1. Quadriformis ratio

Four and more

Whatever god it was, who out of chaos

Brought order to the universe, and gave it

Division, subdivision, he moulded earth

In the beginning, into a great globe,

Even on every side.


Ovid, Metamorphoses, 4

Most people do not have a firm grip of the fourfold way of thinking, although it is a classical concept with a long history. Its roots go back to the early days of mankind.
What is tetradic thinking? A look of incomprehension and raised eyebrows are often followed by a request to explain. Then it is my turn how to tell, in a few sentences, a complete philosophy of life.
Usually I decide, to preserve the light touch of conversation, to give some examples like the four directions of the winds or the four seasons. Sometimes I mention the four temperaments or the evangelists.  Yes, that is common terrain, we know about them. I feel less comfortable as the conversation continues. We are only talking about illustrations and images, which allow a glimpse in the universe of the mind. Even so, the world itself remains hidden.
We all know the spirit of survival, the reflexes which save our life. The prehistoric man knew what to do when he stumbled across a ferocious animal crossing his path. Fight or flight. His reaction was not much different from a modern pedestrian trying to cross a street, while cars are approaching. The distance to the cars and their speed are vital and appropriate action has to be taken in a split second: either cross or stay. You make it or you don’t. A mistake, the wrong choice between yes and no, can make the difference between life and death.
Biological survival is based on dual thinking. The hostile environment constantly requires elementary decisions. We all know that and most of us are pretty good in taking them (otherwise we wouldn’t be here). The dual mind is geared to recognize opposites and stereotypes. The law of the jungle is a simple one. A mouse does not benefit from the idea that not all cats are killers.
One step further brings us out of the jungle with its danger zone and into the realm of rational consideration. Survival is no longer the name of the game – although it still urges constant attention. We can now sit back and think things over. Time is on our side. Decisions can be made in a thoughtful way: the ‘either’ and ‘or’ is joined by a third possibility, the ‘somewhere in between’. Or, like Goethe put it in a more poetic way (in the ‘Westöstlicher Divan’):

Dein Lied ist drehend wie das Sterngewölbe
Anfang und Ende immerfort dasselbe,
Und, was die Mitte bringt, ist offenbar
Das, was zu Ende bleibt und anfangs war.

Triple division offers new possibilities. Life becomes more complex, and sometimes less secure, but it is also more rewarding. The intermediate position adds an element of play to the communication. There is more than survival alone. Ideas, theories and dreamed visions will influence the communication. The threefold frame of mind offers a comprehensible context for the visible and invisible. It is the backbone of intellectual thinking. Its historical crown was shaped in the dialectic philosophy of Hegel (1770 – 1831), with the thesis, antithesis and the synthesis as the representatives of the threefold division.
The matter (of a primary division) is taken here to yet another magnitude: the fourfold division. Now the emphasis of a communication includes the previous steps, but adds another one to it. The survival-techniques and intellectual play gave us our basic existence and a world of technology, but more and more questions remained unanswered. In particular, those connected with human feelings. The deeper questions of life, which are fueled by an unknown source, do not have room for expansion in the confines of lower divisions thinking.
It is acknowledged that the complexity rises again in a tetradic vision. Subjectivity and objectivity are on equal terms and the boundaries between fact and fiction become less clear. The established world of science does not like that situation. Fact should be facts. The interpretation of facts should be unambiguous. Unfortunately, that is not the case (which even the modern scientists have to admit). The observer has transgressed into a dynamic interaction with the observed. The view depends on the width of thinking, with changing limitations.
The tetradic way of thinking caters for the different types of divisions. It is a frame of mind – this must be accepted – which has never been adopted as a philosophical ‘system’. However, its basics are as old as mankind itself. The last book solely dedicated to the fourfold (division) was written nearly thousand years ago by an anonymous Anglo-Saxon monk. It was titled the ‘Tractatus de Quaternario‘ and can be found in the Gonville and Caius College in Cambridge (Ms. 428). Nobody bothered, in the following ages to highlight this manuscript as a monument of a system of thought. The first mentioning by Montague Rhodes James in a descriptive catalog in 1908, and further references by Elizabeth SEARS (1986) and BURROW (1986) did not cause a stir.
These events put the present study into perspective. However, it also deprived our endeavor of its revolutionary character. There is nothing new under the sun, said Salomon, and how right he was. We are obliged to call the present book a ‘rediscovery’ of a world view in which intuition, thoughts, observation and feelings are equally important. Our insight will not be the first and will be not the last: it is a distinct stage in consciousness, which always occurs if the observer is provided with the peace to reach it.
The attentive reader will notice that the above-described development of division thinking – from dual to triple and quadruple – has an evolutionary connotation. The brain provides a conceptual picture, which is capable of producing ever-increasing divisions of a mental universe. The insight of an  observer moving towards higher division thinking is a linear one, but that is only part of the story. Higher division offers the opportunity to leave the straitjacket of the linear and to develop a cyclic outlook. We can learn another language, with words and images derived from four different abstract departments, interacting in a circular way. Together they provide a grammar of a universal language. The Greek physician and skeptic philosopher Sextus Empericus pointed in that direction by saying: ‘Eyes and ears are poor witnesses for men if their souls do not understand the language’.
Communication is not only a matter of close observation, but also of registration in the right frame of mind. Life in general, and perhaps philosophy, in particular, should be an investigation into those structures: where are the boundaries of the concepts used to define and validate our visibility?
The answer lies in the division as the ultimate unity in a communication. And therefore, the number of parts on the outset is of prime interest. Like the Greek philosopher Pythagoras already stated in 550 BC: Number is the base of all. This statement – which is often misunderstood and used in a numerological context – refers to the fact that every type of information-exchange is rooted in a specific number of initial parts, a priori established on the outset of a communication.
A theoretical communication-model, based on a cyclic four-division was formulated in an earlier stage of the study (‘De Vier Landen’ (The Four Areas), 1992; unpublished). Abstract movements were caught in figures and graphs. Neologisms were used to express hitherto unknown concepts, in a spirit of what TATON (1957/1963) called:

‘To name a thing is to create it’.

A graphic representation of a ‘universal communication’ emerged at the end of this cognitive exercise. The graph showed the values of a ‘communication-factor’ (CF) during every conceivable communication. So there was a measure, derived from the initial partition of unity in a cyclic model, to approach the relation between the observer and the observed. This understanding is fascinating material, with some reminiscence to the ‘Philosopher’s Stone’ or the ‘Elixir of Life’ of the alchemists (fig. 1).


Fig. 1 – An alchemical laboratory as seen on an engraving of the middle of the eighteenth century (1747). The laboratory can be regarded as the first ‘scientific’ place of collection. The alchemist aimed at a transformation of the material to higher values, like gold and/or the Philosopher’s Stone (THOMPSON, 1932).

There will be no elaboration of the theoretical side of the model at present. It may be of historical interest, that the term ‘quadralectic‘ (or quadralistic) was first used (by the writer) as a neologism in November 1983 (in an unpublished book called ‘Van God Los’). The nomen of ‘quadralectic’ is closely related to the term ‘dialectic’. This expression signified in Greek philosophy a kind of practical logic and was used in a discussion to understand each other. The dialectic way is based on a dichotomous division in an oppositional setting. The quadralectic condition aims to be a logic tool of the same standard, but in a non-oppositional environment.
The tetradic perception conveys a world of difference in the dynamics of data interchange: not only the facts are important, but also the context in which they are presented. Visibility, as a powerful process, looses its ignorance. What we see and at which particular time is – in a cyclic setting – the key to the whole circle in time. The observer is a player in his or her own observation. It is a matter of energetic interplay from both sides. The active double dialectics can create a new world of insight.
These preliminary observations might have a philosophical touch, but readers should not be scared away. This study will be a preparation of the ground for further investigation in the unlimited land of the four-fold. My background as a geologist gave me a good insight in the scale of time and the floating character of divisions. That’s all. This knowledge remained in the back of my mind when I turned my attention to cultural history, in search of a fossilized system of thought. And when traces were found, it invited to a deeper understanding. This representation is a first report of that journey.


BURROW, J.A. (1986). The Ages of Man. A Study in Medieval Writing and Thought. Clarendon Press, Oxford. ISBN 0-19-811188-6

SEARS, Elizabeth (1986). The Ages of Man. Medieval Interpretations of the Life Cycle. Princeton University Press, Princeton, New Jersey. ISBN 0-691-04037-0

TATON, René (1957/1963)(tr. A.J. Pomerans). Ancient and Medieval Science – from Prehistory to AD 1450. Thames & Hudson, London.

THOMPSON, C.J.S. (1932). The Lure and Romance of Alchemy. George G. Harrap & Comp., London.

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2. The Four Quartets

Burnt Norton House

The turn-off to Mickleton is easy to miss while driving down from Aston Subedge over the B4035. All the attention is focused on the beautiful road, which climbs up the Cotswold-plateau to Chipping Campden.

burnt nortonIt was a late afternoon in July 1991 just after the longest day. The sun still gave its warmth at this time of the day. The first road to the left directed to Attlepin Farm. The road festooned through the park-like landscape. A small ‘Private‘-sign was ignored as a consequence of the importance of the mission. For this was not a simple tourist outing. This was a visit to Burnt Norton House, a place of inspiration for the first poem in T.S. Eliot’s ‘Four Quartets’.

