Neoplatonists called the triad (of effects): the First Limit, the First Infinity and the First Mixture. These stages were associated with remaining, procession and reversion (GERSH, 1973). The theological scholars of the Roman Catholic Church created the Holy Trinity (Father, Son and Holy Spirit) to cover the same three cognitive phases. The visible part of the European culture reached, from 1200 AD onwards, its prominence largely within this frame of mind.

The number of impressions – provided by the effects and leading on a spiritual level to a quantity of causation – tend to increase within any given communication. The triple mental framework, despite its dynamic qualities, will lose its creative potency when a quantity becomes overpowering. There are – at an advanced stage of a dynamic communication – simply too many causations to understand. Everything is connected with everything, and thinking becomes a muddled mass of facts and figures, ideas and rejections, beliefs and sensual pursuits. There is no way out of the forest. Or is there?

The answer is yes: imagine a fresh division. Create conceptual space by re-arranging the impressions in a wider setting. Make it four: cause, effect, causation and consequences. And apply them simultaneously. Try to gain an insight into the process of division itself, as the initiator of understanding. If we can reach that stage, we will find that the consequences also move through their own development, and are not the end of the story. They guide us eventually back to the source again: a cyclic movement without end.

The description of optical illusions by Von Helmholtz gave a – possibly unintended – insight into the mechanism of vision. A white plane with a small black square and a black plane with a white square enter the communication. The observation leads to the conclusion (consequence) that the white square looks bigger than the black square, although  they have the same size. Apparently, the surroundings of an object influences the impression of an observer. Why? What happened to the common mathematical logic of perception?

Part of the answer is found in the concurrency of the observation: the ‘split vision’ adds an extra dimension to the observation. The four optical elements (a large white plane, a small black square, a large black plane and a small white square) are observed in ‘one glance’, so it seems. However, in reality, an incredible quick comparison takes place between the four elements. A ‘new’ visibility is created in this process: not only the mathematical rules related to the (absolute) sizes of black and white planes are applicable, but also the abstract interpretation of the different sizes adds to the outcome of the ‘objective’ perception. In the ‘split vision’ appears the world of division with all its philosophical implications.

Another optical illusion in which the ‘split vision’ is a major agency was named after L. Hermann, who described the phenomenon in an article (‘Eine Erscheinung simultanen Kontrasts‘, Pflüger Arch. Ges. Physiol. 3: 13 – 15) in 1870. A grid of black (or white) squares, with a specific degree of proximity of adjacent areas and a specific line width causes the optical illusion of grey areas at the intersections of the lines (fig. 460; in: MURCH, 1973; p. 224, fig. 6.9).


Fig. 460 – Some patterns to prove, that a new visibility (induced brightness) appears on the junctions of a grid – if the dimensions are chosen properly: A) Grey squares at the intersections of the white lines; B) The same effect against a grey background. C) The effect is lost due to distance; D) A reversal of A) with the same effect.

‘Such effects induced by the grids – often called Hermann grids – are not completely understood’ stated MURCH (1973, p. 225), ‘although the mechanism of lateral inhibition certainly plays a part. According to the theory of that mechanism, the white intersection fixated by the observer activates units sensitive to brightness and evokes an inhibitory response from surrounding units. Since the point at which two white lines cross (intersections) stimulates a greater proportion of inhibited units than the white lines outside the intersections, the input for brightness is weakened at precisely these points. During periods of involuntary eye movements over the pattern, the intersections are actively inhibited by the brightness units processing the fixated intersection.’

The term ‘involuntary eye movement’ might hold, in all its vagueness, the key to modern visibility. It is the movement in the observation and the interplay of opposites in a ‘split second’, which causes the eye, and subsequent the observer, to see things, which are not there.

