Kiviuq was a young inuit living far in the northern lands. The landscapes he knew were made of an everlasting tapestry of snow. Kiviuq really loved to read, but he enjoyed writing even more. He liked to write all kinds of stories: stories about the falling snow of winter, of wandering polar bears and the 24hour day of light in the warm summer.He even wrote modern versions of the ancestral myths of his people. Everyone in the village appreciated reading his stories. Through his words he would bring tears of joy and tears of pain. Kiviuq was happy to bring stories to people and to be an important part of his community.
When the Europeans arrived, they created an alphabet to write in Inuktitut, his native language. It was a very peculiar alphabet made of hooks, probably to go fishing, of toboggans to go hunting and of triangles for which Kiviuq had no idea of the use. Even if Kiviuq was a very talented writer, he had troubles at school. He would write long and beautiful texts, but would always fail his exams because of his calligraphy. His teacher was not happy with his handwriting. His triangles were always slightly tilted on the side, as if pushed by the northern winds. The teacher wanted him to write with equilateral triangles, with three sides of exactly the same length. Sadly enough, Kiviuqwas not able to fulfill his teacher’s expectations. His triangles were always isosceles or scalenes, with all kinds of different sides and sizes. Kiviuq felt the different shapes of his triangles helped express emotions in his stories. For example, a very narrow triangle would express danger, while a flattened smooth triangle would express calmness and the time that lingers on. Nevertheless, none of his readers had complained about the shape of his triangles so Kiviuq kept on with this habit. Now, because of his bad marks, his teacher told him he could never go to university and become a writer. This madeKiviuq really sad.
So Kiviuq started working more and more on his triangles. He would practice for hours and hours drawing them and studying them. The more he would study triangles, the less he would write stories. People in the village started worrying because he would not enchant them with his words anymore. Kiviuq was adamant and decided to become a writer so he kept working hard on his triangles. With all this work, he started having good marks in mathematics and sciences. The better his calligraphy became, the worse his stories were. They became boring; the snow was static and the hunts brought no game. He did not worry about this because he had better marks in writing, even with his boring stories. Although less passionate about it, he really got amazing marks in mathematics and sciences.
So one day he finally went to university to become a scientist. Through his years of study, he learned everything that was to be known about geometry, especially about triangles. He then become known as one of world’s most eminent specialists in triangles which lead him to meet very interesting people all around the world and to see amazing marvels on all continents. He went to visit the great pyramids of Egypt, the Eiffel tour and the great Mont Everest’s peak.He worked with the artist Maurelius Escher for the art of tiling the plane with triangles, with the architect Ieoh Ming Pei for the construction of the pyramid in front of Le Louvre consisting of 666 triangles. He even met Dr. Sierpinski and helped him in his worked on his famous triangle constructed with an infinity of smaller triangles.
One day Kiviuqmet a man named Buckminster Füller. Mr. Füller was really good with triangles as well and together they decided to construct a map of the world using triangles. After months and months of difficult labour, they finally finished the map and exposed it on a giant wall. Proudly, they spent some time admiring the result. It was a very interesting map, it placed the countries in very different places than the usual mapKiviuq knew. In particular, Kiviuq noticed the map placed the North Pole almost in the middle. Kiviuq started remembering his old land and his old friends.
Kiviuq decided to go home after all these years of work and traveling. Coming home, he felt like a stranger. He realised people were really interested in his traveling stories, but none would bring tears of joy nor tears of pain. Kiviuq missed these days when he felt he could reach people’s hearts, so he started to write stories again. In his stories, the snow was not shining like crystal under the sun, the fish tasted like nothing and the polar bears were gone. He had forgot the feeling of his own culture and lost himself. Kiviuq realised he was again really sad. One day, wandering in the village he saw some kids playing in the snow. After some hesitation, he decided to go play with them. He had forgotten the simple pleasure of playing in the snow and the pleasure of building igloos. The more time he spent in his town, the more he was rediscovering the deepness of his culture. One day, when he was ready, he decided to write a story, his own story. He was able again to make the wind dance and the northern lights shine. Moreover, he was now able to place his own culture in a world mixed with myths and sciences. The wind had a soul but as well a precise velocity. It came from and went to places Kiviuq could now describe. Finally he could describe precisely the extremely complex geometry of every single dancing snowflake.Now again, people were reading his story and snow made of tears of joy and tears of pain were to be found in his northern lands.
I have the chance to have three paintings on the walls of the Lighthouse cultural center in Copenhagen for few weeks. To visit, look at the calendar if the space is open, the paitings are available only when the space is open for activities.
The paintings relate to my researches in narratology that can be found in favious articles of the blog. The main useful articles are:
”The blues were played in New Orleans in the early days very, very slowly, and not like today, but in a Spanish rhythm.” Baby Dodds, musician.
When I started to dance and DJ, I quickly became interested in the history of American blues music to find out what it could add to the scene. Being very passionate by nature, I started out with this simple motivation but ended up with the desire to write a book on blues history. In the process, after going through quite an unhealthy amount of documentation, I had the chance to give history classes to blues dancers in Montreal and Toronto. I was then surprised to learn that even long-time dancers or teachers seemed unaware of one of the most important Latin influences on the blues: the Spanish tinge. It reminded me that there’s always a difference between how a dance really looked back in the good old days and what we can see in modern competitions and dancefloors. In between lies a big gap in knowledge and a long list of innovations, adaptations and all natural elements contributing to the evolution of any dance. This article reaches back in history to try to find some hints about blues’ birth and the meaning it could have for dancing.
Back at the beginning of the last century
Blues music probably appeared somewhere in the 1890’s in Afro-American communities and slowly spread around. While the dance itself remained obscure, the music started to appear in partitions during the next 20 years or so. The first white craze for blues music exploded around Handy’s composition ”The Memphis Blues” (recorded in 1914) that Vernon and Irene Castle were using to promote the new dance in vogue: the foxtrot. Although the following video doesn’t show us this particular dance, it does present the general movement of the famous dancers.