Sometimes there was an occasional visitor, the caretaker of the house told me. He even kept a visitor’s book for the select adepts of T.S. Eliot, who found the road to the house and wanted to taste its atmosphere. I signed the book and went along ‘the door that never opened‘ – into the garden – that famous rose garden with the ‘huge tree with figured leaves‘. It was all still there, presumably in the same serene quiescence where Eliot found it in the thirties. Something strange was going on with the time: Time present and time past. Are both perhaps present in time future? And time future contained in time past.

 These opening lines of Eliot’s poem (ELIOT, 1963) became reality. Look: on the heavy stone styles of the entrance to the rose garden were four squares, who (probably) provided the name of the poem. This was living history, at arm length (fig. 2).


Fig. 2 – Marten Kuilman at Burnt Norton House. Visit: 19 July 1991.

Footfalls echo in the memory
Down the passage
which we did not take
Towards the door
we never opened
Into the rose garden

(T.S. Eliot – Four Quartets, 1944)

To be honest: it wasn’t a seasoned T.S.Eliot-connoisseur, who trotted through the fields of the Cotswold. I had recently come across the title of the poem of T.S. Eliot in the library of Birmingham. And I didn’t even see the poem itself, but my attention was drawn to a reference heading an article of Eleanor SIMMONS GREENHILL (1954). She wrote over ‘The Child in the Tree‘ and used a quotation of Eliot’s ‘Four Quartets’ as an introduction. It was only this title, which put me on the trail of the four-parted poem and made me – in due course – an admirer of Eliot’s work.

That very moment on the edge of the Cotswold, in the last glow of the sun descending in the west, I knew – for a short, but momentous long instant – that Paradise still existed. That the ‘Tree of Knowledge of Good and Evil‘ (fig. 3) was just a part of Paradise. Besides this crude division, there was a rose garden and other figured trees, who threw their shadows onto the grass.


Fig. 3 – The Tree of Knowledge of Good and Evil and the Seven Deadly Sins is seen here in an illustration from Boccaccio’s ‘De Claris Mulieribus’ (Louvain, 1487). The symbolism of the (Judeo-Christian) creation-myth is strongly influenced by the two-fold way of thinking. Good and evil, God and Satan, man and woman are part of a world in which a division of incompatible opposites is highlighted (and in which, without saying the first item is good and the second is bad). The acceptance of this Christian imagery had a strong hold on the subsequent development of the cultural history of Europe.

HENINGER (1974) draws an interesting comparison between the world of Dante in his ‘Divina Commedia’ (1304 – 1321) and Eliot in ‘The Waste Land‘ (1922). The world view of Dante (1265 – 1321) is geocentric, within limits and based on order. This view is emphasized in the tripartite of Inferno, Purgatorio and Paradiso, recalled in hundred ‘cantos‘ and written in the ‘terza rima‘ (HARDT, 1973; LOOS, 1984).

Eliot (1888 – 1965) accumulates fragments, parts of personal memory. The parts do not seem to be ordered or bound by limitations. In the ‘Four Quartets‘ (1935 – 1942) is a certain order, but this is of a different character than in Dante’s poem. WHITMAN (1958/1963, p. 107) speaks of ‘some dross, a residue of unavoidable discursiveness which one may as well treat as frankly unpoetic.’

The four locations in Eliot’s ‘Four Quartets’ are Burnt Norton, East Coker, The Dry Salvages en Little Gidding. They can be viewed as four elementary stages in the communication with the world (GARDNER, 1978).

1. Burnt Norton is the Garden of Eden, the Paradise, the rosegarden.

2. East Coker is a village in Somerset near Yeovil. This stage is concerned with cyclical movement of the seasons and birth and dead (fig. 4).


Fig. 4 – East Coker,  a village in Somerset near Yeovil. Photo: Marten Kuilman. Visit: 10 May 1994.

3. The Dry Salvages are to be found near Cape Ann (U.S.A.), where the incarnation took place (fig. 5).


Fig. 5 – The Dry Salvages are seen in the far distance between the the island and the mainland. Photo: Marten Kuilman. Visit: 8 Sept. 1998.

4. Little Gidding is a village in Cambridgeshire, where Nicolas Ferrar founded a Christian commune in the seventeenth century, close to the world’s end (fig. 6) (DRABBLE, 1979, p. 35):

There are other places

Which also are the world’s end, some at the sea jaws,

Or over a dark lake, in a desert or a city –

But this is the nearest, in place and time


Fig. 6 – Little Gidding in Cambridgeshire (England). The fourth and last stage in Eliot’s ‘Four Quartets’, described as ‘a place where prayer has been valid.’ Here the quadrilogue finds its end and beginning again. Photo: Marten Kuilman. Visit: 28 Sept. 1996.

For Eliot this is the place where personal history will be left behind and replaced by a cosmic being for which ‘history is a pattern, of timeless moments’. The quadrilogue ends with the disappearance of the elements and a return to the beginning:

We shall not cease from exploration

And the end of all our exploring

Will be to arrive where we started

And know the place for the first time

This is a picture of the human existence in a cyclic world view. Our presence is a continuous voyage to unknown lands. All we have to do is use our eyes and senses and get hold of some sort of map to lead the way. Or, like HARRIES (1983) put it: ‘Reflection on the facts alone does not suffice to let us understand these facts as parts of a meaningful whole. That requires a creative reading born of faith or love.’

ELIOT, T.S. (1963). Collected Poems 1909 – 1962 (Pp. 187 – 223: ‘Four Quartets’). Faber & Faber, London/Boston. ISBN 0 571 10548 3

GARDNER, Helen L. (1978). The Composition of Four Quartets. Faber & Faber, London/Boston. ISBN 10 0571110487

HARDT, Manfred (1973). Die Zahl in der Divina Commedia. Linguistica et Litteraria, 13, Atheneüm Verlag, Frankfurt/M.

HARRIES, Karsten (1983). Thoughts on a Non-Arbitrary Architec­ture. Pp. 9 – 20 in: Perspecta, 20. The Yale Architectural Journal. Inc., and the Massachusetts Institute of Technology.

HENINGER Jr., S.K. (1974). Touches of Sweet Harmony. Pythagorean Cosmology and Renaissance Poetics. The Huntington Library, San Marino, California. LCCC 73-78049

LOOS, Erich (1984). Der logische Aufbau der ‘Commedia’ und die Ordo-Vorstellung Dantes. Akademie der Wissenschaften und der Literatur, Mainz/Franz Steiner Verlag GmbH, Wiesbaden. ISBN 3-515-04191-5

SIMMONS GREENHILL,  Eleanor (1954).  The Child in the Tree. A Study of  the Cosmological Tree in Christian Tradition.  Pp.  323 – 371 in: Tradition: studies in ancient and medieval history, thought and religion. New York, Fordham University Press, Vol. X, 1954.

WHITMAN,  Cedric H.  (1958/1965). Homer and the Heroic Tradition. Harvard University Press, Cambridge, Massachusetts/W. W. Norton and Company, Inc. LCCC 58-7252/ISBN 0393003132

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3. Fourfold visibility


A practical example of a fourfold visibility will be given here. It involves the so-called ‘variable double stars‘ who appear to the human observer as one star with a varying light intensity. Various reasons may account for this inconsistent luminosity (PETIT, 1987), but we are especially interested in the ‘eclipsing double stars’. In this relative small class of stars the changes in luminosity are due to geometrical circumstances (ASIMOV, 1966/1971; FLAMMARION, 1968). The light-intensity of some of these ‘doubles’ (binaries) is measured and graphical represented in fig. 7. The most famous binary is the star Algol in the configuration of Perseus (Beta Persei). Four phases can be recognized in the brightness of the double star:

1. A Small Double star is right in front of the Large Double star and ‘covers’ the brightness of it. The result is a decrease in total luminescence for an observer on earth and produces the first deep low on the brightness-graph.

2. The Small Double star moves to the right and slips away from the Large Double star (from the point of view of an earthly observer, in reality, it follows an orbit around the LD). Now they are both shining to their full advantage and the nightly observer notices a relative high: a saddle on the graph.

3. However, the Small Double star, continuing its orbit, will disappear behind the Large Double star and the light will be ‘lost’. Since the Small Double star bears his name with proud, this movement will only result in a relative minor diminution of the total brightness. The earthly observer marks the temporary disappearance as a small depression on the graph paper.

4. The Small Double star reappears behind the Large Double star and the maximum luminosity returns. It is a mirror-situation of the earlier phase when both stars were ‘free’ and effectuates in a second high on the brightness-graph.

When the Small Double star continues its orbit around the Large Double star the cycle will start all over again. It is the cyclic interplay between two partners, which makes the link with the present subject.


Fig. 7 – This representation gives the graphs of the luminosity of eclipsing double stars (binaries) according to ASIMOV (1966/1971). They have a general succession in common: a period of brightness (in time) is characterized by three depressions (two deep and one shallow low) and two saddles or highs.

The above-mentioned example represents a cosmic displacement system. However, it also has strong similarities with the interaction in a communication based on a fourfold division.

