Mona Lisa, an Allegory of Seeing / © 1974-2003 by Franz Gnaedinger, Zurich, fg(a), fgn(a) /




MONA LISA, an Allegory of Seeing           


Noi conosciamo chiaramente chella vista e delle veloci operationi chessia, ed in un punto vede infinite forme: niente di meno non comprende se non una cosa per volta. Poniamo chaso che tu lettore guarderai questa carta scritta, e subito giudicherai questa esser piena di uarie lettere, ma non conoscierai in questo tempo che lettere sono, ne che volino dire, onde ti bisogna fare apparola apparola verso per verso a voler notizia d’esse lettere.  (Leonardo da Vinci, B.N. 2038. 28a)  Dicho chellochio portando consecho infinite linie le quali sono appichate overo unite con la sopravenienti chessi partano dalle chose vedute, e sola la linia di mezo d’essa sensuale e quella che cogniosce e giudicha i chorpe colori, tutte l’altre sono false e bugiarde  (W 19148b)  L’occhio a una sola linia centrale, e tutte le chose che vengono all’ochio per essa liniasono bene vedute. Dintorno a essa linia sono infinite altre linie aderenti a essa centrale, le quale son di tanto minore valitudine quanto esse son di magore remotinone dalla centrale  (W 19010b)  L’occhio a in se una sola linia posta in mezo a infinite altre linie aderenti a quella la quale e detta centrale e tutto le spetie delli obietti che venghono all’occhio per essa linia sono perfectamente vedute sella troppa lungha disstantia non le impedisce. Dintorno a essa linia ne sono infiniti aderenti a quella le quali son di tanta magore o minor valitudine quanto sono vicine o remote a tal centrale  (D 10b)  La linia media / centrale / maestra / principale / maestra delle altre linie / abraccia con vera cognitione le chose grandi da lontano come le pichole da presso / si dirizza sempre a tutti quelli obietti di che sa avere certa e vera notizia / L’occhio fa una linia maestra / capace di comprendere i termini / L’ochio manda moltitudine di linie che circundano questa principale di meza / linie di debole comprensione, meno potente die conoscere il vero / onde le chose, delle quali i termini sono giudicati di esse linie, son confuse  (B.N. 2058. 23b, D 8b)  //  Muovesi lamato per per la cosamata come il senso per il sensibbile e consecho s’uniscie effassi una cosa medesima.  //  l’anima, figliola della natura  //  essai chelluomo emodello dello mondo  //  La natura e piena d’infinite ragione che furono mai in esperienza  //  Le spetie del nostro emisperio enitrino epasino con tutti li corpi cielesti per il punto naturale nel quale s’infondano e vniscano nella penetratione e interseghatione luna dellaltra come laltra delluna ... la spetie della luna alloriente elle spetie del sole allocidente in tal punto naturale sono unite ... chi crederebbe chesi brevissimo spatio fussi capacie delle spetie di tutto luniverso o magnia actione qualle ingiegnio potra penetrare tale natura qual linghua fia quella chesplicare possa tal maraviglia cierto nessuna // Inefetto l’omo non si uaria dalli animali senon nell’accidentale chol quale si dimostra essere cosa diuina perche doue la natura finissce il produrre le sue spetie, lomo quivi comincia colle cose naturali affare collaiutorio dessa natura infinite spetie le quali nonessendo necessarie achiben si correggie come fan li animali none disspositio cercarne  (Windsor 19030 verso)  //  Qui Adam, Eva di là – Oh human misery; of how many things you make yourself the slave for money  //  Se tu sprezzarai la pittura, la quale è sola imitatrice di tutte l’opere evidenti di natura, per certo tu sprezzarai una sottile inventione, la quale con filosofica e sottile speculatione considera tutte le qualità delle forme: mare, siti, piante, animali, herbe, fiori, le quali sono cinte d’ombra e lume. E veramente questa èscientia e legittima figlia di natura  //  Non mi legga chi non è mathematicho  //  e se tu dicessi la musica essere composta di proporzion,  o io con questa medesima seguito la pittura




‘Seeing’ may appear to be the simplest of acts: we merely open our eyes and perceive the world around us. However, it is actually a wonderfully complex physiological and psychological process. Let me explain.