A slightly sarcastic but maybe accurate description would be: white people dancing to Afro-American music. Which is, indeed, not a problem by itself. It simply demonstrates the complications of trying to find out what is the ”real” way of dancing a dance. The star couple had a very strict idea of what dancing was all about: no shimmies, no hip movement, no hopping, no dips, no twisting the body and so on since such movements were obviously sinful and disgraceful. (Basically, no fun.) After Vernon and Irene, things could have easily found another path in history and their dance could have ended up being called blues dancing.
What could have changed as well, in similar fashion, are the labels for the music. ”Memphis Blues” can actually be considered a rag, no musician of the time seems to reject fast versions of ”The St. Louis Blues” as being non-blues, and even some songs long after were still labeled foxtrots, like many of Washboard Sam’s recordings. In a similar fashion, Louis Armstrong would explain to Bing Crosby: ”Ah, swing, well, we used to call it ragtime, then blues -then jazz. Now, it’s swing. Ha! Ha! White folks yo’all sho is a mess. Ha! Ha! Swing!”.
Indeed, the multiple variations on this same topic are almost infinite and we end up with the conclusion that it’s very hard to define what blues music is. This forces us to keep two possibilities in mind: either the blues was a very narrow and specific musical trend that has been stretched out for commercial value by recording companies, or it should be considered as a wider variety of music. We’ll probably never get a fully satisfactory answer, but the fact is that the fun part of the debate is hunting down some specific historical components of the music itself.
The question of Latin influences on the blues finds a hint of resolution in various anecdotal contexts. It can be Skip James’ tuning, which he got from a man named Stuckey, who learned it in Europe from Bahamian soldiers during WWI; or, slightly more convincingly, in some songs bearing Caribbean influences, like ”Coal Mine Blues” by Georgia’s songster Peg Leg Howell, which sounds like Mighty Sparrow’s old calypso recordings; or even more explicitly in Clara Smith’s ”West Indies Blues”.
It still remains rather obscure in official American music, and Latin influences might predate the blues and jazz era. In his 1897 publication, Rag Time Instructor, Ben Harney points to some Spanish origins of ragtime music based on the habanera rhythm. Sadly, not much more evidence can be found, either in his own text or in other ragtime publications of the time.
Probably the most famous Latin section of any blues comes in the introduction of the main blues anthem; ”The Saint-Louis Blues” published by Handy in 1914. In his biography he mentions that while playing at the Dixie Park in Memphis, his band went through the habanera section of Will H. Tyer’s ”Maori”. He was impressed by what he described as the natural gracefulness of the black dancers during that section and he suspected something inherently ”black” about this rhythm. He tested his hypothesis by playing ”La Paloma”, the famous Spanish song with its clear habanera section. As predicted, dancers seemed particularly at ease with it so he decided to keep this rhythm in mind for later use. After hiding it in ”The Memphis Blues”, he added it to the introduction part of his well-known blues composition. These few measures are often described as a tango introduction since the habanera rhythm is also very present in Argentina’s national music. It is as well present in various African music. Conclusion; a main Latin element in blues is not strictly Latin.
Where did the rhythm come from?
We know a large quantity of slaves were brought first to the Caribbean for ”seasoning”, basically to prepare them to become good slaves and have more value on the market. Some would stay just a little while, some would stay there forever. The conditions in the Caribbean allowed them to continue to perform music and dances rooted in Africa but evolving in the specific context of the various islands. These dances, like the Bamboula, the Chicta, Calinda and many others involved various hips and shoulders movements, often described as lascivious by white observers.
With the independence of Saint-Domingue in 1804, a lot of white and Creoles fled Haiti and transited to Cuba where their music and dances mixed with the local traditions like the contradanza habanera and others. Then many ended up in New Orleans, some 10 000 already by 1815, where Latin, French and multiple African influences finally all mixed in the pot of the Crescent City and surroundings. Lower class blacks with their very specific dances would often face racial discrimination from whiter, therefore more privileged creoles. Nevertheless, both bore an Afro-Caribbean heritage.
Of the lower class, we know for instance voodoo priest Marie Laveau would hold secret events at Lake Pontchartrain, where dances were different than what you would see publicly at Congo Square. As for the creoles, some famous musicians came out of that social group, like Alphonse Picou, Emmanuel Perez, Sidney Bechet and many others. The 1890’s Jim Crow laws forced some more interactions between black and creole musicians since before Jim Crow creoles lived more as a separate upper class from blacks, while after Jim Crow they were more considered all in the same boat by whites. Of the various traces of such collaborations and influences, one can listen to the Latin-sounding trumpet solo on King Oliver’s Creole Jazz Band version of ”St-James Infirmary”, or simply in the various biguine-like songs of Sidney Bechet and the Haitian Serenaders or the Baby Dodds Trio, sung in creole dialect. (Even if these recordings date from 1938 and 1946, biguine music’s popularity goes back to the beginning of the century, and like the tango, it found a large audience in France).
As segments of the The Spirit Moves by Mura Dehn clearly evoke, both the brothel’s social function of blues and the ceremonial Afro-Caribbean dancing probably influenced the basis of blues dancing in New Orleans. The dancers here were professional stage artists, but they seem aware of the underground background of the dance. Sandra Gibson’s very sensual motions might come from lascivious movements from the red-light district while Al Minns’ performance near the end of the video looks like a slow Afro-Caribbean dance.
In the small jazz revival of the late thirties, when collectors suddenly started to realize or remember that jazz came from people of color, some aficionados decided to go back to its roots and talk to old-time musicians. This period is also the beginning of a long series of records of Latin jazz et Creole jazz by various artists. One of these musicians was no other than one of the most important jazz composer of the century.