Initially, there is a twofold division (the Double star) out of a primeval unity (the cosmos). The Greek meaning of ‘kosmeo‘ is ‘to establish order’ or ‘getting an army into line’. Brightness is the result of a simultaneous action. An observer, who joins the communication at this stage, will be able to recognize four phases: two phases when the Double star appears to be a unity (producing less brightness) and two phases when the Double star shows itself as a real double (marked by maximum brightness).

This mechanism of alternating unity and division can be transposed on to any human communication in a cyclic context. We have to assume that the opposite communication-partner is ‘part of ourselves’ and belongs to our (division) ‘system’. There are two options after the initial (two) division: either my ‘other half’ is static, or it is dynamic.  There are four phases to compare in the latter case, just like the Small Double star (SD) orbiting the Large one (LD).  The ‘brightness’ of a four-fold communication can be expressed – like binaries – in a graphic representation. The graph provides a tool to monitor a communication at any given moment.

This natural phenomenon offers a new dimension of thinking. Observer and the observed are involved in an impelling interaction in which the changing intensities can be measured. And this is all the result of a different appreciation of the starting point, putting an initial two-division in a dynamic and cyclic environment. Visibility – as an intermittent interaction – is the result of fundamental decisions taken on the outset of the mutual exchange of information.

Plutarch resumed the words of Heraclitus as he indicated the dynamic visibility in a communication (KAHN, 1979):

One cannot step twice into the same river,

nor can one grasp any mortal substance

in a stable condition, but it scatters and again

gathers; it forms and dissolves, and

approaches and departs

Those who understand the language of quadruple thinking achieve a much wider perspective in life. Not only in the direct interaction with the environment, which becomes richer and more differentiated, but also in view of the cultural history of which we are a part. Events get a new meaning. It is not only the opposites, the hatred and the wars which determine our historical consciousness, but also – and more preferably – the harmony, beauty and insight.

ASIMOV, Isaac (1966/1971). The Universe. From Flat Earth to Quasar. Penguin Books Ltd., Harmondsworth, Middlesex, England.

FLAMMARION, G.C. (1968). The Flammarion Book of Astronomy (Andre Danjon, Ed.). Readers Union/George Allen and Unwin, Ltd., London.

KAHN, C.H., (1979). The Art and Thought of Heraclitus. Cambridge University Press, Cambridge.

PETIT, Michel (1987). Viariable Stars. John Wiley & Sons, Chichester. ISBN 0 471 90920 3

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4. Tetragonus mundus

The fourfold division


‘Why do we recognize four seasons? Why not two, or three, or five?’ This basic question was posed by BURROW (1986) in his description of the general classifications in the Middle Ages. He did not find a unanimous answer. In the end, he said, it must be the power of numerology and the legacy of Pythagoras (who is always ready to carry the philosophical load if numbers are concerned).

This view is too limited. The fourfold division is more than just a numerological toy or a mythic belief in figures. It is a language of the mind and a way of communication. It is a system in which the complexities of the universe can be assessed and unified in a logical order.

This line of thinking leads to a search for earlier occurrence of quadruple thinking. The descriptive term ‘Tetragonus Mundus’ has an ambiguous meaning in the European cultural history. It points to a corpus of thoughts in which the fourfold division occurs in all kinds of images. Major fragments of Greek philosophy are present, but also remnants of Alexandrian and Syrian interpretations of Egyptian and ‘Eastern’ wisdom. Furthermore, last but not least, a substantial part of the Babylonian and ‘magical’ knowledge has entered the early Christian belief and was transmitted to Europe. In this muddle of intellectual material, in combination with Europe’s own cultural background, the structural element of a division in four parts was not far beyond the surface.

The constant interest in the ‘Tetragonus Mundus’ over ages of human soul-searching never resulted in a ‘system of thought’. People utilized the four-fold, scholars – like Peter Lombard in his ‘Sentences’ (Sententiae) – employed four methods for ordering their material and endless artists created visual references to the quaternary (fig. 8). There is, in short, a constant tetradic background noise throughout history, but it was never realized that this division holds the key to a deeper understanding.


 Fig. 8 – The New Jerusalem described in a vision of the Prophet Ezekiel. An illustration from Nikolaus of Lyra ‘Postille’, printed by Ulrich Zell in Cologne in 1485. The square city and its twelve gates (4 x 3) can be viewed at in four different ways: 1. The anagogical or highest unknown meaning (the Celestial City); 2. The allegorical interpretation of looking at the facts, but referring to something else (the Church); 3. The literal, historical approach (the city of Jerusalem), and finally: 4. The tropological meaning, where the image of Jerusalem includes moral values (the faithful soul) (PILTZ, 1981).

Centuries passed with the murmuring of a tetradic state of mind, being so normal that nobody paid special attention to it. The Holy Fathers of the Church in the third and fourth century and the Celtic missionaries of the seventh and eighth century brought the torch to the Carolinian times and the Scholastics of the twelfth and thirteenth century. Then the light diminished, only to return in a most artificial way in the Renaissance. There it became a thing to be toyed with. This game lasted for at least three centuries, when – at the end of the eighteenth and the beginning of the nineteenth century – the quadruple mind returned as a viable option in the communication.

From that time onwards to the present day, the ‘Tetragonus Mundus’ has been with us, side-by-side with strong dualistic forces, fighting for identity. If a tetradic mind becomes involved in this struggle, it looses out. Because, by its very nature, it is a peaceful way of thinking. It understands the basic mechanisms of strife and cannot propagate its ideology in a forceful way.

Even today there are people, who only have an inkling of what the ‘Tetragonus Mundus‘ is all about. They mention the four elements – fire, air, earth and water – as building stones, know the seasons and the cyclic movements of the universe, but can’t put the fragments together into a unity of thought.

The tetradic imagery – and not necessary the tetradic way of thinking – can be followed from the proto-history to the present. From the caves of Lascaux in Southern France – where human beings decorated their paintings of horses and bison’s with tetramorphic signs – to the outlay of motorway crossing in the shape of a cloverleaf: they are all tributes to fourfold symmetry. The distribution of these signs in time and place is world-wide and indicates an elementary and human way of expression.

The signs might be a reference to the sun and the Central Fire, where all division starts, and to cyclic movements in time.  There is also a division in place, a matter of position and orientation. The graphic utterances often don’t give any clue to the state of mind of its creators. Furthermore, to put any serious inquirer on the wrong foot, tetradic imagery might simply be ‘natural’ or ornamental with no visible connection with a particular way of thinking.

The modern investigator arrives here at a dilemma. Which signs are germane and which are not? The answer must be formulated in an Orwellian way: all signs are relevant, but some signs are more relevant than others are. Or better: the question should not be put in an either-or way. All tetradic signs are, in our present view, part of a wider context. The ‘context’ is directly relegated to the way of thinking and the number of initial parts in the division of the unity (of communication). Only a further investigation into the significance of a sign within a framework of thoughts can give a clue towards its real meaning.

A number of figures with a tetradic imagery is given in fig. 9 a/d. The selection has been arbitrary and only serves the purpose of showing the extraordinary versatility of the sign. Some have strong links with a quaternary way of thinking, like the Mexican and Buddhist examples. Our knowledge of the ancient beliefs of Aztecs and Maya-Indians makes it clear, that cyclic-tetradic thoughts were far advanced and resulted in a complete different approach to reality. Fragments of these thoughts were brought back to Europe by Spanish ‘conquistadores‘ and clergymen, who were born and bred in a world of dualistic thinking in opposites. It is not at all surprising that these men were only fixed on the material rewards of the gold and silver and had no imagination of the spiritual world they destroyed in the meantime.

Other examples of fourfold symmetry go back to an animistic world of shamanism and sorcery (like the symbols from Russia, China, Polynesia and Indonesia). In these worlds, the quaternary images are not necessary part of a distinct system. However, the attention paid to the ‘subjective’ aspects of a communication brings these artifacts of mind-practices in the realm of the unknown realms of the mind. They point to a deliberate attempt to explore areas of human sentiment.

Furthermore, there are stylized and ornamental signs on plates and coins. A direct relation to a tetrad philosophy is difficult to establish. It might be a geometric division, offered by the shape of the object, without reference to any wider implications. A square – like a tile – is tetradic in itself and any decoration tends to follow the given restrictions. These artistic appearances are the most difficult to interpret: how occasional are they? Are these signs part of a wider context?

Natural occurrences, like the images of Chladni’s sand figures, do not give such problems of definition. They show that tetradic imagery can be an regular thing, without any (human) philosophy involved.


Fig. 9a – Some examples of tetramorphic imagery in time and place are given here to show the versatility of the four-fold. 1. Signs for thunder, China (HENTZE, 1967); 2. Boat and sun-symbolism, Scandinavia (GELLING & DAVIDSON, 1969); 3. Chumash painting, Santa Barbara, California (HADINGHAM, 1984); 4. Aztec-calendar, Mexico (ENDRES & SCHIMMEL, 1984); 5. Korean signs for ‘longevity’ (SCHMELTZ, 1891); 6. Design of the Bouriates, Russia (SANDSCHEJEW, 1928); 7. Geometric drawing of the New Hebrides. (DEACON, 1934); 8.  Minoussinsk, Russia. (HENTZE, 1928).