Von Helmholtz, who pioneered this terrain in his ‘Handbuch der physiologischen Optik’ (1866), believed in the overall illumination (of an object) as the base for an observer’s judgement of brightness. The eye measures by ‘one look’ at a dark Hermann-grid (large black squares and small white lines), its brightness and hangs on to this base of reference in the consideration of the details of the object, in this case the smaller white lines. The ‘back to the reference’-idea causes the intersections of the lines to ‘aim for darkness’, resulting in a grey area.

This explanation is of the utmost importance from the present quadralectic point of view, because there are clearly four steps in Von Helmholtz’s process of observation and communication:

1. The overall illumination is assumed as a unity (mainly black);

2. An observation takes place in a ‘split second’ and is based on lower division thinking (the relation between black and white);

3. The oppositional thinking proves wrong, because there appear grey areas. An optical illusion is born;

4. An explanation points to the frame of reference.

A limited investigation, as a matter of interest, was carried out into the nature of the start-position, or the ‘unity’ of the black field. A relation was sought between the proximity of the squares (the line width) and the occurrence of the optical illusion. The aim was to find a fixed ratio between the line width and the width of the (black) squares to the point where the optical illusion would disappear.

The inquiry did not reach its goal to find a ‘break point’. The drawing of grids with different ratios  (fig. 461) led to the conclusion that the occurrence of the induced brightness was, indeed, a ‘shady affair’. The lower two squares should hold the key, but the problem is that the disappearance of the illusion in the square in the right-hand corner (ratio 3 : 4 or 1 : 1.33) is only superficial. With some ‘effort’ (in taking some extra distance and more eye-movement) the illusion effect can still be created. Therefore: the observer can  participate actively in the creation of his own optical illusion. This personal and ‘subjective’ factor makes it impossible to fix a scientific or ‘objective’ boundary.


Fig. 461 – The results of an investigation into the relative size of squares and lines necessary to set up the optical illusion of induced brightness in Hermann-grids.
Top left: ratio (width of white line/width of black square) 1 : 8; top right: 1 : 4; bottom left: 1 : 3; and bottom right: 1 : 1.33. The effect of induced brightness is largely (but not completely) lost between the two bottom squares. An absolute (ratio) value cannot be found, since the number of squares and the distance to the observer appear more important towards the vanishing ratio.

The optical illusion of the Hermann grid, which is a reality within a subjective boundary, provides an example of modern quadralectic thinking. Many ‘illusions’ are based on the confusion, which sets in, when the timing of an observation is exposed. The unity of ‘one glance’ is, in essence – a multiplicity of ‘involuntary eye movements’ in a process of comparison. We have to accept that observation is governed by dynamic division thinking, including its ‘subjective’ components.

ROBINSON (1968) distinguished ‘distorting illusions’ and ‘size illusions’. The first group encompasses illusions related to shaping (Zöllner, Hering, Poggendorf) and the second group deals with width (Müller-Lyer, Oppel) and size (Ponzo, Titchener). MURCH (1973, p. 235) added a third category of the vertical-horizontal illusions, caused by a different mechanism than the previous ones. All three categories have their ‘timing’-aspect in common: left and right eyes are involved in a process of constant comparison of the individual elements of an object, guided by the knowledge and experience of the observer. If this cognition or understanding is frustrated, it is called an ‘illusion’. The actual frustration or misinterpretation can be explained in terms of division thinking. The (optical) illusion is the name given to a perception of which the observer thinks that it takes place one quadrant, but in actual fact occurs in another quadrant.