When Jelly Roll Morton crossed the door at the Library of Congress in 1938, he was already going downhill in his career. After benefiting from the instant money and gratification of his popularity in the music world, he had to see publishers, record companies and white musicians make a fortune from his music. The 20-hour long interview he was going to give to Alan Lomax in the following months ended up becoming one of the greatest jazz interviews ever made, even considering the various inaccuracies.
Of course, we need to challenge Morton’s main claim of having invented jazz music in 1902, but the interview still provides a lot of interesting information. One of the key moments of the oral history he left us is the precise description he gives of the influence of the certain rhythm and in creation of blues and jazz, starting from an example we already know:
‘’Now take La Paloma, which I transformed in New Orleans style. You leave the left hand just the same. The difference comes in the right hand –in the syncopation, which gives it an entirely different color that really changes the color from red to blue. Now in one of my earliest tunes, New Orleans Blues, you can notice the Spanish tinge. In fact, if you can’t manage put tinges of Spanish in your tunes, you will never be able to get the right seasoning, I call it, for jazz.’’
This “Spanish tinge” is the habanera rhythm. It is believed he has learnt it from a Spanish guitar teacher he had in his teens. It seems plausible since there was also a strong Latin presence in New Orleans and the rhythm had already spread around (as we can hear later in singer Lydya Mendoza and others). Its direct influence on blacks and creoles seems to have stuck to piano since guitarists from New Orleans don’t share this musical figure, nor can it be found in Harry Oster’s various recordings made in Angola prison years later. Morton’s use of seasoning relates most likely to food, but since it shares the meaning of adding value it makes it all slightly disturbing.
What is so fascinating in this interview segment is that you can clearly hear the drift from a Spanish song to the blues, all of this simply by changing the complementary syncopation of the left and right hands, which gives the music a particularly appealing kick that makes you want to move. It is also interesting to notice in the interview that he starts from the Spanish tinge and slowly modifies it into another closely related rhythm known as the Charleston rhythm, which is the one we can hear on his recording of ”New Orleans Blues”.
As Morton points out, it is for him also an essential part of jazz composition: he uses it for instance in ”The Craze”, ”Creepy Feelings” and others. The Spanish tinge probably traveled with some early piano players as far as New York. Willie the Lion Smith and James P. Johnson remember Jack the Bear playing the salty song ”The Dream”, again with the same rhythm.
It is difficult to know why the Spanish tinge doesn’t appear on more records from the 20’s. It might be that for recording companies, its Latin flavor made it less suitable for the race record market. In any case, we know it spread at least a little at the time, as the recording of ”Tia Juana Man” by Ada Brown testifies. Not surprisingly, the various musicians on the song are also New Orleans old timers like Albert Nicholas, Barney Bigard, Johnny St-Cyr, George Mitchell and Luis Russell.
The Spanish tinge almost disappeared when New Orleans pianists moved up the Mississippi to reach Chicago. Luckily though, some pianists like Doug Suggs and Little David Alexander were using it in the Windy City. It appears on many of Jimmy Yancey’s recordings, from ”At the Window” to his interpretation of Leroy Carr’s classic ”How Long Blues”.
From there, the multiple exchanges and cutting contests between musicians might have worked to spread and modify its structure. Its influence can still be heard on some random recordings like the piano line of Jazz Gillium’s ”Gonna Take My Nap”, some Champion Jack Dupree (from Louisiana) or Sunnyland Slim’s songs and multiple other recordings. It also appears in closely-related patterns on Memphis Slim’s left hand, and he gave us to understand that such a form was also practical, since it was possible to play it while holding a cigarette!
It also reached the West Coast as it can be heard on multiple recordings by Lloyd Glenn: ”Southbound Special”, his interpretation of Yancey’s composition ”Yancey Special”, on ”Savage Boy”, and so on. Its use on ”Old Time Shuffle” might indicate that the rhythm has been used for a long time. (The shuffle was danced on plantations, but other song labeled as shuffles, like ”T-Bone Shuffle” or ”Ballroom Shuffle” don’t use the Spanish tinge.)
Indeed, once you started spotting the Spanish tinge or its offshoots, you see it everywhere, especially in the R&B era and Jump Blues/Jive where variations of it can be found at different speeds. Ruth Brown, Clarence Garlow, Professor Longhair, Percy Mayfield, Howlin’ Wolf, John Lee Hooker, Chuck Berry and others were influenced by it, at least for some of their songs.
What does this mean for dancers:
Since this asymmetrical rhythm seems to have been present for a long time in the blues and related music, it is natural to think a specific step could have followed it. In a Latin dance like salsa, the asymmetry is reflected in the steps. From the lead’s perspective it goes: Left foot front, right foot under, left foot under followed by its anti-mirror right foot back, left foot under, right foot under. The equivalent for follows is right foot back, left foot under, right foot under followed by its anti-mirror left foot front, right foot under, left foot under. It is very plausible that similar steps were used in New Orleans more than a century ago. Of course, a possibility would be the mirror version of salsa steps starting with the left foot back followed by right foot front. (for the lead)
The same holds for half of that series of steps, which is also a structure present in current dances, and that can be simplified as step, step rock step. The rock step here holds the same function of following the asymmetrical structure of the music (it can be seen as half the mirror version of salsa steps while for follows as half the normal salsa steps). The rhythm here can be understood as the superposition of two patterns, binary and ternary. We’ll see how it could be thought in terms of dancing.
In that video, from the lead’s perspective, the beat at 1 could be for the rock step behind, on the left foot and the step under when the red spot hits 2. As a result, the beats at a and & are used by the right foot to balance back the motion of the left foot.