Fig. 9b – Examples of tetramorphic imagery in time and place. 9. Pyramid of the Maya, Yucatan, Mexico. The building period of the temple cities is between AD 317 – 650 (Tzakol-phase in pottery) (HENTZE, 1967); 10. Borobudur, Java, Indonesia. Build as a Bhoeddist-temple, eighth century A.D. (KROM, 1930); 11. Bowl, prov. Honan, China (HENTZE, 1967); 12. Ceramics from Susa I, Persia (HENTZE, 1928) ; 13. TLV-board, Shih divination, Han Dynasty, China (KAPLAN, 1937); 14. Plate from Sighisoara (Germany) Sighisoara-Wietenberg culture (MELLINK & FILIP, 1974); 15. A ‘mandoedoe‘, Sumatra, Indonesia (SCHNITGER,  1939).


Fig. 9c – Examples of tetramorphic imagery in time and place. 16/left: Penny of the Quatrefoil-type, c. 1018-24; right: Penny of the Short Cross type, c. 1030-35 (BACKHOUSE et al., 1984); 17. Knot of Salomo on a tile in Bebenhausen, Germany (1228)  (WIENAND, et al, 1977); 18. Graffiti in Martin’s church in Little Waltham (Essex, England) (PRITCHARD, 1967); 19. The Fuller Brooch. Late ninth cent., British Museum, London (BACKHOUSE et al., 1984); 20. The Sutton Brooch. First half eleventh cent. British Museum, London (BACKHOUSE et al., 1984); 21. Exlibris, Altzella. Univ. Library of Leipzig, Ms. Nr. 675  (WIENAND, et al, 1977).


Fig. 9d – Examples of tetramorphic imagery in time and place. 22. The name of Allah. Calligraphy in quadratic Kufi, Baghdad, 12/13th cent. (BRENTJES & RÜHRDANZ, 1979); 23. Chinese incense-clock, 1329 (WRIGHT, 1968); 24. Bowl from the Cyclads, Greece (KERENYI, 1950); 25. Chladni’s sandfigures, caused by vibrations on a metal plate (SOMERVILLE, 1834 in: ALIC, 1986); 26. The road to hell, Aztecs. Codex Borgia (ENDRES & SCHIMMEL, 1984); 27. Diagram of the world with the central mountain, the Meru. West-India, seventeenth century, Ajit Mookerjee, New Delhi (RAWSON, 1973).

BURROW, J.A. (1986). The Ages of Man. A Study in Medieval Writing and Thought. Clarendon Press, Oxford. ISBN 0-19-811188-6

PILTZ, Anders (1981). The World of Medieval Learning. Basil Blackwell, Oxford. ISBN 0-631-12712-7

Fig. 9 a – d:

1. HENTZE,  Carl (1967).  Funde in Alt-China. Das Welterleben im altesten China. Sternstunden der Archäologie. Musterschmidt Verlag, Göttingen.

2. GELLING, Peter & DAVIDSON, Hilda Ellis (1969).  The  Chariot  of  the Sun – and other Rites and  Symbols  of  the Northern Bronze  Age.  J.M.  Dent,  London.

3.  HADINGHAM,  Evan (1983).  Early Man and the Cosmos. William Heinemann Ltd., London (redrawn).

4. ENDRES,  Franz Carl & SCHIMMEL, Anne-marie (1984). Das Mysterium der Zahl. Zahlensymbolik im Kulturvergleich. Eugen  Diederichs Verlag,  Köln.

5.  SCHMELTZ, J.D.E. (1891). Die Sammlungen aus Korea im Ethnogr.  Reichsmuseum zu Leiden. Internationales Archiv für Ethnographie, IV (1891), p. 105.

6. SANDSCHEJEW, Garma (1928). Weltanschauung und Schamanismus der Alaren-Burjaten.  Anthropos, Vol.  XXIII. In: HENTZE, Carl (1922). Mythes et Symboles Lunaires. Editions ‘De Sikkel’,  Antwerpen. fig. 58, p. 83.

7.  DEACON,  A.  Bernard (1934).  Geometrical drawings  from Malekula  and other islands of the New Hebrides.  The Journal  of the Royal Anthropological Institute of Great Britain and Ireland.  Vol.  LXIV, pp. 129 – 176.

8. HENTZE, C. (1922). Op. cit. p. 180;

9. HENTZE, C. (1967). Op. cit.

10. KROM, N.J. (1930). Baraboedoer. Het heiligdom van het Boeddhisme op Java. H.J.  Paris,   Amsterdam;

11. HENTZE, C. (1967). Op. cit.

12. HENTZE, C (1922). Op.  cit. fig. 14.

13. KAPLAN, Sidney M. (1937). On the Origin of  the TLV Mirror.  pp.  21 – 24 in:  Revue des Arts Asiatiques, XI.

14.  MELLINK,  M.  & FILIP, J. (1974). Propyläen Kunstgeschichte. Band 13. Propäen Verlag, Berlin.

15.  SCHNITGER,  F.M. (1939).  Forgotten Kingdoms in Sumatra. E.J. Brill, Leiden.

16. BACKHOUSE,  Janet;  TURNER, D.H.; WEBSTER, Leslie (Ed.)(1984). The Golden Age of  Anglo-Saxon Art 966 – 1066.  British Museum Publications Limited, London.  ISBN 0-7141-0532-5 (fig. 212/215).

17. WIENAND, Adam; BICKEL, Wolfgang & COESTER, Ernst (1977). Die Cistercienser. Geschichte. Geist. Kunst. Wienand Verlag, Köln.

18. PRITCHARD, V. (1967). English Medieval Graffiti. Cambridge University Press. LCC 66-11034.

19. BACKHOUSE, Janet et al. (Ed.) (1984). Op. cit. fig. 11.

20. BACKHOUSE, Janet et al. (Ed.)(1984). Op. cit. fig.  105.

21. WIENAND, Adam; BICKEL, Wolfgang & COESTER, Ernst (1977).  Die Cistercienser.  Geschichte.  Geist.  Kunst. Wienand Verlag, Köln. ISBN 3 87909 074 2

22. BRENTJES, Burchard & RÜHRDANZ, Karin (1979). Mittelasien. Kunst des Islam. VEB/E.A. Seemann Verlag, Leipzig.

23. WRIGHT, Lawrence (1968). Clockwork Man. Elek Books  Ltd., London. Source: BEDINI, Silvio A. (1963). The Scent of Time – A Study of the Use of fire and incense for time measurements in Oriental Countries. The Transactions of the American Philosophical Society,  Vol.  53,  Part 5,  Aug. 1963.

24.  KERENYI, Karl  (1950). Labyrinth-Studien. Labyrinthos als Linien-reflex  einer mythologischen Idee.  Albae Vigiliae.  Neue Folge, Heft X. C.G. Jung  zum  fünfund-siebzigsten Geburtstag 26. Juli 1950 gewidmet. Rhein Verlag,  Zürich. Or: PURCE, Jill (1974). The Mystic Spiral. Journey of the Soul. Thames & Hudson, London. ISBN 0 500 81005 2

25.  SOMERVILLE, Mary (1834). Of the Connections of the Physical Sciences.  John Murray,  London. In: ALIC, Margaret (1986). Hypatia’s  Heritage – A History of Women in Science from Antiquity to the late Nineteenth Century. The Women’s Press, London. ISBN 0-7043-3954-4

26. ENDRES, Franz Carl & SCHIMMEL, Annemarie (1984). Das Mysterium der Zahl.  Zahlensymbolik im Kulturvergleich. Eugen Diederichs Verlag, Köln. ISBN 3-424-00829-X

27. RAWSON, Philip  (1973). Tantra – De Indiase cultus der extase. De Haan, Bussum.  ISBN  90 228 3973 9; A larger illustration  in: MICHEL, George  (1977). The  Hindu Temple.  An Introduction to its  Meaning  and  Forms. Paul Elek, London. ISBN 0 236 40088 6

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5. Regularity through the ages

Regular patterns

The reconnaissance journey of the fourfold division leads into the world of regular patterns. STEVENS (1980) wrote a comprehensive book on the phenomenon of symmetry, dealing with the basic ingredients of repetition in two-dimensional patterns. He distinguished four types of movements to obtain multiplication of patterns:

 —————–    1. translation         from     b    to     b

—————–     2. rotation                            b    to     q

—————–     3. reflection                         b    to     d

—————–     4. glide reflection                b    to     p

These movements can be recognized in the basic fourfold division of a communication. The translation is the Prime Originator in the first quadrant. The prime shift generates space and the opportunity for comparison. The rotation in the second quadrant has a cyclic overtone and adds a different dynamic element to the exchange of information. The reflection in the third quadrant is a confrontation with boundaries, a view in the mirror. The glide reflection in the fourth quadrant is an elaborated combination of translation and reflection.

The essential feature in the recognition of patterns – including the mind patterns – is the so-called fundamental region. STEVENS (1980; p. 57) gave the following definition: ‘a fundamental region is the region of minimum area that can be repeated without gaps of overlaps to make a complete pattern.’ LAUWERIER (1988) called this unit a ‘primitive cell’, to indicate the repeating element. It is instructive to look at the anatomy of a fundamental region and the way it is created. Because in the end the human memory is just a set of (mind) patterns, constructed from various fundamental regions.

