Next Article
Front Page
Email Author
Comment

About Face Again - Part Two
Twisting, Twifoldry, and Twins in the World of Picasso

© John Fudjack and Patricia Dinkelaker - February, 2000


Section Three - Three Motifs in Picasso's Work
skip to
section four

There are three motifs that surface time and again in Picasso's work. Together they constitute a three-pronged attack on non-paradoxical structure. Each motif, like a separate strand in a single braid, takes a turn at getting the upper hand as it runs circles around its partners and then, in turn, is enveloped by them. We have already alluded to all three in Part One, but disentangle them here for closer inspection. They are -

  1. A 'serpentination' motif - which involves a TWISTING (or TWINING) of figures, front to back and vice versa. In Part One we suggested that this and related techniques in cubism were an attempt to express the fact that figure ultimately stands in a paradoxical relationship to ground. While figure will always, at some point, inevitably collapse back into ground in a classic 'enantiodromic' shift it also incessantly separates itself out from ground. Figure and ground are two incommensurable 'opposites' out of which the basic structure of our experience is nevertheless forged.

  2. A 'split screen' motif - in which incompatible styles of painting divide the work into left and right halves that are then brought into a higher synthesis by virtue of the complex and ultimately paradoxical ways in which frames are nested in each half. We shall call this technique TWIFOLDRY. What is ultimately being explored is not unlike what Bergson, Yeats, Read and others called 'interpenetrating orders', which we discussed in depth in About Face. In the present paper, in a section devoted to Twifoldry, we offer examples of this motif in Picasso's work.

  3. A 'sleeper-watcher' motif - in which two parties are depicted in the
    Contemplation, 1904
    painting - one of whom is asleep and being watched by the other. Or is it the sleeping figure who dreams the watcher into existence? Under the auspices of this motif Picasso explores what happens to the relationship between 'subject' and 'object' in the course of a structural inversion - a topic on which the two previously mentioned motifs remain silent.

    Insofar as watcher and sleeper can be conceived as mirror images in alternate realities, we shall call them TWINS - because in certain societies, according to anthropologists and mythologists, the appearance of the 'twin' or 'double' brings this kind of relationship to the fore. 12 When properly understood, this motif affords revelations about Picasso of the most personal and intimate sort, which we turn our attention to in Part 3 of this paper.

In all three motifs Picasso is exploring inversion - how to literally turn things 'inside out'. But in each motif he approaches the subject from a slightly different vantage point. In the sophisticated forms of Serpentination that he created, which utilize subtle figure-ground reversals, and involve nested frames of reference that interpenetrate in complex ways in his Twifolded canvases, not only are the objects that he paints turned inside out, the pictoral space itself suffers a radical inversion.

When, furthermore, these inversions are experienced in a very personal way, as the sleeper-watcher paintings suggest possible, it is as if the object thereby becomes capable of being experienced as subject. And vice versa, the subject becomes a watched 'object'. 13 When, in other words, what we are in relationship to is another human being - as the sleeper-watcher paintings suggest that it can be - there is, in the process of inversion, an exchange that takes place at such a profound level that in us the dead can come alive, and we, in their stead, die. To this feature of the inversion process we return in Part Three. In the meantime, we take a closer look at the second motif.

Section Four - The Twifolded Canvas
skip to
footnotes

To get a quick sense of what Twifoldry involves, have a look at the following diagram, by Douglas Hofstadter. 14 It accomplishes a synthesis similar to what is intended by Bergson when he talks about 'interpenetrating orders'.

Please look at the word to the left and take a minute to consider what name is being spelled.

Actually, Hofstadter is spelling two names - 'J.S. Bach' and 'Fermat'. But he does this in a rather interesting way. Each of the large letters in the diagram is made up of smaller letters. And in order to be successful in discerning either name one must shift one's attention from one level of organization to the other at a midway point during the process of reading from left to right.

In order to see the J, S, and B in the name 'J.S. Bach' one has to zoom IN to the level at which the small letters appear. But in order to read the last part of the musician's last name ('ach') one must zoom OUT and attend to the larger letters.

One must also jump from one level to another to read the name 'Fermat'. But in this case we do it in reverse order and start with the larger letters on the left side to see the first syllable of the mathematician's name ('fer'), and then shift attention to the smaller letters on the right hand side of the illustration to read the last syllable of his name, 'mat'.