Most explanations of optical illusions are centered in the biological field, concentrating on the action of the retina in combination with signals, received from the brains. This approach resulted in the ‘retinal inhibition theory’ (GANZ, 1964; ROBINSON, 1968) and the ‘perspective theory’ (THIERY, 1896; TAUSCH, 1954; GREGORY, 1963, 1966), using terms like ‘lateral inhibition’ and the ‘after-effect’. MURCH (1973, p. 235) concluded that ‘geometrical illusions most probably arise from a number of sources and no single theory is able to account for them all.’ The knowledge and understanding of visual illusions has greatly increased since the early pioneers entered the terrain, but there still remain interesting questions in the generation of the phenomena. For a good introduction see: ‘An Overview of Illusions of Brightness’ by Akiyoshi KITAOKA (2008) (Dept. of Psychology, Ritsumeikan University). psy.ritsumei.ac.jp/~akitaoka/McCourt2008mytalke.html


Fig. 462 – The Logvinenko illusion. Although gray diamonds are identical, there appear to be light-gray ones and dark-gray ones (LOGVINENKO, 1999).

The philosophical explanation of geometrical observation with visibility-in-division-thinking needs further elaboration. A visual fallacy is the result of a way of seeing, just as other observations. However, optical illusions give a rare glance into the mechanism of division thinking and illustrate, in a graphic way that observations in different quadrants lead to dissimilar results. That insight is of the greatest importance, and not only for philosophers.


Fig. 463 – The optic illusion as a medium of artistic expression. To the left: The painting ‘Four Quadrants’ of Malewitsch, 1915. To the right: ‘Black Quadrant‘, around 1913.

The fascinating background of visual deception was taken up, at the beginning of the twentieth century, by artists like Kasimir Malewitch (1878 – 1935), Paul Klee (1879 – 1940), Piet Mondriaan (1872 – 1944) and Victor Vaserely (1908 – 1997). They tried to evoke the hidden world behind the optical illusions. Malewitsch’s painting from 1915, titled appropriately the ‘Four Quadrants’, is a good example of the fresh spirit of artistic discovery (fig. 463).

The deliberate reduction of form to a plane necessitates the observer to imagine the invisible invisibility of unity. The (visual) illusion hides a world of truth, the lingering in division thinking, which cannot be detected under normal circumstances of observation. The above-mentioned painters introduced the realization of the ‘split-vision’ and its psychological and philosophical importance in the early twentieth century. It was the last form of visibility to conquer: the circle of visibility has – at least in the art of painting – been closed.

The world of painting was not an isolated case. It was  the crowning point of a continuous development of the soul, which became visible as early as the eighth century. Isaac Ben Solomon Israeli (850 – 932) wrote his ‘Book of Definitions‘ and distinguished four types of questioning (SIRAT, 1985):

———————–   1. existence        – if something exists

———————–   2. quiddity         –  what something is, the essential

———————–   3. quality            –  how something is

———————–   4. quarity            –  why something is

And the Dutch philosopher Frans Hemsterhuis (1721 – 1790) – who belongs together with Erasmus, Spinoza en Geulincx to the most important philosophers of the Low Countries – said the same thing in his recognition of the four fundamental powers of the soul (in: ‘Aristaios of over de Goddelijkheid’, 1779; PETRY, 1990):

————————   1. imagination           –  unsorted collection of ideas;

————————   2. reason                    –  comparison of thoughts, their substance;

————————   3. willpower              –  will and action;

——————–   4. moral principle   –  connection with other humans and the ability .                                                                                to see the other as oneself.

and in recent times the quadralectic philosophy – as it has been explored here – has the same point of departure.

Communication is, in the beginning and end, a mere ‘visio‘, in which different ways of observation come together. The human consciousness is based on a ‘split vision‘, which gives way to a process of comparison. The division in the ‘visio‘ is the ultimate measure for the depth of a communication and – finally – for the meaningful existence of mankind itself.

Tetradic thinking is a form of virtual reality. The divisions in our mind are real, even if we are not aware of them. They are used all the time in the exchange of information with the world. Any philosophy or practical psychology has to state its primary division, and preferably the position of the observer within the context of that division. Make sure, that no position is rated ‘higher’ than another. Value every position in its own right.