Four steps are necessary to outline the smallest unit, or to ‘create our own fundamental region’ (STEVENS (1980, p. 177)(fig. 10):

 1. An irregularity on a horizontal line between A and B; any shape is permitted. ESCHER (1958/1986, p. 118) recommends a simple contour line without too many deep incisions.

 2. The shape of the horizontal irregularity is repeated on a parallel line in C and D; it is essential for the creation of a pattern that the shape is the same as between A and B;

 3. An irregularity on the vertical line, starting at A and ending at C; again any shape is permitted.

 4. The shape of the vertical irregularity is repeated on a parallel line between B and D; now a geometric form is created, which is called a fundamental region: the region of minimum area that can be repeated to obtain a whole pattern.

These are the four distinct (mind)actions required to build-up a fundamental region, which forms a pattern by repetition.


Fig. 10 – The four phases in the development of a fundamental region, the minimum area that can be repeated to obtain the whole pattern.

STEVENS (1980) and LAUWERIER (1988) referred to the Dutch mathematician and graphic artist Maurits C. Escher (1898 – 1972), who plays a major role in the (re)discovery of visible regularity and symmetry in the early decades of the twenties century. Escher’s book ‘Regelmatige Vlakverdeling‘ is an excellent introduction to his work (ESCHER, 1958/1986).

ESCHER (1959/1970, p. 11) described the regular cover of a plane as ‘the richest source of inspiration I ever discovered. And it is still not exhausted. The symmetry drawings show how a plane can be divided in or filled up with regular shapes, who border each other without overlap. The Mores were masters in this art. They have, particular in the Alhambra in Spain decorated the walls and the floors by  congruent, multicolored pieces of majolica, which fits perfectly. What a great pity that they were not allowed to make portrayals! They only used figures with an abstract-geometric design. Not a single Moorish artist has ever, as far as I know, ventured (or maybe he never imagined) to use natural or recognisable figures like fishes, birds, reptiles or human beings as elements in a division of a plane. This restriction seems to me unbelievable, because it are just the recognisable elements in the patterns which is the reason of my continuous interest in this domain.’

Escher finds a response in the modern computer-generated graphics. The fascination to create regular patterns within certain boundaries is the common interest. William KOLOMYJEC (1976) rendered homage to Escher in an article on the appeal of computer graphics. He uses a computer program to draw ever-smaller fundamental regions in a square or circle (fig. 11). The dynamics are of a rotational nature and focuses on the size-reduction towards the edges. Escher has also ventured this terrain in several pictures, like ‘Vierkantslimiet’ and ‘Cirkellimiet I – IV’.


Fig. 11 – Escher in the round. Computer-graphic by William Kolomyjec, 1975. The use of computers to generate regular patterns has contributed to the popularity of the Escher’s graphic work, which was born in an artistic craftsmanship.

The modern attention to the generation of patterns can be seen in a long history of division (in any imaginable sense) and the visible aspects of it (again in the whole specter of visibility from mathematics to art). The most notable preoccupation came to the surface in the Renaissance (fig. 12).


Fig. 12 – Piero della Francesca’s painting ‘The Flagellation of Christ’ in the Galleria Nazionale delle Marche in Urbino, Italy.

Marlyn ARONBERG LAVIN (1972) said of the floor of Piero della Francesca’s painting ‘The Flagellation of Christ’: ‘The geometric floor pavements in the front and back bays of the Praetorium, disguised by their perspective foreshortening, are similar in design to certain kinds of Renaissance horoscope drawings. Recent studies have shown these pavement designs, in their reconstructed state, to be special purveyors of mathematical symbolism. The eight-pointed stars that form their central motifs, are the astrological sign for the planets’ (in: WITTKOWER & CARTER, 1953) (fig. 13).


Fig. 13 – The reconstruction of the eight-pointed star in the floor of the Praetorium in Piero della Francesca’s painting ‘The Flagellation of Christ’. Ducal Palace, Urbino (Italy).

A Dominican priest, Dominique Douat, again raised the interest for patterns in the early eighteenth century. He published a treatise ‘Methode pour faire une infinite de desseins differents avec des carreaux mi-parts de deux couleurs par une ligne diagonale’ (Paris, 1722). It is an elaboration of an idea put forward in 1704 by another Dominican clergyman, Sebastian Truchet. He published in the ‘Memoires de l’Academie Royale des Sciences’ a number of possibilities to arrange square tiles, which were divided by a diagonal line into two coloured parts (GOMBRICH, 1979; LAUWERIER, 1988).

Truchet took a pair of bi-colored tiles, and studied their position and orientation. He found ’64 combinations de deux Carreaux mipartis de deux couleurs’ (Table I in Truchet’s publication; fig. 14). The four different stages in the position of a two-fold division (ABCD) has similarities with the example of the binary stars in a cyclic (rotational) environment.


Fig. 14 – Sebastian Truchet gave sixty-four possibilities to arrange a pair of bi-colored tiles in his ‘Memoires sur les Combinations’ (‘Treatise on Combinations’), 1704.

The significance of Truchet’s arrangements (for the present investigation) lies in the graphical treatment of combinatories. The handling of a two-fold division (or bi-colored tiles) in a topological manner is ‘a kind of metaphor for the hierarchy of separations and connection in all things.’ (SMITH & BOUCHER, 1987; p. 378)

Quadralectic thinking is closely related to this figure of speech: it is a topological approach to communication and an effort to valuate the relation between the ‘invisibilia’ and ‘visibilia’ in a mathematical way. In this case ‘hierarchy’ must not be regarded as a scale from high to low, but as a measure of distance between the partners in a communication. In the end this measure is a simple number, which acts as an expression of visibility.

ARONBERG LAVIN, Marilyn (1972). Piero della Francesca: the Flagellation. University of Chicago Press. ISBN 0-226-46958-1.

ESCHER, Maurits C. (1958/1986). Regelmatige Vlakverdeling. De Roos, Utrecht (De Roos Foundation), Museum Meermanno-Westreenianum, ‘s-Gravenhage.

– (1959/1970). Grafiek en tekeningen. Koninklijke Uitgeverij J.J. Tijl N.N., Zwolle.

KOLOMYJEC, William (1976). The Appeal of Computer Graphics. Pp. 45 – 51 in: LEAVITT, Ruth (Ed.) (1976). Artist and Computer. Creative Computing Press, New Jersey/Harmony Books.

LAUWERIER, Hans A. (1988). Symmetrie. Regelmatige structuren in de kunst. Aramith Uitgevers, Amsterdam. ISBN 90 6834 032 8

SMITH, Cyril S. & BOUCHER, Pauline (1987). The Tiling Pattern of Sebastien Truchet and the Topology of Structural Hierachy. Pp. 373 – 385 in: Leonard (Journal of the International Society for the Arts Sciences and Technology), 20th Anniversary Special Issue. Volume 20, Number 4, 1987. Pergamon Press, Oxford/New York.

STEVENS, Peter S. (1980). Handbook of Regular Patterns. An Introduction to Symmetry in Two Dimensions. The MIT Press, Cambridge, Mass./ London. ISBN 0-262-19188-1

WITTKOWER, Rudolf & CARTER, B.A.R. (1953). The Perspective of Piero della Francesca’s ‘Flagellation’. Pp. 294 – 302 in : Journal of the Warburg and Courtauld Institutes. Vol. 16 (1953).

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6. The four senses

Divina Quaternitas

The best introduction to the four-fold way of thinking can be found in the excellent Ph.D. thesis of ESMEIJER (1973/1978), titled ‘Divina Quaternitas’. She gave a comprehensive and illustrated survey of the occurrences of the ‘ordine quadrato’ in the Western European culture. The book is concerned with the medieval ‘quadriga mundi’ (the four-fold world), which is an important part of the present field of investigation (fig. 15).


Fig. 15 –  Four-fold symbolism is prominent in the church of Monreale (Sicily); Photo: Marten Kuilman, 2012).

Focal point in the exegesis (of Bible texts) is the handling of the four ‘senses’, the Latin expression for the ways of observation or the basic viewpoints in a communication:

————————-    Historia                        the (historical) fact

————————-    Allegoria                       the deeper meaning

————————-    Tropologia                   the moral meaning

————————-    Anagogia                      the higher meaning

These four different types of ‘seeing the world’ were established by Augustine, bishop of Hippo (354 – 430) in his ‘De Doctrina Christiana’ (PUSCHMANN, 1983), but their home ground can be found in the Nile-delta in Egypt, in the Alexandrian melting-pot of ideas at the beginning of the Christian era. The fourfold interpretation was well-known in the medieval ‘memoria technica’ and summarized in the following expression (MÂLE, 1910/1961, p. 139):

 Littera gesta docet, quid credas allegoria

Moralis quid agas, quo tendas anagogia

ECO (1985/1991, p. 218) – in relation to the use of the four senses in the work of Dante (in the ‘Convivio’ and Epistola XIII in the ‘Letter to Cangrande della Scala’; see also de LUBAC (1959/1964), Partie II, Tome II, p. 321) – attributed this distichon to Nicholas de Lyra or Agostine of Dacia. He called the theory of the four senses a ‘manner of interpretation, which was very common during the whole of the mediaeval culture’.