The manner in which the words and letters in the above illustration are organized with respect to each other is reminiscent of the t'ai-gi-tu ('yin-
yang') diagram, in which the white area that comprises the left-hand side of the diagram has a black center, while the black area on the right side has a white center. In a previous article we've discussed how the plot in O'Henry's story about the Christmas gift was organized in a similar fashion. Shakespeare, Hegel, 15 and others have also explored structures of this type. And we saw in Part One of this paper how Federighi's 'Three Graces', and Escher's 'Day and Night' achieved similar effects.

In none of these examples, however, is a shift in STYLE, MANNER, or REPRESENTATIONAL MODE used to create the impression of polar opposites. In the exploration of this particular method it would be Picasso who would specialize.

But it is not Picasso who invented the technique, if Robert Witkin is right. In recent years 16 he demonstrated how van Eyck's 'Marriage of Giovanni' (1434) manifests what he calls 'two presentational codes' in 'a single work'. [Click here for a pop-up window with a (133k) rendition of the painting from www.artchive.com.]

The two halves of the painting ... are not treated formally in the same way. There is a distinct shift in 'visual language' as one moves from one to the other. The left-hand side of the picture develops a language of tonality, subtle variations in light and shade, while the right-hand side develops a language of color.

... Read as a text, the peculiarity of van Eyck's painting of the marriage lies in the way in which van Eyck seeks to reconcile two mutually exclusive modes of symbolling, two levels of abstraction, within a unified aesthetic work, thereby rendering important social values in ways characteristic of a transition between two types of society (page 142).

It will be remembered, from Part I, that it is Witkin's thesis that social structure is made visible in art styles. So his conclusion here that Van Eyck's twifolded canvas should reflect deep social tensions should come as no surprise. But this is not what concerns us here. Nor will we address why, if such a technique is characteristic of 'modern' art, it should occur in this 15th century painting - although Witkin's thesis presents some intriguing theoretical options.

What will concern us here is how Picasso utilized the technique, and to what end.

Picasso first began to twifold his canvases in the period which resulted in Les Demoiselles D'Avignon (1907). Between 1905 and 1906 he had become increasingly aware of the relationship between foreground and background in painting, and was presumably immersed in this problem at the time of painting Desmoiselles. Cezanne and Greco, who had experimented with the distortions resulting from a conflation of figure and ground, were Picasso's progenitors in this regard. Often cited as influences on him during those years, they are also acknowledged as having had an impact on this painting in particular. 17

Demoiselles is the work that is now believed by many art historians to be the single most important piece of 20th century art. At the time, however, it repulsed not only Picasso's critics, it unsettled his closest friends. Considered 'horrific' by friend and foe alike, it occasioned not only ridicule, but concern for the painter's mental health.

Apollinaire murmured revolution; Leo Stein burst into embarrassed, uncomprehending laughter; Gertrude Stein lapsed into unaccustomed silence; Matisse swore revenge on this barbaric mockery of modern painting and Derain expressed his wry concern that 'one day Picasso would be found hanging behind his big picture'. [Huffington, Picasso: Creator and Destroyer, 1988 (New York: Simon and Schuster), page 93]
For years the portrait sat with its face turned to Picasso's wall. Ironically, it was not even exhibited in the grand 1932 retrospective that would inspire Jung's contempt.


Click here for larger file, in pop-up window
Nevertheless, Demoiselles was the 'turning point in Picasso's career,' according to Choucha, and 'led to the development of cubism.' And it is cubism that 'represents the most radical innovation in Western art since the Renaissance.' 18

We could present a detailed analysis demonstrating that this painting is usually viewed in terms of a display of rival styles that split it vertically down the middle. But this is not really necessary, as we shall use another twifolded canvas of Picasso's - produced nearly 50 years later - to examine the technique that we are interested in exploring here. And, in any case, we will find ourselves returning later in this paper to the interesting psychological circumstances in which this painting came to be. At this point, however, we are more concerned with trying to gleen some insight into the inspiration that lay behind the basic impulse that led Picasso to the idea of shifting styles mid-painting.

We have come to believe that in Picasso there is a recognition that order, if it is to succeed in dispelling disorder, must be constructed on a foundation of disorder - as strange as this notion might at first seem. This is not simply to say that order can emerge OUT of disorder, but that if we look deeply into what order is, we find at its very center, as its ESSENCE - disorder. To ensure harmony one must, therefore, give chaos a central place - a place of honor, as it were.