In follow’s perspective, the beat at 1 could be for the right step in front and when the red spot hits 2 the right foot steps under. The left foot then serves to balance that motion at a and &.
The asymetry of the rhythm can appear in simpler forms in the dance. From the Memphis Slim’s song previously presented, the slow dance could very well include the asymetric basic steps: left-right-left, pause, right-left-right, pause.
Of course, we’ll never know exactly how much such asymmetrical steps were commonly used since it was easier to make money from people’s recordings than from filming their dance events. We know from some songs that many steps were used in country blues-related music, like the grizzly bear, the chicken scratch, the turkey trot and many others. A step behind in a kind of a salsa-ish or rock-step-ish fashion seems also very probable.
We have some footage of jazz dancers in big cities like New York and Chicago, which includes some rock steps, but there’s pretty much none for blues dancing, especially in areas like New Orleans and the Piedmont. It is very plausible that Piedmont dancers were either tap dancing or doing a mixture of bouncing steps, like some stationary Charleston. Maybe some bouncy rock steps were there as well. Blind Blake did mention the Charleston in ‘’Dry Bone Shuffle’’ and Blind Willie McTell’s wife remembers dancing the Charleston in a club on her husband’s music. It is difficult to know how much this was similar to the Geechie and Gullah’s dances that inspired James P. Johnson to compose his ‘’Charleston’’ that made the dance popular through the show Runnin’ Wildin 1923. To go full circle, we can notice that the Spanish tinge can be obtained from the Charleston rhythm by substituting the second dotted crochet (or quarter) by three eighth notes, the first of the three remaining silent to obtain the rhythm. It is exactly the reverse process of what Morton does in the interview for Lomax at the Library of Congress. How these rhythms influenced the dancing remains an interesting question since Piedmont blues represents a fair deal of blues recordings and records travel fast. For instance, it could have reached the Mississippi delta since a guitarist like Robert Johnson knew how to play covers of Blind Blake, Blind Boy Fuller or Blind Willie McTell. Such a dance would also be well suited for other upbeat music of the time like the Mississippi Sheiks and others.
As much as the Spanish tinge seems to have sometimes only few occurrences throughout a song, at various speeds, some rock step equivalent might very well have been part of the various steps that created the blues or all related dances of black people at the time. Even more interesting, is that we can use the Spanish tinge to link both some parts of blues and jazz music and even, maybe, dancing. Nevertheless, it remains difficult to estimate which proportion of recordings really contain direct influences of the Spanish tinge. If it was a steady left-hand pattern of some early 1900’s compositions, it evolved into various forms or was absorbed in other songs as only an occasional punctuation. In all cases, it’s hidden there in many songs.
Finally, it reminds us that there is a lot we don’t know about blues dance history. Whatever we learn about how to dance will always remain useful indications, hints and advice, not an absolute. Dances grew as combinations of various steps, often invented by long-forgotten folks in juke joints, rent parties or other dancing venues. The same holds for present-day dancing, where everyone’s personal touch can add to how we dance. In any case, we should keep in mind learning to dance well with the people in our own community. The center of the scene are the dancers with their valid preferences, points of view and skills. The only absolute criteria we should always keep in mind are to be safe and respectful, with a touch of bluesiness to wrap it all up.
Thanks to Andrea Rosenberg, Debbie Carman, Evelyne Batoula and Dominique Perras Saint-Jean for editing and suggestions. Thanks to Felix-Antoine Hamel for sharing musical passion and knowledge. Thanks to all the dancers I enjoyed sharing a moment with on the dancefloor.
Borneman, Ernest. Jazz and the Creole Tradtition. Presented to Penrose Library, University of Denver
Broven, John. Rhythm & Blues In New Orleans. Pelican Publishing Company
Brown, Scott. James P. Johnson; A Case of Mistaken Identity. Scarecrow Press
Emery, Lynne Fauley. Black Dance from 1619 to Today. Dance Horizon Book
Fernández, Raúl. Latin Jazz: The Perfect Combination/La Combinación Perfecta. Chronicle Book, 2002.
Goffin, Robert. La Nouvelle-Orléans, capitale de jazz. Éditions de la Maison Francaise
Gray, Michael.Hand Me My Travelin’ Shoes: In Search of Blind Willie McTell . Chicago Review Press
Guralnick, Peter. À la recherche de Robert Jonhson. Traduit par Nicholas Guichard. Le Castor Astral
Handy, W.C. Father of the Blues. MacMillian
Herzhaft, Gérard, Americanaè Histoire des musiques de l’Amérique du Nord. Librairie Arthème Fayard
Kubik, Gerhard. Africa and the Blues. University Press of Mississippi
Lomax, Alan. Mr. Jelly Roll: the Fortunes of Mr Jelly Roll Morton, New Orlans Creole and ”Inventor of Jazz”. University of California Press
Lornell, Kip. Introducing American Folk Music; Ethnic and Grassroot Traditions in the United States. McGraHill
Reich, Howard and William Gaines. Jelly’s Blues: The Life, Music, and Redemption of Jelly Roll Morton. Da Capo Press
Rosemain, Jacqueline. Jazz et Biguine: Les musiques noires du Nouveau Monde. L’Harmattan
Schuller, Gunther. Early Jazz: It’s Roots and Musical Development. Oxford University Press
Today is a great day for narratology. It is so for the simple reason that we agreed to make things more complicated, beautifully so. It doesn’t matter how many Oscars end up in the hands of Denis Villeneuve for his sci-fi movie Arrival, what matter is its inescapable presence.
Storytelling, in all forms, has always been a way to shape our minds. Stories need to be entertaining in order to stand out from all the available ways to occupy ourselves, but also need to be challenging so we can step out of our habits, and learn to think something new.