1) We see what we look at, and we see what we know


When we open our eyes we see a multitude of varied and shifting colors. Using our minds we turn them into objects: one of the reds I see becomes my pullover, another my blanket, and yet another a ring binder. I can only see the spines of the books on my shelf, but – as I know my books well - I have a visual image of their respective covers. You may see only your sleeping wife’s cheek and her hair on the pillow, and yet you see the whole person so familiar to you. If a tourist asks me the way to the Chagall windows in the Zurich Fraumünster, I virtually see the alleys and streets threading between the mass of houses; the church I know from all sides and at every hour of the day; the small chapel; its tall windows; and the colors in Chagall’s beautifully stained glass. Even if a house is hidden behind foliage, with only a few stones and bricks visible, we nevertheless perceive a hole structure and not a ruin. .When I turn my head I see a moving kaleidoscope of colors, and yet all the objects in my room, the walls included, stay firm, for I know that they remain still while I move my eyes. I see only one side of the objects in front of me, and yet I see whole objects, for I know them from all sides. What I see and what I know and what I believe come together in my visual perception.

     Ask someone to draw a can, and he will most probably render the lid as a kind of circle rather than an ellipse, for he knows intellectually that the round piece of sheet metal is formed in a circle. Beginners in artistic drawing have to forget much of what they know in favor of what they actually see. They learn to see anew: they discover the wonderful, stimulating garden of opposing and blending colors, lights and shadows which are the common objects of our daily life as they appear to our eyes. Once I read about a blind man who underwent a successful operation on his eyes. Afterwards, could he see immediately? No: the man was lost in a sea of colors, lights and shadows, and it took months for him to move securely in this disturbing world. As children we learn how to see, how to use our senses, bodies and limbs; we experience our surroundings and gradually acquire a reliable knowledge of the world. It is this knowledge, combined with an inborn, which allows us to perceive the world ‘simply’ by opening our eyes.


2a) We dispose of a ‘drop’ of concentrated and moveable attention in a wide field of stationary attention; 2b) We perceive the thing to which we apply our moveable attention, while everything else remains in a more subconscious background; 2c) the more moveable attention an object attracts, the larger this object appears, and when two objects of different sizes attract an equal amount of the moveable attention, they seem to have the same size


Look at the moon rising above the horizon: its beautiful, round, orange form attracts all of our moveable attention and thus appears much larger than in a photograph. However, when the moon is seen high on the sky we have to raise our heads to see it, and are no longer secured by the firm objects along the horizon. We see a small disk of light swimming in an ocean of dark air, we may feel slightly dizzy, and are therefore no longer able to concentrate wholly on the moon – this, in my opinion, is why we now perceive its size as a camera might. Or you may observe a herd of grazing cows. Now and then a cow jumps, and then something peculiar happens: that cow appears to be much nearer than her companions, if only for a fraction of a second. The jumping animal attracts all of our moveable attention at once, and therefore appears much larger than the other cows, yet as we know that they are more or less the same size, our minds turn large into near. Imagine a row of people sitting at a table. Those nearby should appear large, those farther away small. Yet we perceive all human faces to be about the same size, for each face attracts roughly the same amount of moveable attention. Only a face very close up appears very large (for example when kissing), and only people very far away appear truly small (or tiny when seen from a tower, or wandering along a hill). Thus the drawings of children and so-called naïve artists often demonstrate a peculiar perspective that may be called a perspective of attention.

     The moveable drop of high attention may focus on a visual contrast, a sound, a touch, or another stimulus provided by one of our senses (there is no outer-sensual perception, but we may have more senses than we know). It may move from one contrast to another; it may zoom in on a tiny spot of the highest concentration; it may expand to a larger field of weaker attention. It may even be absorbed by the field of stationary attention – and immediately spring to life again, for example if, while reading or daydreaming, we suddenly hear a door slam.


3a) We see sharply by focusing our eyes on a given object; 3b) we see clearly by looking straight at a given object


Many years ago I made a series of drawings by means of an unusual procedure. I fixed my eyes on a delicate but strong, small but clear point of contrast in my visual field and tried to depict and the surrounding objects as unclearly as my eyes rendered them. What happened? As long as my moveable attention was busy exploring the point of contrast everything was fine. But after the spot had received lengthy and careful examination, my moveable attention, restless, fanned out in all directions and tried to tug my focus towards another promising point near the first. I refused to give in, and kept my look steadily on the same tiny spot. My moveable attention made another urgent plea to move on, which I ignored – so it freed itself from my focus and explored my visual field on its own! At this juncture something odd began to happen. The objects around me began to lose their shapes; a dark portion of one object joined the dark portion of another; shapes began joining and melting; and instead of the common objects I perceived a peculiar world of shadows and lights that had a strange life of their own. But when I could retain my focus no longer and finally released it, it joined my moveable attention, and all the objects were restored at once to their proper states and looked firm and steady as never before! Thus: as long as we move our eyes we revive and update our transient knowledge of the many casual details of our respective surroundings; yet if we fix on a single point in the visual field over a long period this transient knowledge fades away and we begin to see what we see with our eyes alone, without the help of our minds.