In her biography of Picasso, Francoise Gilot - who was not only a fellow painter, but also Picasso's friend, mistress, wife, and mother of two of his children - tells the following story. It starts with a comment that she has made about a painting that he is working on -

I asked him if he thought the green was balanced by the violet. 'That's not the question,' he said. 'I don't need to know that. I'm not trying to make this first proposition more coherent. Right now I'm interested in making it more disturbing. After that I'll start to construct - but not to harmonize; to make it more deeply disturbing, more subversive.'

'So far it's instinctive, you see. Now I've got to set down something that goes further than that, something a good deal more audacious. The problem is, how can I shake up that first proposition? How can I, without destroying it completely, make it more subversive? How can I make it unique - not simply new, but stripped down and lacerating? You see, for me a painting is a dramatic action in the course of which reality finds itself split apart. For me, that dramatic action takes precedence over all other considerations. The pure plastic act is only secondary as far as I'm concerned. What counts is

the drama of that plastic act, the moment at which the universe comes out of itself and meets its own destruction.'

' ... So my purpose is to set things in movement, to provoke this movement by contradictory tensions, opposing forces, and in that tension of opposition, to find the moment which seems most interesting to me.' 19

Instead of looking to see how the urge described in the above passage displays itself in Demoisselles, we skip ahead 50 years to a work that shows a similar methodology and intent - 'Canvas 0' in the 'Women of Algiers' series (1955).

In it there is also a left-right split, on which is built a 'symmetry of cancellations', in the words of art historian Leo Steinberg. This occasions what he calls a 'reciprocal counterchange' that takes place between the resulting halves. Although Steinberg's analysis of the painting is much too long to include in full, it is a very clearly rendered account, fascinating in its own right. So we have decided to present it as if it were one extended passage, although what follows is actually a composite of excerpts. The original from which these are taken runs quite a bit longer.

The passage speaks for itself. But for the benefit of those not familiar with this type of examination we should mention that Steinberg's ability to identify specific characters or objects in the painting ('the smoker', 'the servant', or the 'beacon of coffee') derives not from some special super-human capacity to see what escapes the rest of us, but from a knowledge about the painting gleened from earlier sketches made by the artist, and other similar sources of information.

Sleeper [the figure at the lower right] and Smoker [the figure at far left] in Canvas O do not really differ in style - they are both in Picasso's style of 1955. But they differ in representational mode, one faceless, schematic, the other comparatively naturalistic. They differ too with respect to their housing; if this were the inside of an accordian, I could imagine it compressed on the right, pulled out on the left. And they differ in the kind of claim they make on the beholder. While the slumbering one heaps her limbs about her unconscious center, the other stares back at the spectator as if to match his curiosity; his opposite pole. One could call them iconic and narrative, but with the narrative mode paradoxically more abstract. Which of the two is the more solidly rendered is hard to say. The alert Smoker offers a single perspective, all frontage and no turns permitted, as enjoined by the fixed point of view. And she is drawn without tonal gradations, all contour and local color. Whereas the Sleeper's colors function as tonal values, making forceful suggestions of turning planes, like a folded construction. The two figures define one another in their distinctness, but also in their affinity. One is all aspect - the other, all aspects in one. Yet we know them to be interchangeable.

It is clearly more than a matter of juxtaposing 'representational' and 'Cubist' styles, or abstract and realistic. For the schematic Sleeper, who looks at first sight like the shook fragments in a kaleidoscope, is at least as stereomatic in structure as a Japanese origami; and the Smoker is after all a painted emblem in the style of a playing card. Which has more volume - a paper sculpture or a Queen of Hearts? The variables which for us define truth to appearance are infinite, and Picasso learned long ago that they could be separated and shuffled.

For instance: in a classical work such as the Three Graces, all the figures are understood to be exactly or nearly alike; in terms of style, all are brought up to the same level of finish and anatomical specificity. They are distinguished only by orientation, i.e. in showing either their fronts or their backs. But their likeness to one another assures us that each figure actually possesses both a front and a back view, whether she shows it or not. The depicted figure is, as it were, endowed with two variables - a given anatomical character and a given orientation in space. Now Picasso reshuffles the mix. Having compounded both orientations in one, he feels free to vary the figures in anatomical specificity, leaving one of two interchangeable women in a schematic mode. The one with the pretty face has no other aspect; while her sister of many aspects has no particular face. Yet they are understood to belong, like the Graces, to one sisterhood. (230-231)

... the Smoker's apparent three-dimensionality turns out to depend entirely on those two distant elements at the back: the beamed ceiling that radiates stylishly from her hat and the molded jamb of the door that runs into the trim of her shoulder.