1. The literal meaning (or historia) is based on an empirical approach to reality. Only the visible aspect counts.

2. The deeper meaning based on the allegory. This word is derived from the Greek ‘allos’ (other) and ‘agora’ (marker/forum, creation) and has the connotation: to say something, but mean something else (HAWORTH, 1980). ALLERS (1944) spoke in this context of an ‘example by translation’.

3. The moral meaning or tropologia is an observation supplemented with certain values. ‘Tropos’ is a way of turning (‘conversio’) an expression into another meaning. In the vocabularies of the exegesis, it was a practical synonym with ‘allegoria’ (de LUBAC, 1959/1964; Partie I, Tome II, p. 551/552).

4. The highest meaning or anagogia reaches into the unknown. Within the communication is a distinct area, where the observer is excluded to verify the facts. Nevertheless this area is considered to be of importance and plays a role in the communication-as-a-whole.

The afore-mentioned four ‘senses‘ are placed in a particular sequence. Or, as Henry de LUBAC (1959/1964; I, II, p. 416) could say: ‘La formule classique, celle du quadruple sens, est au fond de structure plus simple‘ (The classical formula of the four senses has, in fact, a simple structure). First the facts, then other facts, followed by the moral of the story and finally the unknown. This is  an empirical approach to the environment. The human observer is placed in the middle of the known world, like the medieval scientist, who thought that the earth was the centre of the cosmos. In a modern (quadralectic) understanding, there has to be a rearrangement to fall in line with a neutral perception.

I.   First Quadrant        –   anagogia     –  the invisible invisibility

II.  Second Quadrant   –   allegoria      –  the invisible visibility

III. Third Quadrant      –   historia        –  the visible visibility

IV. Fourth Quadrant   –   tropologia    –  the visible invisibility

This positioning is scaling new ground. There is, as far as I known, no publication, which associates the above-given classification of the ‘senses’ with a specific type of visibility in a communication. And the mutual position is the heart of the matter. The four ‘senses‘ can be seen in a numerological way, as individual members of a linear visibility, but also as the outcome of an interaction in a cyclic communication. In the latter case, the sequence of names is not haphazard. They point to a distinct phase in a cross-exchange. The ‘senses‘ bear (in a quadralectic interpretation) a topological message and must be viewed in their right perspective.

CAVINESS (1983) gave a well-documented survey of the fourfold ‘visio‘ at the end of the twelfth century. She, like NOLAN (1977), did not place the senses in a particular sequence, but they have something to say about order. Instrumental in their approach was Richard of St. Victor – the Magnus Contemplator – who died in 1173. He was correctly regarded as a key-figure in the interpretation of the spiritual attitude in the later Middle Ages. His utilization of the four ‘senses‘ marked the historic schism between the old, non-hierarchical (four-fold) way and the new, hierarchical (two-fold) way.

At the end of the twelfth century the four, individual – but interrelated – ways of seeing were gradually moulded into a scheme based on opposites: low, simple, down to earth, visible versus high, difficult, heavenly and invisible:

                                                         Low/easy reach/weak

————————–    1. Corporal view    –   the visible world

————————–    2. Mystical view     –   the spiritual world

————————–    3. Figurative view  –   the moral world

————————–    4. Anagogic view   –   the visionary world


Richard of St. Victor’s ‘orbis quadratus’, so vividly described by Barbara BRONDER (1972), moved to a lower division, if the opposites of low-high, human-godlike and body-soul are emphasized. The fourfold ‘visio‘ only returned to a new understanding after six hundred year of  Renaissances, Ages of Reason and Romanticism (STAUDINGER LANE et al, 2009).

SEARS’ book ‘The Ages of Man‘ (1986) will be mentioned next to the work of Esmeijer. The medieval division – not only fourfold – was here the well-researched area of interest. Sears gave many examples of division in ages, cosmological speculations and pointed to the symbolism of figures. She drew the conclusion, that ‘the quadripartite life was defined within a cosmological system developed in antiquity and subsequently transmitted to the Middle Ages’ (fig. 16).


Fig. 16 – The four ages of man in Aldobrandino of Siena’s ‘La regime du corps‘ (1287). In:
BURROW (1986).

The relation between the ‘antique tetradic thought’ and the mediaeval equivalent was painted from Pythagoras as a source (JOOST-GAUGIER, 2006), to Empedocles (four elements), Hippocrates (four humors), Ovidius (Metamorphosis) to Theon of Smyrna (the tetractys as ordering-principle), Ptolemy (Tetrabiblos) and Antiochus of Athens (with a tetradic program). Followed by Ambrosius (the ‘syzygy’ of elements, ‘virtues cardinales‘), Isidore of Seville (Liber de Natura Rerum) and the ‘quarternarius’ of the Anglo-Saxon monk Byrhtferth, with a culmination in the anonymous, early twelfth century, publication of the ‘Tractatus de Quaternario’, concerned with the ‘force and power of the number four’ (fig. 11). This sequence is, to a certain extend, rather eclectic and suggestive, but proves, nevertheless, that the tetradic interest has a long and colorful history.

CAPLAN (1929) and SMALLEY (1931) took up the importance of the four senses as a guiding line in the Scholastic period. The latter, in her article on ‘Stephen Langton and the Four Senses of Scripture‘ stated that ‘the multiple interpretation, its technique, and its value to those who used it, are just beginning to be discussed.’ She referred to Cassian, who gave a clear definition in his ‘Collationes’ (XIV, 8). Guibert of Nogent (MIGNE, 1844/64, PL. CLVI, col. 26) gave the example of the four meanings (senses) of the word ‘Jerusalem’ (fig. 4), but there is no proof that Guibert was the originator.

The first influences of lower division thinking became manifest in the second half of the twelfth century. Stephen Langton, whose class notes lectures in Paris are preserved, never expounded in the Fourth Sense (anagoge). He followed Hugh of St. Victor, who also used a threefold division. The anagogic sense gradually merged with the allegorical and the moral. ‘Langton was grinding the corn (farina) of the Scriptures into the bread of tropology’ (SMALLEY, 1931; p. 69), but ‘it does not occur to him to distinguish between the teaching of Scripture and his own ingenuity’.

The most comprehensive research in the medieval ‘senses‘ was performed by Henri de LUBAC (1959/1964) in his four-parted study ‘Exégèse médiévale. Les quatre sens de l’Ecriture’. This erudite book was written with a firm knowledge of even the most obscure medieval writers and provides a wealth of facts in relation to the fourfold way of thinking in the Scholastic period. He demonstrated the important role of Beda (c. 673 – 735) in the establishment of the theory of the four ‘senses‘ (LeGOFF, 1984/1987, p. 163). De LUBAC (1959, Part I, Tome II, p. 422) speaks of Beda as ‘le premier auteur qui nous offre pour ainsi dire un tableau developpe du quadruple sens‘ (the first author who offers us so to speak a developed overview of the four senses).

ALLERS,  Rudolf (1944). Microcosmus – from Anaximandros to Paracelsus. Pp. 319 – 407 in: QUASTEN, Johannes & KUTTNER,  Stephan (Ed.).  Traditio,  Vol.  II. Cosmopolitan Science & Art Service Co., Inc.      New York, 1944.

BRONDER, Barbara (1972). Das Bild der Schöpfung und Neuschöpfung der Welt als ‘orbis quadratus’. Pp. 188 – 210 in: HAUCK, Karl (Ed.) Frühmittelalterliche Studien. Jahrbuch des Instituts für Frühmittel-alterforschung der Universität Münster, Band 6, Walter de Gruyter, Berlin/New York.

BURROW, J.A. (1986). The Ages of Man. A Study in Medieval Writing and Thought. Clarendon Press,Oxford

CAPLAN, Harry (1929). The Four Senses of Scriptural Interpretation and the Mediaeval Theory of Preaching. Pp. 282 – 290 in: Speculum 4.

CAVINESS, Madeline H. (1983). Images of Divine Order and the Third Mode of Seeing. Pp. 99 – 120 in: Gesta, XXII/2. The International Center of Medieval Art.

ECO, Umberto (1985/1991). Wat spiegels betreft (‘essays’) (tr. Aafke van der Made). Uitgeverij Bert Bakker, Amsterdam. ISBN 90 351 0863 9

ESMEIJER, Anna C. (1973/1978). Divina Quaternitas. Een onderzoek naar methode en toepassing der visuele exegese. Proefschrift Rijksuniversiteit Utrecht, 29-06-1973. Also as:

–   (1978). Divina Quaternitas. A Preliminary Study in Method and Application of Visual Exegesis. Van Gorcum, Assen/Amsterdam.

GOFF, Le, Jacques (1984/1987).  De cultuur van middeleeuws Europa (La civilisation de l’Occident medieval). Les Editions Arthaud, Paris/Uitgeverij Wereldbibliotheek, Amsterdam. ISBN 90 284 1521 1

HAWORTH, Kenneth R. (1980). Deified virtues, demonic vices and descriptive allegory in Prudentius’ Psychomachia. Adolf M. Hakkert, Amsterdam. ISBN 90-256-0823-X

JOOST-GAUGIER, Christiane L. (2006). Measuring Heaven. Pythagoras and His Influence on Thought and Art in Antiquity and the Middle Ages. Cornell University Press, Ithaca and London. ISBN 976-0-8014-7409-5

LUBAC, de, Henry (1959/1964). Exégèse médiévale. Les quatre sens de l’Ecriture. Tome I – IV. Editions Montaigne; Aubier, Paris.