4) Thanks to the wonderfully complex organization of our visual system we see a complete and ever-renewed picture of the world we live in: an image we construct from a few impressions and which we influence by our feelings, needs and desires


When we are in love, the world seems bright and shining; fresh, as if wet; or in warm, soft pastel shades. But when we suffer through love the same world may appear gray and closed. We notice what corresponds to our being and mood, and as we fabricate our image of the world from a few impressions we color it according to our needs, wishes and feelings.


With great difficulty I managed to keep my focus on a single point of contrast for up to twenty minutes. It was hard work! You will realize this if you try it for yourself. It was even harder to draw the unclarity that I perceived peripherally, for my hand, incorporating subconscious knowledge, freely produced all kinds of forms and shapes. One afternoon a model was invited to our drawing class. She stood naked in front of us, and I carried out one of my experiments: I looked straight into her eyes and drew her body as unclearly as it appeared to me. She noticed my strange behavior and smiled charmingly. It was my best drawing ever (unfortunately, someone else ‘stole’ it from me). In those weeks I had a casual look at a reproduction of the Mona Lisa – and was immediately fascinated by her smile: I looked into her eyes, steadily and firmly as I had during my experiments, and there was a real smile! Then I looked at her lips, and the smile disappeared. I looked into her eyes again, more systematically. When I looked into her left (the eye in the center of her head), the smile returned: a kind smile, a loving smile, full of warmth and understanding. I repeated my experiments over several weeks, and I saw many other smiles, which seemed to depend upon my own: sometimes critical, at times closing and withdrawing; at others kind and embracing  

     When I looked into her eye, the shadows of the lips and those of her rounded cheeks blended until I could no longer really discern the corners of her mouth; yet as I knew that the corners were there I placed them somewhere in the field of blending shadows, and as these places lie to the sides of the actual corners of her mouth and slightly above them, her lips seemed to extend toward the sides, while the ‘stroke’ of her mouth turned into a bow, and so she seemed to smile. Yet when I looked at her lips in order to catch her smile it was gone. For I could see her lips clearly again and discern the corners of her mouth from the shadows on her cheeks. A surprisingly moveable smile, capable not only of moving its appearance but also of altering its expression! The smile appeared especially beautiful when I smiled myself. Try this: look slightly from below at a reproduction of the Mona Lisa’s face in its original size, relax your lips, keeping them free of any emotion, and then draw in the corners of your mouth. As easily as you are smiling now, the shadows on the woman’s face will blend, and your smile will reinforce hers to bring forth her most beautiful and loving.

     When I was carrying out these experiments I had the impression that the Mona Lisa’s changing smile was kind of an answer to me, her viewer. Why? As we don’t really see her lips we project our feelings onto those vague shadows. My happy surprise evoked a loving, understanding smile. Other feelings evoked different expressions that in some way reflected my state of mind.

    I read what several authors had written on the Mona Lisa’s smile, and developed the impression that a description of the smile could be a description of its author. Leonardo had, after all, said that a picture must be like a mirror. His painting of the Mona Lisa was an accurate portrait of a Florentine beauty, and is, moreover, a mental mirror of the viewer’s nature and feelings.

     When we gaze into a normal looking glass we see our own faces; yet when we look into the very special ‘mirror’ offered to us by a great artist – such as the Mona Lisa by Leonardo da Vinci – we see our feelings appear on the face of a stranger, such that we do not recognize them as our own; rather, we have the impression that she can read our minds and souls, and that she, in order to understand us better, adopts our feelings for a while and radiates them in her smile. We are surprised, feel recognized, and in reaction to this new feeling her smile once again changes, into the smile of someone we believe able to read our very souls.


I was convinced that Leonardo da Vinci had carried out similar experiments to my own, and was of course pleased to find evidence of it. Over the years I have found seven passages in his surviving written works that mention rays of vision and say that only the central rays are strong and true, while those surrounding weak (debole) and deceitful (bugiarde). Studying the Mona Lisa painting, I could see how the soft shadows of the cheeks lead our focus and moveable attention to the left eye, and how the shadow of that eye leads them right inside it, a clear and delicate point of contrast pleasing to our moveable attention. And why does the Mona Lisa have no eyebrows? This may have been the current fashion, but it may also have been one that pleased Leonardo, as lack of eyebrows lays more emphasis on the eyes. Leonardo made very sure, in all ways, that we would look into the eyes of the Mona Lisa, and doing so be rewarded by a loving smile.