Given this striking cooperation in her behalf between doorway and ceiling, it comes as a shock to observe that these features positively refuse to cooperate with each other. The doorway jars too destructively with the ceiling's perspective that it cancels it out. It re-flattens the space toward the right to meet the different needs of the Sleeper. (232)

[While, with respect to the Sleeper] if the formal principle here appears to be that of Cubist process, it is scotched by the fourth figure in the design - the Servant. Her headlong rush into the cave, preceded by her beacon of coffee, pierces the Sleeper's surround as if, ignorant of Cubist principles, she mistook this black ambience for a literal rendering of deep shadow space.

The picture is built on a symmetry of concellations. On the right, Servant and coffee pot, together with the striped curtain coulisse at the margin, restore to the Cubist space the suggestion of stage-like depth. On the left, a stated perspective of depth is annulled by flat contradiction. While the left half of the picture shows an essential illusionism confounded by elements of forbidding flatness, the opposite half invites elements of disruptive illusionism to break up an essential abstraction. A concord of contradictions; it is the principle of reciprocal counterchange - now expanded to a symphonic scale - which had been with Picasso ever since his composite front-and-side faces, wherein the profile portion was charged with a facing eye, and the en face with the eye in side view - a mutual exchange of competences designed to make the parts lock.

Cubism here plays a powerful role - but a role assigned within a scenario. It becomes one of the several space-determining forces whose clash on thepictorial stage may be likened to the clash of doom-determining forces intragedy. Old fixed-point perspective is another such force, and, finally, the simultaneous sighting of one body from dispersed vantages is a third - a space-determinant whose sheer possibility depends on the co-existence of the two others.

Cubism, in its original break-up of pre-faceted solids, had indeed furnishedthe tools for Picasso's subsequent interweaving of aspects. But that problem, and the goal it pursued, Cubism neither solved nor confronted: the vision of a thing reappearing, as it were, successively in the same contoured place; its incarnation in simultaneous aspects, each itself flat as a baroque brushstroke is flat, but in combination a record of sensuous perception. Look at the Servant, compounded of a full back view behind and a full front view before - as if we had stayed with her form as it turned. And the Sleeper, whose repeated left side turning over we are given to know with a kind of prolonged intimacy. As Picasso finally renders the Sleeper's form - her body a coincidence of front and back within palpable contours - the familiar distinctions between right and left, between things here and hidden, between now and anon, settling and rising, all dissolve into forms of awareness whose affinities suggest both further abstraction and fuller possession. (233)

Steinberg gets it right when he moves from a discussion of the alternate approaches to 'perspective' that are available to the artist to a discussion of the 'forms of awareness' that are manifest in this object, the painting. What might easily be projected outward, into physical space, is reclaimed as a structural attribute of the consciousness of the artist who created the piece, and the consciousness of the observer who is capable, by virtue of forms of awareness that she shares with the artist, of enjoying the piece. Steinberg wants us to see this piece from the inside out, as it were - as the manifestation of a most profound insight, on the part of the artist, into the structure of his own awareness.

The 'concord of contradictions' about which Steinberg speaks reminds us of the manner in which, in Bergson's system, each of the mutually exclusive (yet interpenetrating) orders of existence are prone to 'interruptions' that link them to the other. The higher order synthesis that is thus achieved relies not on making the lacunae disappear, but in bringing them into a juxtaposition that ultimately CONNECTS.

This can be put in a slightly different way - not only is there a collapse, in the Picassian world, of the familiar distinctions between 'right' and 'left', 'back' and 'front', and so forth - those very distinctions are somehow enfolded into the collapsed product. We've tried to illustrate how this sort of thing might be conceived mathematically in our 3-D Puzzle, in which bilateral ribbons are supercoiled in such a way as to produce a unilateral ribbon or 'mobius strip' - a figure that continues to marvel contemporary DNA researchers. 20


- Continue to Part 3 -

Footnotes and References

12. In a section on 'Twins' in an extensive work on Jung and Picasso entitled Hegemony and the Harlequin, we consider the work of anthropologist Victor Turner (The Ritual Process), and comment -