JUNG (1921/1967; p. 357), in Book VI of his ‘Gesammelte Werke’ about the ‘Psychologische Typen‘, provided a helpful illustration. Carl Jung was, despite his interest in the magical world of four-division and alchemy in general, a man who was thinking in terms of opposites. He wrote in a letter in 1947: ‘What we call life is just a short episode between two great secrets which are in essence one’.

And he gives more of his division-thinking away in one of his last letters (10th of August 1960), written some month before his death, in which he stated: ‘It is possible that we look at the world from the wrong side, and that we could only find the right answer by changing our position, and look at her from the other side, that is to say, not from the outside but from the inside’ (JAFFÉ, 1977).

Carl Jung saw a fundamental opposition (‘fundamentalen Gegensatz’) between the ‘extravertierten‘ and ‘introvertierte‘ personalities and ‘rationalen‘ and ‘irrationalen Typen‘. This type of division pointed to the familiar terrain of dualism: outside and inside, visible and invisible. However, more interesting ground was broken – from the quadralectic point of view – in the sub-division of these types. The following description of four different groups of ‘personalities’ or psychological types of human beings was given:

————————–   1.    intuition      (Intuition)

————————–   2.    thoughts     (Denken)

————————–   3.    reality         (Empfinden)

————————–   4.    feelings      (Fühlen)

Some people are, in Jung’s view, primarily guided by intuition. Others have creative ideas. A third type is fixed on the material, while a fourth is steered by emotions. He used his (sub)division as a means to place persons in various categories.

One step further is an enhancement of this process by bringing the division back in a single person or individual mind. Psychological types are, in that situation, no longer attributed to a group of persons, but exist in a single human being. It is a matter of emphasis and importance (the ‘visibility’) of the various aspects, fluid in time and place, which will determine the ‘type’ at any given moment.

This last step brings the world of tetradic thinking within reach. The psychological types reflect four different kinds of visualizations within the human mind and provide a framework for thinking in general:

1. The first type of seeing – intuition – operates in an inaccessible area of the human mind. Primary decisions are made without any rational foundation. They spring out of an invisible invisibility.

2. The second type – thoughts – is partly visible. That is to say: thoughts can be registered and even put on paper. However, it is hard to see thoughts as material units. Therefore, we bring them together in an area where the invisible visibility is dominant.

3. The third type of seeing – reality – is easy to demonstrate. It only focuses on the real matter, the dream of an empiricist: the world of the visible visibility.

4. The fourth type – feelings – deals with a way of seeing, which passes again in obscurity. We all know what feelings are. They are very close to thoughts, but they carry a heavier load. They are enriched thoughts. We are in the realm of the visible invisibility.

If we do agree with the classification of psychological types and its interpretation in terms of visualization, we are much closer to the meaning of a quadralectic communication. It is hard to say, if Jung himself was aware of this approach.

Starting-point in any communication is the mapping of ‘mind-quadrants’. The different types of information should be stored according to their specific form of visibility. The quadrants are the references in our dealings with the world. A short description of the conceptual contents of the quadrants will be given here, to express the quadralectic point of view one more time (with a reference to Carl Jung) :

1. The first quadrant (I) leaves room for the unimaginable. Every observer is free to give the chimeras a name. Every hope and expectation, under the denominator of unity or belief, originates here or can be projected onto this area. It is also the place where the a-priori decisions in a communication are made, including the primary division to valuate the visibility. JUNG (1921/1967, p. 480) described the registration of an observation in this quadrant as: a ‘Wahrnehmung auf unbewusstem Wege vermitteln‘ (to reach an observation by the unconscious). The importance of such an ‘unconscious’ communication in the first quadrant (I) must not be rated higher then the observations of other quadrants, even if the ‘roots’ of many later decisions can be found here.