Many movies have been able to satisfy both sides of the balance. Some by the order the story is been told, as for movies from Pulp Fiction (Tarantino, 1994) to Memento (Nolan, 2000) and some reached similar effect by the inner story structure like Primer (Carruth, 2004), Looper (Johnson, 2012) or Triangle (Smith, 2009).
What makes Arrival particularly interesting is the prominent place given to a language itself, a language that allows for more intricate patterns and storytelling process.
Indeed, being presented on the other side of the mirror, within the diegetic world itself, this language permits time traveling by itself, understanding both future and past events at the same time and acting coherently with all of them.
Sadly, there is no proof such a language exist in our world. What we do have though is languages that help us understanding stories as groups of logical interactions of groups of events, just like in the classic publicity against drug abuse where one works more, to make more money, to make more drugs, to work more.
Of such languages are mathematics and the way we can use them to represent and understand stories, our own stories, and our own patterns. Such simple examples can easily be drawn and analyzed with graph theory and topology.
A conclusion we can drawn from Arrival, both in its content and its worldwide popularity, is that we might need to accept the fact that we need to learn something new to be able to solve, as humans, the the various pressing worldwide problems that could lead to our extermination, or at least mass decimation.
The fact we all seemed to touch so many of us could simply be the fact that it addresses the conclusion we have all already made, somewhere in our self-regulating surviving minds: we need to find solutions, solutions to problems deep enough that it could involve restructuring the way we tell ourselves, as a species, our own story, through mass media, through education, through thinking.
The fact that Arrival is there tonight might be a very indirect way of admitting it. Movies that create intricate story structures are a strong first step, since they are also a language. Arrival stands clearly as its own pertinent example. It’s entertaining enough that masses want to watch it, and complex enough so that we need to make links ourselves, conclude ourselves, think a step further.
The real arrival that is needed is not the aliens’ one, it’s the arrival of new languages, new paradigms.
The epitome of art as resulting from automated process can be find in fractals. We will discuss two examples to underline two major components of the automated process; the structure, or skeleton, implied by the automata, and the theoretically possible infiniteness of its application in time and space. Fractals are geometrical figures defined by Mandelbrot in Les objets fractals (1975) in an attempt to describe the geometry of nature. These objects are often use by iterated processes and are self-similar for certain scale factors. Similar to the comments by Mandelbrot in his article Fractals and an Art for the Sake of Science, we can distinguished between two types of fractals, the organic and inorganic ones. The organic fractals identifies by and obvious similarity with nature whereas inorganic share the structural quality of independence of scale but keep evident traces of man’s hands.
Inorganic fractals usually results from a defined automated iterated process. For instance, the Koch curve is obtained by infinitely adding triangles on the middle thirds of each segments of its constitution. Surprises arise when one realises the same figure can be obtained from different automated processes. The Thue-Morse sequence is a sequence of zeros and ones built iteratively in such a way to avoid any triplet repetitions. It is constructed by the infinite concatenation of the complement of a binery sequence. The sequence is constructed as follow: 01, 0110, 01101001, 0110100110010110 etc.
It has been shown it was intrinsically related to the Koch curve: by assigning directions values to the digits of the sequence, it is possible to obtain the iterations of the Koch curve (Ma and Holdener, 2005). In a similar fashion, by assigning another set of instruction to the digits it has been demonstrated that the same Thue-Morse sequence can serve to obtain a tamil kolam (types of ritual figure drawn with sand or rice powder) (Allouche, Allouche and Shallit, 2006). The three entities, the Thue-Morse sequence, the Koch curve and this particular kolam are simply different interpretation of a common genetic code hidden in their automated iterative processes. (Figure 8) An automated process could, therefore, generate three or more different objects that we, from a visual point of view, consider distinct.
Figure 8: A Kolam and its equivalence as Koch curve and Thue-Morse sequence
Finally, we present a set of fractals named Julia sets and the Mandelbrot set. Again, it all roots back to the idea of representing complex numbers in a plane. Complex numbers are have two components, and real part on which we add an imaginary part, or equivalently a multiple of the square root of -1, denoted i. We usually write them a+bi. To reprensent them on the plane, we give the real component value to the x-axis and the imaginary part to the y-axis. Therefore the point (3,4) represent the complex number 3+4i. This representation helped understanding the way complex numbers multiply themselves and led to studies of conformal mappings as previously seen.
With this coordinate equivalence for a complex number, we can represent complex numbers on the plane as vectors with a length and a direction. The new vectors obtained from the multiplication has an angle equals to the sum of the previous vectors and a length equals to the product of their length. As a result, a number bigger than one will spiral out to infinity if multiplied by itself an infinite number of times. At the end of the First World War, the Academy of Science of Paris promised a prize for the better paper on complex numbers’ dynamic. From his hospital room where he cured his injuries, Gaston Julia wrote many important papers on the topic. He defined his set by the set of complex numbers not diverging to infinity when iterated in rational functions. For C a nonzero complex constant, the Julia set of quadratic forms f(z) = z² + C forms a fractal. Indeed, at the time, Julia did not have the tools to visualize the complexity of these sets. When the computer entered universities in the 60’s and 70’s, researchers started to code programs that would automatically generates Julia sets. The results started to evoke, even if only slightly, how rich wew the images Julia was trying to draw decades ago.