On the subject of twins, Turner has the following to say:
In many societies, twins have this mediating function between animal and diety: They are at once more than human and less than human. Almost everywhere in tribal society they are hard to fit into the ideal model of the social structure, but one of the paradoxes of twinship is that it sometimes becomes associated with rituals that exhibit the fundamental principles of that structure; twinship takes on a contrastive character analogous to the relationship of ground to figure in Gestalt psychology. Indeed, one often finds in human cultures that structural contradictions, asymmetries, and anomolies are overlaid by layers of myth, ritual, and symbol, which stress the axiomatic value of key structural principles with regard to the very situations where these appear to be most inoperative. (Turner, page 47)
In some cultures either one or both of the twins are put to death, in other cultures they are 'permanently assigned a special status' (48).
This may be regarded as another instance of a widely prevalent social tendency EITHER to make what falls outside the norm a matter of concern for the widest group OR to destroy the exceptional phenomenon. In the former case, the anomalous may be sacralized, regarded as holy. Thus, in eastern Europe, idiots used to be regarded as living shrines, repositories of a sacredness that had wrecked their natural wits. They were entitled to food and clothing from everyone. Here the anomaly, the 'stone that the builders rejected', is removed from the structured order of society and made to represent the simple unity of society itself, conceptualized as homogeneous, rather than as a system of heterogeneous social positions.

... The paradox that what is good (in theory) is bad (in practice) becomes the mobilizing point of a ritual that stresses the overall unity of the group, surmounting its contradictions. (49)

In this context we think of Jung's recommendation that the individual who seeks self-realization - 'individuation' - ought to take her personal 'stumbling block' as the 'cornerstone' of her personality. The implication is that the fourth, 'inferior' function of the individual will be the 'sacred' one, and should be made 'central' in some important fashion - in other words, that the individual should reconfigure her personality in a LIMINO-CENTRIC fashion. Anomaly, paradox, and 'absurdity' present one with access to the 'spiritual' world, and are hence quintessentially SACRED - the 'fool' in one paradigm is the 'wise man' in another, incommensurable paradigm. What the above paragraph by Turner implies is that society NATURALLY (without 'deliberation' or rational planning - ie, 'unconsciously') organizes itself in a limino-centric fashion - around those events, like the birth of twins, that are 'absurd' or anomalous. Here we have a picture of the community GATHERING around the absurd (much like they gather around the premature death of a child): the ANOMALOUS becomes the concern of the community - an action by which social 'anomy' is ritually 'contained'. (Fudjack and Dinkelaker, 1997)
back to text

13. This is not unlike when, according to Sartre, we are subjected to what he calls the 'look' of the 'other'.
back to text

14. Douglas Hofstadter, Godel, Escher, Bach: An Eternal Golden Braid, 1979, (New York: Random House), page 331.
back to text

15. Hegel says:

In love an individual has the consciousness of himself in the consciousness of an other; he has divested himself of himself, and in this mutual self-divestment he has gained himself. ... The true essence of love consists in the surrender of consciousness of self, forgetting oneself in another self, and yet having and possessing oneself for the first time in this perishing and forgetting.

This 'divestment', in which what is given up is gained, constitutes a contradiction or paradox. Hegel's point complements the one made by Glassman and Fields about the individual's need to surrender the 'Self' in order to find it.

The notion of 'divestment' (uncovering, denuding) is interesting - as it implies the attaining of an undifferentiated core stripped bare of detail. But the 'love' in which this is accomplished is not only 'unity', but 'multiplicity' as well:

... Love is a differentiating of two, who yet are absolutely not different for one another. The feeling and consciousness of this identity is love..." [Hegel, in Albert Hofstadter's book, Agony & Epitaph: Man, his art, and his poetry , 1970, (New York: George Braziller), page 238.]
So whereas the Buddhist formulation tends to stress the SPIRITUAL (i.e., N-based) nature of this accomplishment and treat love, kindness or compassion as auxiliary, Hegel is here emphasizing its interpersonal, FEELING (i.e., F-based) nature, and permitting the 'wisdom' component to play a subsidiary role. Although both appear to be manifestations of an NF approach - the Buddhist path might thus be considered Nf, while Hegel seems to be plotting an nF course.

Insofar as the 'consciousness' of which Hegel speaks involves apprehending the paradox at the heart of love, it could be considered a developed level of consciousness, one in which its liminocentric nature is acknowledged and 'realized' (in both senses of the word: made real and comprehended).
back to text

16. Robert Witkin, Art and Social Structure, 1995, (Great Britain: Polity Press), page 142.
back to text

17. In Cezanne, as Witkin mentions, 'there is often the inclusion of different viewpoints, simultaneously within the same painting' (156). It should come as not surprise, then, that according to art critic Roger Fry it is Cezanne who emerges as the predominant influence in Picasso's Desmoiselles, in which a Cezannian 'linking together [of] foreground and background planes' occurs (see William Rubin, page 28).
back to text