2. If a belief becomes more solid (conscious) and can be visualized as a model (or theory), we find ourselves in the second quadrant (II). Jung calls it a ‘Denken nach dem Objekt und den objektiven Daten‘ (thinking towards an object and the objective data). He referred to the relation of an observer and the observed that is: the situation after the primary division-decision was made. Still everything is not clear, and there are awkward questions left (for instance: ‘what is objective?’), but we are able to work with it and communicate its contents (especially if the background of the conceptual frame is not further questioned).

3. The material phase of a communication is all too familiar. The domain of the third quadrant (III) is entered with our conception in the womb and ended with the last breath. In the meantime, there are the conscious experiences of life, the touching of the earth beneath our feet, and being an inextricable part of the cosmos. Jung describes it as a ‘Subjektive Wahrnehmung‘ (subjective or selective observation), for many people the only ‘real’ one.

4. Finally, the visible matter will reach a level at which our understanding falls short. It becomes too much. We are lost in the all-embracing visibility. At that stage, we have entered the fourth quadrant (IV). Jung pointed to the dominant feelings: ‘Die Bewertung durch das Fühlen erstreckt sich auf jeden Bewusstseinsinhalt‘ (the valuation by  feelings encompasses all the conscious levels). Belief offers in this situation a new perspective, but the invisibility will rise as well, leading to oblivion.

Every participant in a communication partakes in a cycle of visibility, and every point of view is governed by a place on that circle. This very basic statement holds for every human being and has been known for a long time. The classical author of an epigram gave the following summary of life (MACKAIL, 1890/1906, p. 275, Life XXXII/1906, p. 297, Life XXXIII; and in a slightly different form in: BRANDON (1965):

————————-   The Sum of Knowledge:  I was not

————————-   I came to be

————————-   I was

————————-   I am not: that is all

————————-   and who shall say more, will lie:

————————-   I shall not be

This epigram was written on a Hermes statue in the Museum of Bologna. Communication is an aspect of universal being that has no limitation in place and time. It exists all the time, between all parts of the universe, but the connections are not always perceptible.

The only way to understand a complexity lies in a frame of reference, which provides the means to compare information and makes valuation possible. The initial division will define our role as a knowledgeable participant of the universe. The reference-frame itself is, in the end, a reflection of the law that greater complexity generates greater accuracy. That is what multiple division thinking is all about.

The conclusion has to be a daring one: let the subjective be part of our thinking. The traces of tetradic thinking can open up a comprehensive way to historical understanding. Any modern scientist has to acknowledge that the empirical truth might not be the whole truth. A good investigator ought to know that the results of research – and in due course its ‘truth’ – are subject to the limitations posed by division thinking. That is to say: the initial, a-priori number of compartments in the conceptual substratum used to order the material and valuate its presence is a vital ingredient in the approach to the world.

Any type of science, which categorically eradicates subjectivity and feelings from its terrain of questioning, does not reach into the realm of a full quadralectic approach. Those scientists who remain in their given boundaries (because they are not able or ready to take the mental step beyond them) have their own point of view and there is nothing wrong with that. Within their own limits, they can be right. As long as they don’t condemn others with a wider outlook.

I will end this introduction to four-fold thinking in a creative way, setting an example of how the European cultural presence can be interpreted in a modern way. I assume that my readers are, by now, familiar with the characteristics of its cognitive frame. The speculative and subjective elements have to be accepted (and discussion should remain open).

The following presentation is a possibility, like all other ways of recording history. There simply is no ultimate truth in history (or even in the presence or future, for that matter). Everybody who reads more in it, or is in search of this truth, does not understand the quintessence of the quadralectic thinking:

1. The First Quadrant (I), or the period of an invisible unity, started in ‘Europe’ at the beginning of the Christian calendar and ended around the year 600 AD. No single form of division thinking was prominent. The influence of unity- and dualistic thinking (from the Persian Mithraism and early Christianity) was strong. It provided clarity in confused times. However, there also were the remains of tetradic (‘Hadrian’) thinking, left over from the aftermath of the Roman Empire. This mature cultural setting joined forces with the heritage of the Central-European Celtic people.