When the eye of a woman is given so much importance, it must have special significance. What if the Mona Lisa is not only a lively and accurate portrait of a Florentine beauty, but also an allegorical embodiment of seeing? This woman is looking at us, and we are looking at her. She is looking and being looked at, thus seeing both in the active and passive form. We see with our eyes, and for this we need light. Her eyes occupy the center of the upper part of the painting, while the brightest part of the painting area, namely the shine on her breast, lies just above the very center of the painting. The dark green-brown hem of her garment may bee seen as the horizon, while the bright shining spot may represent the rising sun, and the lines of the veil thrown over her left shoulder would symbolize the trajectories of the sun traveling over the sky. We see the Mona Lisa sitting in a dark room, close to the bright opening of a small balcony. If the lateral columns were still there, we would have the strong impression that it were a window. We stay inside the dark chamber, looking out: the chamber may well symbolize the chamber of the eye. If so, the opening of the balcony would represent kind of a pupil (square instead of round). The windows are to the house what the eyes are to the body. Inside the virtual eye, the Mona Lisa would occupy the so-called natural point hailed by Leonardo: namely that wonderful interface of the eye where all rays of light received from seen objects meet and join at one single point. The woman is turning towards us from the picture plane. Her legs are parallel to the balcony, but the upper part of her body is rotating in our direction, her face first: but her eyes have already reached us, and gaze straight out of the picture at the viewer, joining the plane of vision with the direction of view. Her upper part body forms a kind of circle, and her head forms a second, smaller one around her left eye in the center of the upper part of the painting.


To the left and right of the woman mountains, lakes and river valleys can be seen. They form an almost prehistoric landscape which, in its dreamlike breadth, may symbolize nature itself. It has well been noted that in the painting we are looking down upon the left lake, but are on the same level as the right one. You will recall the two perspectives that we found in the Last Supper. Here we have another pair of perspectives, and they communicate to us the same thing: that we can never really understand the world and life by looking at it from just one point of view. Leonardo himself concurred with the antique understanding of the living being as a tiny cosmos. The Mona Lisa, by representing the natural point in the eye, may refer to the ancient belief of microcosm within macrocosm (this belief may appear strange belief, but the modern theory of fractal geometry teaches us that the same forms may appear on the highest and deepest level of structures). In the picture we have nature; a living being; and ‘artificial’ objects: a veil, a dress, a stool, a balcony, a house, roads, a bridge. The woman is depicted in obvious balance with the nature surrounding her, while the artificial objects occupy a modest place and accompany and suit her; they also form a kind of artificial pupil (window) in an artificial body (house). All of this may be a symbol of art as the harmony of the artificial world with life and nature. And yet the house lifts the Mona Lisa high above nature: it hovers above the landscape in the background, much as would a balloon. This may symbolize that our artificial trappings lift us higher than the lives and necessities of common animals. In a mysterious and frequently mistranslated passage, Leonardo wrote that humans are distinguished from the animals by our use of tools. In one of his drawings he depicted objects falling to Earth. Above he wrote: Adam here, Eve there; and below: Oh human misery; of how many things you make yourself the slave for money. In other passages of his writings he praised the human eye. It would not be too difficult to find a quote for every element of my above interpretation.


May I also mention that Leonardo’s writings are full of geometrical sketches? He considered his art to be a science, and said that no science can do without mathematics. Furthermore he praised proportions, and compared his way of putting together a picture with that of composing a piece of music: e se ti dicessi la musica essere composta di proporzione, o io con questa medesima seguito la pittura – and if you say that music consists of proportions, I as a painter also use that means (namely proportions).