Figure 8: Julia Set
Nowadays, colors are added to these figures to produce marvelous pictures. On top of having to compute a great number of points in the complex plane in order to obtain a single picture, by a step by step focusing figures on the border of these sets we can obtain fractal zooms. In theory, these zooms could produce infinitely many different forms and could last forever, such is the complexity of Julia sets. (Figure 9) A French mathematician, decided two classify the Julia sets into connected and disconnected ones. His classification led him to another infinitely complex set now dubbed the Mandelbrot set on which infinite zooms are also possible. The exploration of fractals led mathematicians into trying to define three dimensional versions of Julia sets and the Mandelbrot set. Difficulty arises and multiplication for complex numbers are to be represented. Since the complex numbers are defined on two dimensions, the real and imaginary one, the complex multiplication can be visualised in the plane, the representation of two complex numbers would need four dimensions. Fortunately, Paul Nylander have found a way to represent such mapping in three dimensions and three dimensional fractals based on this operation have arised, as for example the Mandelbox and the Mandelbulb. (Figure 10) As for the planar versions of these fractals, three dimensional fractals are never fully seen since they are infinitely intricate. Although, with the arrival of 3D printers, there’s been many attempt the represent some fractals such as the Sierpinski triangle. As well, it worths mentioning Tom Beddard’s work with fractal sculptures with lasers.
Figure 9: Inside of the Mandelbox, image by Krzysztof Marczak
Fractals apply naturally to arts, but they can also find specific technical application. For instance, there are many attempt to construct virtual landscapes based on fractal oriented programs. This tradition find its roots in the work of Voss who was developing programs to generate infinite maps, which has been done as well by Mandelbrot.
Again, the question of author seems problematic. The multi-level architecture behind these zooms starts with the definition of complex numbers, the idea of the complex plane, the studies of Julia, the long story behind computers and their programs, and then gigantic calculations made by the computers to generate a fractal zoom. The choice of the zoom’s point and the applications of specific colors is what is left to the last person involved in line, the artist. In that case, what is the fractal, where does it stands between discovery, invention and piece of art. Mandelbrot put it in these words: ‘’ Thus fractal art seems to fall outside the usual categories of ‘invention’, ‘discovery’ and ‘creativity’.’’ (Mandelbrot 1993, 14)
If again, no simple solution can be drawn from these various examples, they can still be linked to the idea of author. The idea of authorship is not only to refer to the existence of a creative process, but as well to contain a certain mark, a certain signature proper to the author. Of course, in the previous examples, elements of signature could be grasp at different level, in the choice of topic for a conformal mapping photograph, in the program’s style of coding lines, in the idea behind the proofs a the different theorems leading to these constructions. Since traces of authorship could be found at all these level, it shows that the notion transcend the simple binary separation between what is art and what is patentable. I urges as well that research centers such like universities to allow more permeability between areas of sciences and arts and include more classes on cross-disciplinary classes where students and researchers from both groups can meet and work together not only to solve problems, but to propose new ones as well.
Félix Lambert (First ideas presented at Harvard in 2013, first draft for this paper finished may 2015)
 For the detailed history, the reader is referred to Michèle Audin’s work: Fatou, Julia, Montel: The Great Prize of Mathematical Science of 1918.Springer, 2011.
Allouche, Gabrielle, Jean-Paul Allouche et Jeffrey Shallit. 2006. « Kolam indiens, dessins sur le sable aux îles Vanatu, courbe de Sierpinski et morphismes de monoïde ». En Ligne : Annales de L’Institut Fourier, Tome 56, n°7, p. 2115-2130. Consulté le 07/02/12. http://aif.cedram.org/item?id=AIF_2006_56_7_2115_0
Arnold, Douglas N. and Johnathan Rogness. 2008. « Möbius Transforms Revealed ». Notices of the American Mathematical Society, Vol. 55, Nu. 10, p. 1226-1231.
Audin, Michèle. 2011. Fatou, Julia, Montel : The Great Prize of Mathematical Sciences of 1918, and Beyond. New York: Springer, Lecture Notes in Mathematics 2014, History of Mathematics Subseries.
Bouton, Charles. 1902. « Nim, a Game with a Complete Mathematical Theory ». Annals of Mathematics, Second Series, Vol. 3, no. 1. P. 35-39.
Calaprice, Alice, Ed. 2000. The Expandable Quotable Einstein. Princeton: Princeton University Press.
Frampton, Hollis. 1970. Zorns Lemma. In A Hollis Frampton Odyssey, DVD 1, 59 min. Criterion Collection 2012.
Ma, Jun and Judy Holdener. 2005. « When Thue-Morse Meets Koch ». Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society, vol. 13. n°3, p. 191-206.
Mandelbrot, Benoît. Les Objets Fractals : Formes Hasard et Dimension, 4th Ed. Paris: Flammarion, 1995.
Mandelbrot, Benoït. 1993. ‘’Fractals and an Art for the Sake of Sciences’’. In Michel Emmer Ed. The Visual Mind: Art and Mathematics. Cambridge: MIT Press, p.11-14.
Munkres, James R. Topology. 2nd Ed. New Jersey: Prentice Hall, Inc., 2000.
Schattschneider, Doris. 1992 (1990). Visions de la Symétrie: Les Cahiers, les Dessins Périodiques et les Oeuvres Corrélatives de M.C. Escher. Traduit de l’américain par Marie Bouazzi. Paris : Éditions du Seuil.
Smit, B. de and H.W. Lenstra Jr. 2003. « Artful Mathematics: The Heritage of M.C. Escher». Notices of the American Mathematical Society, Volume 50, nu 4, p. 446-451.
Stillwell, John. Geometry of Surfaces. New York: Springer-Verlag, 1992.
In between these poles of mathematic as subject, as structure and as narrative construction stands the automated processes. In the last 60 years or so, computers have galvanised and specialised the precision of the relations between the abstract mathematical procedures and visual content. Indeed, it has been possible by means of automated processes, especially in the construction of geometrical operations. The rest of the article focus on structures defined on abstract art instead of figurative or narratives as in the case of Last year in Marienbad. As a result, we are interested in authorship in arts and sciences from a double perspective: as creator of an aesthetic geometrical result and as inventor of an abstract structure. A clear and simple example of such a problematic objects can be found in the Ulam spiral. Bored during a meeting Stanislaw Ulam started to organise numbers in a spiral and in this structures some patterns seem to appear for prime numbers. This simple object of number disposition leads to beautiful imagery when focusing on the prime numbers disposition and to some new mathematical results about these prime numbers.