Different forms of division thinking existed together in this first, undefined period of the European cultural history. None of these forms claimed superiority. The results were a state of non-identity called the Dark Ages. This darkness was not caused by a lack of events, but rather by an incapacity of the history-writers to understand the cognitive setting of the cultural exponents at that particular time. The physical traces of any culture are only a part of the story. They are linked to the equally important time span (before and after the visible visibility) of apparent absence.

2. The Second Quadrant (II) lasted from 600 – 1200 AD, but the first ‘visibility of Europe’ as a cultural unity occurred some one and a half century later, during the reign of Charlemagne (768 – 814). It was in that time that the name of ‘Europe’ became known for some sort of political unity.

The start of this process was arbitrary chosen in the year 750 AD, when Pepin, the father of Charlemagne, questioned Pope Zacharias (741 – 752) about worldly power and had himself crowned as king of the Franks a year later. The English missionary Boniface, who had taken a special vow of loyalty to the Pope, performed the ceremony. Boniface was the founder of the abbey of Fulda with a stricter rule than the Benedictines. His name lives on as one of the four Ice Saints (Marmatius, Pancratius, Servatius and Boniface, with their name days on the 11 – 14th of May). Boniface was killed in Frisia in 754, but it was owing to him that German Christianity was papal, not Irish (RUSSELL, 1945; p. 394).

An early European power play developed in the latter half of the eighth century, with Pepin and later Charlemagne, the Longobards (Desiderius), the Popes Stephen II (752 – 757) and afterward Hadrian I (772 – 795) in the ‘Patrimonium Petri‘ in Rome, the Saxons (Widukind), the Turkish Avaren in the Danube area and the Moors (Abd al-Rahman) in Spain acted as colorful players, all searching for influence and boundaries.

The end of the Second Quadrant was chosen around the year 1200, when the tetradic way of thinking gave way to lower forms of division-thinking (without being fully eradicated). The ‘Twelfth Century Renaissance’ – named by HASKINS (1927) – marked the beginning of a complete different approach towards human behavior.

It is remarkable for this period (600 – 1200) that a conscious base for visibility and representation was virtually absent. Belief (in God) for the many and material powers for the few generated the cultural energy and expression. It was searching rather than knowing. Acceptation rather than questioning. A typical manuscript like the ‘Tractatus de Quaternario’, which was written in the spirit of the time, could have been the hallmark of an era, but remained instead anonymous and did not reach a general acclaim.

3. The Third Quadrant (III) was defined between 1200 to 1800 AD.  The visible visibility of the European culture became widespread and the identity was well established. The period itself was reigned by lower-division thinking. Dante’s ‘Divina Commedia’ was the poetic link from the world of tetradic to triple division thinking. The latter was fully developed some hundred years later in Nicolas of Cusa’s ‘De docta ignorantia‘, which lifted the dynamic triple-division to the new standard (and declared the four-fold option as impossible).

The year 1500 AD marked – right in the middle of this period (the Third Quadrant) – the pivotal point (PP) in the European cultural history. It was a moment of great and lasting changes. The invention of book printing and gunpowder opened worlds of intellectual and physical power, which had never been experienced by so many. The discovery of the Americas and the Indies led to a geographical expansion with unheard-of possibilities, information, opportunities, material wealth, but also new drugs (tobacco) and (venereal) diseases.

The development of the European culture from the Pivotal Point onwards is seen by many – scholars and laymen alike – as the expression of its authentic identity. The number of historical studies and commentaries on events and personalities after 1500 AD is far greater than before that date. The reason could be found, on the one hand, in the relative lesser distance from the observer, but on the other hand, also in the conceptual setting of the (three) centuries following this particular point, characterized by the increase of oppositional thinking.