18. (Choucha, page 31-33) Choucha has the following to say about Cubism -

[Cubism] is a highly complex style, as multifaceted as the different views of reality it tried to represent simultaneously upon canvas.... Most analysis focuses upon its formal qualities. We will examine one particular facet - its esoteric and poetic qualities, which seemed to be more obvious to critics contemporary to Picasso, judging from some reviewers' comments. Andre Salmon called Picasso an alchemist prince, an apprentice sorcerer; Jacques Riviere wrote that he 'strayed into occult researches where it is impossible to follow him.' Blaise Cendrars wrote that cubist painting reminded him of black magic rites, 'they exhale strange, unhealthy, disturbing charm: they almost literally cast a spell. They are magic mirrors, sorcerer's tables.'

Cubism developed out of aesthetic experimentation - it is scientific in its dissection of matter, surfaces, and ordinary appearances, which are then reconstructed upon canvas. It has been suggested that the scientific hypothesis of the fourth dimension may have been an influence upon cubism (Gibbons, 130-147). Howard Hinton had promoted the concept widely between 1880 and 1907, and it had also attracted much attention from occultists who saw this as the scientific verification of the idea of the astral plane. ... Cubism's other scientific parallels included recent atomic discoveries which demonstrated the 'emptiness' of matter, and the philosophy of Henri Bergson, who overturned concepts of time and space. He claimed that different planes coexisted simultaneously.

... Altogether, there seems to have been a convergence of occultism, science, art, and philosophy. Apollinaire discussed how science, art and religionwere beginning to move together. This idea was common amongst nineteenth century occultists such as Edouard Schure and the Theosophists.

An interesting passage in the light of what we have previously said - about dimensionality, art and science, the influence that Henri Bergson's notion of 'interdependent orders of existence' had on the modern artist, and the manner in which the modern artist may, in turn, have anticipated not only physics but Jungian psychology and typology.
back to text

19. Picasso's thoughts inspired Gilot to make the following comment:

I told him if he had never painted a single picture in his life, we would probably have known of him as a philosopher. He laughed. 'When I was a child, my mother said to me, 'If you become a soldier you'll be a general.If you become a monk you'll end up as the Pope.' Instead, I became a painterand wound up as Picasso.'
We cite this epilogue, not only because it shows that Picasso was, in fact, interested in the more profoundly philosophical aspect of his work, but because the anecdote about his mother's comment shows that her attitude toward him was one of 'unconditional regard', to use the phrase that Carl Rogers made famous. It was a 'beneficial' double-bind: whether you do X or not-X, you win.

The negative double-bind states: whether you do X or not-X, you lose. These are similar in structure, of course, but whereas the former is psychologically experienced as liberation, the latter is suffered as a form of psychological imprisonment. Based on extensive research that we've done into the biographies of Picasso and Jung - both of whom, by the way, we see as introverted intuitives - we conclude that whereas Jung was made to suffer negative double binding at the hands of his mother, Picasso enjoyed positive double-binding. Jung, as a result, withdrew from interpersonal contact (and preferred 'thinking', over 'feeling', as an auxiliary function), while Picasso had a highly developed feeling function.
back to text

20. 'This is the remarkable, even somewhat enigmatic property of the Mobius strip, which happens to have only one side and hence is referred to as a unilateral ribbon,' says Maxim D. Frank-Kamenetskii, a biotechnology professor who is 'best known for his contribution in the field of DNA topology, supercoiling, and unusual structures'. Not only does it have one 'side' (conflating 'back' and 'front'), but one 'edge' (conflating 'right' and 'left')(Unraveling DNA - the Most Important Moleculeof Life, 1997, Perseus Books, Reading Mass, page 9)

DNA is often looped into closed circular arrangments (called ccDNA) - bilateral ribbons that are joined at the ends. It has furthermore been discovered that these loops sometimes have KNOTS in them. DNA technicians have as a result become interested in learning how to tie such knots. About this, for instance, Frank-Kamenetskii says -

'A molecule is not a rope whose ends you can tie by hand. You can, however, try a different technique: make the ends of the molecule cohesive, with the reasonable hope that the molecule may form a knot it its ends chance to meet' (112)'.

But there may be another, quite surprising way of tying such a knot, as we've shown in the answer to our 3-D Puzzle. As demonstrated there, if you slice a higher-order mobius strip (i.e., a unilateral ribbon with 3 or more half-twists) up the middle, you get a bilateral ribbon in knotted closed-circular form!
back to text

Beginning of This Paper

Back to Front Page