The signature over the aesthetic constituents being often available, we need to address the question to find the source of the structure and its authorship.
In order to comprehend this relation tied between a creator and an automated process, we need to distinguish between the different tributary relations linking an artistic visual object and an abstract automated process. It is important to underline the implied relation might appears in both directions; an artistic object can be obtain by applying an automated process, and oppositely, an automated process can be discovered by trying to solve an artistic problem. Both sides of this equation share the common ground of creation and the results, no matter what is the original paradigm, lay on shared space of double probability: the result stands in the midway between pure technicality and art. The next step of application of the automated process is fundamentally unpredictable. For this reason, the automated process is in equal rights as much an invention as an artistic creation. Of course, once a seed bloomed, layers and layers of artistic objects, related automated process, solutions to various problems and, finally, new problems might add to the complexity of the object. We study some examples in the following paragraphs.
A practice of tiling the planarity of a wall or a floor is maybe as old as architecture itself. There exist infinitely many ways to tile the plane, but these can be grouped in finite sets when restrictions are added or when classifications are needed. If we restrict the tiling to congruent tiles, then a classification is made possible by considering reflections, rotations, translations and glide reflections of the original tile. The artist Maurelius C. Escher studied these different patterns of tiling and tried to find all possible patterns. Escher found an article by Polya and Haag on crystallography giving the complete classifications of such tilings and Escher based his next experimentations on these observations. Even if Escher have found by himself almost all the patterns, it still give a good example of an abstract mathematical problem including automatic process related to an art object. In this case, the automated process constitutes of applying infinitely many translations, rotations, reflexions or glide-reflexions, to cover a space harmonically.(Figure 4) (Schattschneider, p.23-30)
Figure 4: Polya’s representation of the wallpaper groups. Source: Visions of Symmetry, p. 23
The story does not end here. Of course, different types of tessellations not involving congruent tiles have been explored as a legacy to Escher’s work and covering problem, like the Penrose aperiodic tiling and fractal tilings. The problem even evolved to include other surfaces; mathematicians and artists have explored the tiling of the sphere and this led even to tessellations on other surfaces as the hyperbolic plane or the projective plane. (Figure 5) Therefore, the creation of the tiling problem is double, it includes the eventual creation of a mathematical knowledge as much as of series of artistic creations. Moreover, it creates the space of discussion in which both disciplines challenge each other.
Figure 5: Jos Leys Hyperbolic 1
A similar story is hidden behind conformal mappings. Conformal mappings are functions that project images between surfaces, possibly from itself to itself, by preserving angles of intersection between lines. Conformal mappings arises as a main interest in the study of projections and the complex plane where they naturally arise as differentiable functions. A commonly used conformal mapping from the sphere to the plane is called the stereographic projection. To obtain this projection, we can imagine we set a sphere on a plane, and from the North Pole, i.e. the more distant point from the sphere, we traces rays crossing the sphere at a point and reaching the plane at second point. The stereographic projection is obtained when mapping the whole sphere to the plane in that respect.
In the last decades, photographers like Alexandre Duret-Lutz have used projection in order obtain pleasant photographs offering different spatial perspectives. The application of the stereographic projection lead to very peculiar pictures dubbed wee planets. In these photographs, objects are grotesquely deformed while keeping an overall readability due to the conformity of the projection. Ususally, the horizon surrounding the camera morphs into the circumference a small planet on the picture, resulting in pleasant cartoonesques scenes. (Figure 6) Modern photography contains more peculiar pictures calling for stronger mathematical notions. (Lambert, 2012)
Figure 6: wee planet Alexandre Duret-Lutz
The study of functions in the complex plane led August Ferninand Möbius to the definition of Möbius transforms, a group of conformal mappings constructed from translations, rotations, dilations and inversions (which inverts the inside and outside of a circle before rotating it). These functions are conformal and they can all be link to the stereographic projection through some motions of the sphere. (Arnold and Rogness) For instance, to obtain the inversion, it is equivalent to rotate the sphere upside down before applying the stereographic projection. The use of Möbius transformations is also recognisable in the photographs of Duret-Lutz, especially when the sky stands as a disk in the middle of the picture as a result of the inversion. Interestingly, artists are now applying similar techniques to video, Ryubin Tokuzawa. (http://www.ryubin.com/panolab/panoflash/#)
Other conformal mappings have been explores by photographs like Seb Pzbr or Josh Sommers. The utilisation by Sommers and Pzbr of a special composition of conformal mappings comes, though, from outside the mathematical discipline. In 1956, Escher worked on the highly complex Printing Gallery. The conformal mapping he tried to develop was so elaborate he could never finish his work, leaving a blank space in the middle. Half a century later, Lenstra and his team finally modeled the transformation Escher had in mind and, with the help of computers, they filled the blank spot. (Smit and Lenstra) The transformation, usually named the Droste effect -after on old advertisement using a self-referential figure- is now used by photographers to propose a wide range of new imageries, from self-portrait to the representation of abstract architecture. (Figure7)
Figure 7 : Droste effect on architectural desing
The story of such photographs lies on multiple layers on each of which part of the authorship is diluted. It comes from a rich balance of complex numbers, functions, projections, Escher’s vision and programmers that integrated this process in code to obtain the results on photographs. This automated process and results from a 300 years old long dialogue where the authorship was constructed.
It is of prime importance to underline the presence in these pieces of art of the automated process: without the programs applying the conformal deformations, some photographs and videos, could never have existed. The creations, unreachable solely by humans, exist at the very limit of the creator’s capacity. It is the result of a tremendous collaboration where the sum worth more than the parts.