4. The year 1800 was chosen – again on a subjective base – as the commencement of the Fourth Quadrant (IV). The material development continued, but an invisible component was introduced in the interaction, hidden in the multiplicity: one couldn’t see the wood for the trees. However, this apparent loss created the opportunity for a new space experience. The lines of Fraunhofer (which were visible in the spectral analysis of light) opened up a fascinating cosmic world and allowed judgements on far-away galaxies. This was the outcome of the new visibility, based on axiomatic assumptions, and reaching into the Four Corners of the universe. It provided understanding, which had been lacking for so long, because the frame of understanding had been too simply and rigid.

Joseph von Fraunhofer (1787 – 1826) is, in my view, the most creative scientist ever, since he developed a way of seeing the invisible by means of his spectral lines. I like to stand in his shadow as the developer of a new way of seeing by means of a shift of four-divisions and the finding of a ‘Doppler’ effect in the visible realm rather than in the auditory field.

The early twenty-first century is situated in a well-developed Fourth Quadrant. Some two hundred years of mature and fruitful thinking did pass. The understanding of nature and its use – for good and bad – reached an unprecedented level. However, it  was also the time of two World Wars, which originated in Europe and were firm indicators that dual thinking – with concern for boundaries, identities, power struggle and fight – was far from over. Those who describe and look at history from this violent point of view – like it was taught in the days of my primary school – are guilty of violence themselves.

Fortunately, a wider form of division thinking seems to have taken the upper hand at this moment – half a century after the last World War and the ending of the Cold War. The present writing might therefore be the result of the spirit of time: a genuine effort to reach for peace and understanding.

The position of an observer in a Last Quadrant has a ring of Doomsday-thinking. Practical (quadralectic) scholars would even be able to calculate the ‘Day of Disappearance’ of the European cultural history with mathematical precision based on dual symmetry. The time span between the first visibility (750) and the Pivotal Point (1500) is mirrored and leads to the conclusion that the European civilization is set for the final stage in 2250 AD. Now, that looks like a true eschatological message! Those who interpreted facts this way can join the long line of fortune-tellers and millenarians, who saw that the end was near and called for immediate action to save our soul.

It is hoped for, that the intentions of this present writings are now so far understood that such exclamations are treated with care. That they are recognized as dualistic interpretations of (subjective) boundaries. That is permitted, or stronger, it happens in communications all the time and provides the building stones of inter-human relations. However, the ‘facts’ must be given their right place within the context of tetradic thinking: they are part of a dynamic-cyclic system of visibility.

This observation will be the conclusion at this moment. Looking back to the point of departure (which is also a point of beginning, as Eliot told us), we have to agree with a statement of JUNG (1921/1967, p. 470), when he questioned himself for the number of psychological types:

‘Why did I choose the four types of basic functions is hard to say and there is no a priori reason of it, only to suggest that this view has been formed in the course of a years-long experience.’

Visibility is, in all subjectivity, also a matter of experience.

Heemstede, 1998/2002/2011/2013

A Note to the 2002 Edition: The first edition of the book ‘Four. A Rediscovery of the ‘Tetragonus mundus’ (issued in middle of 1998 in approximately four or five copies) was the translation of the book ‘Vier. Tetragonus mundus. Een studie over het vierdelingsdenken als historische wereldvisie’ (1995). The reason for the issue of this new, English edition found its origin in the technological specifics of the first editions (Dutch and English). They were conceived on a computer with a, now old fashioned, DOS configuration. Reproduction became a hassle and also the letter type of the earlier edition needed a renewal. The DOS-files were converted to Windows and the letter type was changed into Aerial 11. No further editing took place and the pages numbers have been left unchanged (except in the Bibliography and Illustrations) (Heemstede, 23082002).

A Note to the 2009 Edition:  Two copies were produced without changes (Heemstede, 10062009)

A Note to the 2011 Edition:  The book was corrected with the application program White Smoke©. Some improvements and additions in the text were added. The bibliography was corrected and brought up to date (Heemstede, 05072011).

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