MÂLE, Émile (1910/1961). The Gothic Image. Religious Art in France of the Thirteenth Century (tr. Dora Nussey). J.M. Dent & Sons, London, Glasgow/Harper, New York (1958).

MIGNE, J.P. (1844/64). Patrologiae cursus completus sive bibliotheca   universalis… omnium s.s. patrum… Series secunda in qua prodeunt  patres… ecclesiae latinae… (= Patrologia latina; PL.), Paris.

NOLAN, Barbara (1977). The Gothic Visionary Perspective. Princeton University Press, Princeton, New Jersey. ISBN 0-691-06337-0

PUSCHMANN,  Rosemarie  (1983).  Magisches  Quadrat und Melancholie  in  Thomas Manns Doktor Faustus:  Von der musikalischen Struktur  zum  semantischen Beziehungsnetz. AMPAL Verlag, Bielefeld. ISBN 3-922986-07-2

SEARS, Elizabeth (1986). The Ages of Man. Medieval Interpretations of the Life Cycle. Princeton University Press, Princeton, New Jersey. ISBN 0-691-04037-0

SMALLEY, Beryl (1931). Stephen Langton and the Four Senses of Scripture. Pp. 60 – 76 in: Speculum, VI (1931).

– (1968). L’Exegèse biblique. Pp. 273 – 293 in: GANDILLAC, de, Maurice & JEAUNEAU, Eduard (Ed.) (1968). Entretien sur la Renaissance du 12e siècle. Décades du Centre Culturel International de Cerisy-la-Salle, Nouvelle serie 9; Mouton,Paris/La Haye.

STAUDINGER LANE, Evelyn; PASTAN, Elizabeth & SHORTELL, Ellen M. (Ed.) (2009). The Four Modes of Seeing. Approaches to Medieval Imagery in Honor of Madeline Harrison Caviness. Ashgate Publishing Limited, Farnham, Surrey (UK). ISBN 0754660109

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7. Mappa mundi

The fourfold division in place


The division in place is elementary. Where are we? Where is our position? This is the type of questions that every human being poses in an early stage of existence. Knowledge of the location is of prime importance. Or, like GROENEWEGEN-FRANKFORT (1951) put it in connection with the visual art: ‘The problem of space is a problem of relation. ‘Only with an active knowledge of the place and its possibilities are we able to survive’ (fig. 17).


Fig. 17 – The fourfold-division in place as a natural scheme. The classical ‘Tetragonus mundus‘ is often attributed to the Greek philosopher Aristotle (384 – 322 BC). He, indeed, summarized the scheme and moulded it in a well-developed philosophical system, but the roots are older. Four directions enclose a square world, divided in three parts (Asia, Africa, and Europe). South (Auster) is placed at the top and East (Oriens) to the left, which is in defiance with the present map drawing. The picture originates from a ninth century manuscript of Beda’s ‘De Natura rerum’ (Clm. 210, fol. 132v; Bayerische Staatsbibliothek, München) (ESSLING, 1909; BROWN,  1978).

To find a location is a matter of division of space and subsequent valuation of distances. The first positioning is based on a division-concept. The early history of geographic orientation in the Europe has many examples of a strong awareness of the importance of division, in particular, the three- and fourfold variety.

Konrad MILLER (1926/1986) referred in a contribution to the Arabian cartography (‘Zur Geschichte der arabischen Kartographie’) to the possible origin of the three-fold division of world-space. He placed the source of the first cartographic division (in Asia, Europe and Africa) at the joining point of these cultures in the Aegean Sea, on the isles of Rhodos and Samos and at Milete in Asia Minor.

This observation was supplemented by the remark that the Asian cultural influence pointed to a four-fold rather than to a three-fold division: the names of Europe and Africa do not occur in the Bible, and Asia is only mentioned as a Roman province. Furthermore, the Arabs in the Middle Ages did not use a three-fold division of the world. They imagined a cup-like appearance symbolized by the old-Sumerian sign (4000 BC) of two concentric circles: the inner circle as the earth, surrounded by sea and an outer circle. The ‘Four Corners of the Earth’ were known as ‘kanephot‘. They were the seat of the four winds and divided the world in four parts.

The history of the ‘mappa mundi‘ (world-maps) was divided in four subperiods by WOODWARD (in: HARLEY & WOODWARD, 1987; part 3: Medieval Mappae mundi, p. 286ff):

1. The ‘patristic period’ (of the Church fathers) from the beginning of the fifth to the end of the seventh century;

2. The period from Bede (672/73 – 735) to Lambert of Sint Omer (ca. 1100);

3. The period from Henry of Mainz (1100) to Richard of Haldingham (1300);

4. The last period, starting around 1209, was dominated by Franciscan monks. Roger Bacon (c. 1214 – 1294), John Duns Scotus (c. 1265 – 1306) and William of Occam (c. 1290 – c. 1349) were influential.

Unfortunately, Woodward did not state the criterion on which this division was based. It seems, likely, to be the ideological environment in historical time: a ‘classical’-patristic group, a missionary group, a cosmological group and finally a Franciscan group, all spaced over a period of nearly thousand years from 400 – 1400. Within these groups, the variety of forms is considerable. Some points of interest – with particular attention to the number of divisions in each group – will be highlighted here:

1. The world views of Crates (of Mallos), Macrobius, Orosius and Isidore of Seville are labeled together. They are typified as the ‘patristic group’ because they are also used by the Church fathers. Crates (c. 150 B.C) embodied the ‘older tradition’ of a four-parted world. In this view, the globe had four occupied quarters, bearing the names: Perioikoi, Oikomene, Antoikoi and Antipodes (fig. 18). Oikomene was the term originally used in the Greco-Roman world to refer to the inhabited earth.


Fig. 18 – The world-view of Crates of Mallos, c. 150 BC, a grammar scholar from Pergamum. He envisaged four inhabited quarters on the globe: Perioikoi, Oikomene, Antoikoi and Antipodes.

Macrobius (c. 395 – 436) used an elaborate method, with a Pythagorean undertone, to arrive at a five-fold division of a circle. Some hundred-and-fifty ‘mappae mundi‘ using Macrobius’ scheme are known. They became models for many medieval mappae mundi (ANDREWS, 1925).

The following quadruple procedure was used, which became known – some four hundred years later in the time of Rabanus Maurus (died 826 AD) – as the ‘ratiocinationis quadrivium‘ (fig. 19):

a. Division: Lines of 24, 30 and 36 degrees are drawn in a quarter-pie of a circle; this procedure results in a division of 4/60-, 5/60- and 6/60th parts of a quarter circle,

b. Definition: Lines are drawn (in a half-circle) through the intersection of the angle-lines with the circle (A, B). The result is a horizontal three-partition (of the quarter circle).

c. Demonstration: The mirror image of the half-circle three-division results in a six-division of a full circle.

d. Resolution: The mirror-plane is now left out and the full circle is divided in five parts, build up of the following division of the circle: 36, 30, 48 (24 + 24), 30 and 36 degrees.


Fig. 19 – The four stages, later known as the ‘ratiocinationis quadrivium’, to arrive at a ‘Macrobian’ world map, based on a five-division of a circle.

The writings of Paulus Orosius (c. 383 – post 417) had a great influence on later map makers, but no specific maps of this Christian historian are known. The knowledge was derived from his book ‘Historiarum Adversum Paganos Libri VII’ (Seven Books of History against the Pagans’), which was dedicated to St. Augustine (354 – 430).

Isidore of Seville (c. 560 – 636) was the most influential member of the ‘classical’ group. He lived around the same period as Gregorius (c. 540 – 604). His books ‘De natura rerum’ (612 – 615) and the ‘Etymologiarum sive originum libri XX’ (622 – 633) became the sources of references of much later mediaeval knowledge. He was influenced by Ambrose, Augustine, Boëthius, Cassiodorus, Lucretius, Lucan, Macrobius, Orosius, Pliny the Elder, Sallust, Servius and Solinus. Some six hundred and sixty ‘Isidorean’ world maps in manuscripts remained (DESTOMBES, 1964).

Often the ‘Isidorean’ maps were – without much criticism – associated with the so-called T-O maps, who got their name from a three-fold division of a circle: Asia in the upper half, Europe (left) and Africa (right) share the lower half (fig. 20). The biblical association with the sons of Noach – Sem, Japheth and Cham – was generally known.

Careful classification is essential here: most of these maps were, according to DESTOMBES (1964) and ARENTZEN (1984), copied between the eleventh and fourteenth century, thus in a different frame of mind with regards to the number of divisions used in a communication.