Take me, thus, take a look at the underlying geometry of the Mona Lisa painting. If the columns on the sides of the balcony were still there, the inside format would be 4:3, a much-used proportion during the Italian Renaissance. The unit is given by the breadth of the woman’s head on the height of the small fold in her hair veil (nearly on the level of her eyes). Move one unit to the left and one to the right of her head and you obtain the original width, while the height measures 4 units. The left line of the balcony sill divides the height 4 units in the ratio 2:3 (or into 1.6 and 2.4 units). Draw two arcs of the radius 4 units around the lower corners of the inside format. Now draw a circle, whose radius should measure 1,5 units, around the center of the inside format. It will seize the edge of Mona Lisa’s hair. Draw another pair of arcs with a radius 4 units around the upper corners of the inside format. The four arcs will touch the circle in four points: the very points where the two diagonals of the inside format 4:3 cross the circle. These four points are very important in the geometry of this composition. Their horizontal distances measure 1.8 units, their vertical distances 2.4 units, and their oblique distances 3 units. Now imagine circles around all four points. Their radii shall measure 0.1 / 0.2 / 0.3 / 0.4 / 0.5 / 0.6 / 0.7 / 0.8 / 0.9 / 1 / 1.1 / 1.2 / 1.3 / 1.4 / 1.5 / 1.6 / 1.7 / 1.8 / 1.9 / 2 / 2.1 / 2.2 / 2.3 / 2.4 … units. Apply the circles as very fine and precise grooves on a transparency and place this over a reproduction. If seen from below and from the sides the circles joining into S-lines. Two vertical lines follow the cheeks of the woman, cross on her breast and follow the upper arms, while a horizontal S-line explains the shifting of the landscape. It is all as if the figure of the woman were created by the circles; as if Leonardo, who had carefully studied all motions of the water, had anticipated the wave-nature of light and matter! Moreover, four circles meet at the point where the parting in her hair begins: a very close double point on the height 3.5 units (half a unit below the upper edge of the inside format). The left radii measure 0.8 and 2.8 units, the right ones 1.1 and 2.9 units, while the horizontal distance of the two very close points measures 1.8 minus the square root of 0.55 minus the square root of 1.12 = 0.00007962… units. The numbers are based on the equations 55 = 8x8 – 3x3 = 28x28 – 27x27 and 112 = 11x11 – 3x3 = 29x29 – 27x27. The very small distance marked by the double point may, once again, symbolize the two mental perspectives of our human life: coming very close but never really meeting.


May I plead for the reconstruction of the original inside format 4:3? One might simply add a pair of narrow panels and color them according to the sky, the columns and the balcony. A new frame should have a strength of one unit and measure 6 by 5 units (inside measurements 4 by 3 units). Paintings by Leonardo, Raphael and others gain much when displayed in their original formats. Leonardo’s Last Supper is a ruin of a painting, in that it has lost much of its original color. However, its composition has survived undamaged and still has a strong effect on its viewers.



Geometry: Non mi legga chi non è matematico / e se tu dicessi la musica essere composta di proporzione, o io con questa medesima seguito la pittura 

For comparison: Isabella d’ Este, reconstruction of the original format 4:3 (grid 8x6 small units); radius of the circle of the head 1 small unit, radius of the circle of the upper part of the body 2 small units, radius of the central circle around the first circles 3 small units, radius of the arc of the hair around the left upper corner of the original format 4 small units  Villa Farnesina




PS. The explanation for the smile of the Mona Lisa has been found by Marcel Duchamp (some 100 years ago), by me (1974), by Dr. Margaret Livingstone, Harvard neurologist (2001); and presumably by several others at various times. In an old movie the Toulouse-Lautrec character declares that the charm of the Mona Lisa’s smile lies in her eyes ... Those interested in Marcel Duchamp might have a look at his L.H.O.O.Q: the very fine endings of the moustache curl precisely towards the Mona Lisa’s pupils. Pronounce the ominous letters L.H.O.O.Q in English and you hear the word look. Pronounce them in French and you hear the well-known iconoclasm elle a chaud au cul. Read the letters in a spirited manner, namely by adding aire (air), and you get elle a chaud occulaire. In the elaborate artistic language of Marcel Duchamp, air symbolizes the mind or spirit (ruach in Hebrew, pneuma in Greek, spiritus in Latin) while warmth means life in the sense of vivacity. Taken together we can read the five letters as follows: Mona Lisa appears perfectly alive when you look into her eye ... The wit of Marcel Duchamp’s famous joke is that while seemingly making fun of the Mona Lisa he was actually proposing the first real interpretation of Leonardo’s masterwork! You may also consider the enigmatic title of Marcel Duchamp’s Grand Glass:





‘Mona Lisa, looked at by so many people, loves me, smile at me, Marcel Duchamp ...’


Readers of German may refer to my online-essay  DIE SYMBOLSPRACHEN VON MARCEL DUCHAMP UND PABLO PICASSO




Bibliography: I published my interpretation of the Mona Lisa in various photocopied papers and books from 1974 on, for example: Mona Lisa ein Gleichnis des Sehens, Zürich, 1989 (copies are kept in several Swiss libraries); Amphitrite und Poseidon im Salon der Villa Farnesina, oder “Alles ist gleich, alles ungleich”  Zürich, 1974 - ?? (this project for an electronic book,. some 400 pages, also contains a brief interpretation of the Mona Lisa, and won me a prize by the university of Zurich, dies academicus 1994); Geometrische Bildanlagen in der Griechischen Antike, im Mittelalter und in der italienischen Renaissance, 1979-2001 (again with a brief interpretation of the Mona Lisa and a reconstruction of the painting’s original size)




Geometry in art (large files)   gia.htm