Fig. 20 – Three-fold division in map-making. 1. A T-O-map of Brunetto Latini, early fourteenth century (Bodleian Library, MS Douce 319); 2. From an eleventh century manuscript of a ‘Commentary on the Apocalypse of Sint John’ by Beatus of Liebana (Bibl. Nat. Paris, MS Lat. 8878, fol. 7r); 3. From a twelfth century manuscript of Bede’s ‘De natura rerum‘ (Bibl. Nat., Paris, MS Lat. 11130, fol. 82r); 4. World map of Isidore of Seville, 1492; printed by Gunther Zainer, Augsburg.

Von den BRINCKEN (1970, p. 264) states firmly that ‘eine Vierteilung nach den vier Weltreichen Daniels ist übrigens Kartographisch nicht belegt‘ and also: ‘Die T-Karte ist die weitaus verbreitetste Form der mittelalterlichen Karte.’ In a general sense this statement is true, like earlier figures in DESTOMBES’ catalogue (1964) of the ‘mappae mundi’ (between AD 1200 – 1500) confirm. The T-O-map was sanctioned by the Church and fitted in the three-fold way of thinking, which manifested itself in Europe after the year 1200. Scrutiny  of the total number of maps might actually show that the majority of T-O-maps were drawn in or after the twelfth century.

2. The second period in the drawing of ‘mappae mundi‘ is indicated from Bede (672/73 – 735) to Lambert of Sint Omer (c. 1100). They are loosely grouped as ‘missionary’. The world map of Bede is preserved in fifteen original manuscripts, and hundred-and-seventy-five secondary versions are known. In this same period Pope Zacharias (pope from 741 – 752) had a world map painted on the wall of the Lateran Palace (now lost). Charlemagne (described in the ‘Vita Karoli Magni‘) possessed three silver tables with pictures of Constantinople, Rome and the ‘whole world’. Another source of pictures came from the ‘Commentary on the Apocalypse of Saint John’ by Beatus of Liebana (between 776 – 787), copied in many ‘scriptoria‘ in Spain and Southern France (KLEIN, 1976). Towards the end of this period the encyclopaedic ‘Liber Floridus‘ of Lambert of Saint Omer provided many pictures of mediaeval conceptions (DEROLEZ, 1968) (fig. 21/22).

augustusFig. 21 – A miniature of Emperor Augustus with a mappa mundi in Lambert of St. Omer’s ‘Liber floribus’ . c. 1120. In: ARENTZEN (1984).

The ‘Liber Floridus‘ was composed in the years before 1120 and offers a broad survey of all types of divisions: fol. 19v en 20v mentioned the six periods of the world, fol. 24r the winds, fol. 24v the five zones of the earth and fol. 25v gives the positions of the moon, winds and elements (eight-division). Fol. 88r figures the elements and the year (Annus) on the ‘altare Dei‘, situated at the ‘abissus‘. Fol. 221v depicted the zones of Macrobius and the ‘Sphera Platonis‘. Finally, on fol. 241, a ‘Europa Mundi Pars Quarta‘ is visualized (fig. 22/23). These examples show that no particular division thinking is dominant in the ‘Liber Floridus‘.


Fig. 22 – Europa in four parts. Liber floribus. Gent, Universiteitsbibliotheek, Ms 92, f. 241r. The numerological character (of the number 4, pars quarta) is obvious in this incomplete map.


Fig. 23 – The original ‘Liber floribus‘ (1121) in the Stadsmuseum Gent (STAM) exhibition ‘The world in a book’ from September 30th 2011 – January 8th 2012 (photo: Marten Kuilman, 27 November 2011).

3. The third period was fixed – according to Woodward (in: HARLEY & WOODWARD, 1987) – between Henry of Mainz (1100) and Richard of Haldingham (1300). This ‘cosmological group’ reflected a whole new approach to world and cosmos. Charles Homer HASKINS (1924/1960) called this renewal the ‘Twelfth Century Renaissance’. The tetradic way of thinking reached a new level of consciousness, but with ever-stronger numerological undertones. The Crusades to the Holy Land resulted in a fresh geographic awareness and the translations of the Arabs and Greek documents opened up long-forgotten worlds. This knowledge resulted in ever more elaborate maps.

The famous Ebstorf-map (1235) (fig. 24) and the Hereford-map (around 1290) showed, in the climax of the ‘allegorical’ cartography in the thirteenth century, a wealth on details and did not belong to any distinct type of division.


Fig. 24 – The central part of the Ebstorf mappa mundi shows the square city of Jerusalem. The original map was destroyed in World War II during the bombing of Hanover. A set of black and white photographs of the original map was taken in 1891. Several color copies were made before its destruction in 1943 (Wikipedia).

4. The last period has an overlap with the previous one and runs from 1209 to 1400. It is dominated by the work of Franciscan monks. Influential names were Roger Bacon (c. 1214 – 1294), John Duns Scotus (c. 1265 – 1306) and William of Occam (c. 1290 – c. 1349). The small, but very important book ‘De Sphaera‘ of John Sacrobosco (John of Holywood) is dated between 1220/30 and is therefore earlier than a book with the same title by Robert Grosseteste (1175 – 1253). The idea of a spherical world was well established. It contradicts the opinion of nineteenth century authorities that the earth of the thirteenth century scholars was thought of as flat.

An exceptional relict of the fourfold, ‘conceptual’ approach to map making can be found in a schoolbook for young clergyman, printed in 1475 in Germany (fig. 25).


Fig. 25 – A late conceptual, four-fold world view. From a woodcut in the ‘Rudimentum novitiorium‘, published in 1475 by Lucas Brandiss in Lübeck. The top of the map is to the east (Oriens) with the four rivers of Paradise flowing from it. At the center, the Holy Land Palestine (and Judea), but no special reference is made to Jerusalem. The land Ophir is in the south (to the right, Auster) and the Pillars of Hercules are visible to the bottom of the map (west). In: STRAUSS (1981).


The ‘Rudimentum Novitiorum‘ (Lübeck, 1475) in a colored version. The book was re-issued from new blocks from Paris in 1488 and Lyons in 1491. In: SHIRLEY (2001).

All these endeavors in map making reflected the basic need of humanity to locate themselves in a seemingly boundless space and draws attention to the problem of relation. In the end, it is the concept of division, which shapes this relation, and which makes it possible to bring structure in an otherwise incomprehensible environment.

The four directions – North, East, South and West – are now generally accepted as the basic division of place. They are – which is much less understood – the living examples of the possibility to think in a structural-spatial fourfold pattern.

ANDREWS, Michael C. (1925). The study and classification of medieval mappae mundi, Pp. 61-76 in:  Archeologia, Oxford, LXXV (1925-26).

ARENTZEN, Jorg-Geerd (1984). Imago Mundi Cartographica. Studien zur Bildlichkeit Mittelalterlicher Welt- und Oekumene Karten unter besonderer Berucksichtigung des Zusammenwirkens von Text und Bild. Wilhelm Fink Verlag, Munchen. ISBN 3-7705-2258-3

BRINCKEN,  von den, Anna-Dorothee (1970). “…Ut describeretur univer-sus orbis“.  Zur Universal Kartographie des Mittelalters. Pp. 249 – 278 in: ZIMMERMANN, Albert & HOFFMANN, Rudolf (Ed.). Miscellanea Mediaevalia. Band 7: Methoden in Wissenschaft und Kunst des  Mittelalters. Walter de Gruyter & Co, Berlin.

BROWN,  Hanbury  (1978).  Man and the  Stars.  Oxford  University  Press, Oxford. ISBN 0-19-851001-2 (Ann Ronan Library).

DESTOMBES, Marcel (Ed.) (1964). Mappemondes AD 1200 – 1500. Catalogue préparé par la Commission des Cartes Anciennes de l’Union Geographique Internationale. N. Israel, Amsterdam.

ESSLING,  Prince d’ (1909).  Les Livres a figures venitiens de la  fin  du XVe Siecle et du Commencement du XVIe.  Librairie Leo  S.  Olschki, Florence/Librairie Henri Leclerc, Paris.

GROENEWEGEN-FRANKFORT, H.A. (1951). Arrest and Movement. An Essay on Space and Time in the representational Art of the ancient Near East. Faber and Faber Limited, London.

HARLEY, J.B. & WOODWARD, David (1987). The History of Cartography. Vol. I: Cartography in Prehistoric, Ancient and Medieval Europe and the Mediterramean. The University of Chicago Press, Chicago/ London.

HASKINS,  Charles H. (1924/1960). Studies in the History of Mediaeval Science. Frederick Ungar Publishing Co., New York.

– (1927). The Renaissance of the Twelfth Century. Harvard University Press, Cambridge, Mass.

MILLER,  Konrad (1926/1986).  Mappae Arabicae. Beihefte zum Tübinger Atlas des Vorderen Orients, Reihe B, Geisteswissenschaften, Nr. 65; GAUBE, Heinz (Ed.)/Dr. Ludwig Reichert Verlag, Wiesbaden. ISBN 3-88226-293-1

SHIRLEY, Rodney W. (2001). Mapping of the World. Early Printed World Maps 1472 – 1700. Early World Press Ltd., Riverside (USA). ISBN 0 970351801

STRAUSS, L. (Ed.)(1981). The Illustrated Bartsch,  80, Part I: Anonymous Artists, 1457 – 1475. Abaris Books, New York. ISBN 0-91-3870-50-1

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