 For a clear introduction to the topic the reader is invited to consult John Stillwell’s work: Geometry of Surfaces, Springer, 1992.
There is a process involved behind every artistic and scientific productions. These processes can evolve, change directions and motivations, but at some point when the exact procedure is defined, automated processes can be constructed. The automated procedure is then available for others to be experimented and modified in order to find new applications and results. As this extra step is taken, an extended distance appears between the original creator of the process and the final result. Although, as pointed out by Einstein, when great specialisation is involved, the scientific and the artist merge into one identity (Calaprice 245) We show in this article that this double position between art and science is particularly present when creating automated processes. When creating abstract trends of patterns and procedures, the full extent of its applications rarely stands at reachable glance. On the other hand, the creation of subdivisions as copyright and patents leads the path for creators to think about the exact applications for their creations prior to their concretisation. This paper will explore the problematic involved in such a subdivision, especially in the paradigm of modern automated technologies. Various examples involving conceptual mathematic models, automated processes and visual art will be discussed in order to clarify the problematic.
As a first step, we compare different movies implying some mathematical concepts: Zorns Lemma (1970) by Hollis Frampton, Last Year in Marienbad (1961) by Alain Resnais and Pi (1998) by Darren Aronofski. These movies use different strategies to include mathematical concepts. The movie Pi is emblematic of the use of mathematics as a topic within its diegetic world. In this case, some concepts can be explained to the audience; the mathematical concepts are use in quotations since they don’t interfere with the structure of the movie itself. To a certain extent, these concepts could be changed for others and the structure would remain intact. As an example, the relation between the stock market and the value π could be exchange for the golden ratio to obtain a similar movie. It would remain an excellent movie with outstanding visuals aesthetic, only part of the semantic would be altered since the myth around pi differs largely from the myth around the golden ratio. These perceivable modifications would be linked to these specific numbers’ reputation outside the movie. For instance, the golden ration often being related to beauty, its use would charge scenes with a different emotional impact than the profoundly anxious and neurotic feeling that underline the whole movie. The value π does not work as a framing structure, it adds a mythological symbolism to its content and mark the film with a peculiar color coherent with the movie’s topic.
The film Zorns Lemma proposes a different appropriation of mathematical concept as a main constituent of art’s paradigm. The Zorn lemma is an important result in the foundations of modern logic and axiomatic set theory. It states that for a strictly partially ordered set, if every ordered subset has an upper bound in the original set, then the latest has a maximal element. The lemma has been proved independently by Kuratowski and by Bochner in 1922, but its popular appellation sticks to Zorn who proved it in 1935. (Munkres, p. 70)
The movie does not make apparent use of the lemma itself, although Frampton explicitly works its visual content from a set theoretical approach: groups of letters are combined as different sets to form words. As an example, in the second section groups of words appear ‘’organized alphabetically into sets of twenty-four and conforming to the Roman alphabet by combining i and j with u and v.’’ (Jenkins, p. 21) In this case, the abstract frame is calked from of a given field; set theory. Secondly, the object has a similar background question; how to organise elements of a set? In this case, the question is organise letters from the alphabet. The Zorn lemma appears as more than a mere abstract reference and its substitution for another theorem would note guarantee its correspondence with the movie structure. A title linked to the Pythagoras theorem, Fermat’s theorem or Gödel’s theorem would not be suitable references for Frampton’s work since we could not see a correspondence between the movie’s structure and the results of these theorems.
Figure 2: Last Year In Marienbad (Alain Resnais, 1961)
A slightly different approach is explored in Alain Resnais’s Last Year in Marienbad. In this film, the main character, interpreted by Giorgio Albertazzi, often plays the game of Nim -sometimes called the game of Marienbad after the movie– and asserts that by starting first this would ensure him victory. On the mathematical side, the game was proved to be solvable, meaning that there is an algorithm leading inevitably to victory. (Bouton, 1902) The victorious pattern is presented multiple times during the movie and its logic is scaled to the overall frame of interplay with memory between to two main characters. The solvability of the game is implied in the movie as the dry output of destiny: the inevitable reconstitution of the forbidden, and maybe false, memory. The hunt for this blurred memory is ended before it started as the game of Nim is won before every game. As a result, the equivalence relation between the mathematics of the game and the movie’s structure is constructed by narrative means.
Figure 3: Time Structure of Last Year in Marienbad by Resnais
 It was also called Fan-Tan at the beginning of the 20th century (Bouton, 1902)
A first step in dealing with paradoxes is then to accept their double existence as being true and false. Outside academic studies, it is a natural process implied in the appreciation of any narrative art. As it has been described about cinema, there is a point where we accept the false to be true, as if trying to find a proof reducto ad absurdum. This process is called the suspension of disbelief (Walton, 7). Proposed first by the poet Samuel Taylor Coleridge, it has extended in the study of literature, cinema and videogames to name a few. This refers to any action where the false is taken as possibly true in order to appreciate a narration and facilitate the immersion process. Youssef Ishaghpour describes the duality implied as the reality of the image and the image of reality (Ishaghpour, 8-11). The suspension of disbelief is therefore a way to conciliate this duality in order to appreciate the fiction.
The difference when working in a fictional environment rather than in a scientific one is that every time a contradiction or unearthly events appears, it is usually taken as an extension of the perceived diegetic world: when the staircase optical illusion appears in Inception, it is accepted as part of the fictional world. Instead of defying physical rules, it is simply accepted as a new information contained within the fiction. Again, as in Kierkegaard or Bohr’s vision, it is an extension of the paradigm. The same happens for multiple worlds’ diegetic construction such as previously mentioned in movies like ExistenZ or Avalon. The multiple ontological world, very similar in their nature to the Russell’s paradox are accepted as such. The suspension of disbelief catalyses the conceptual acceptation of such constructions and even changes them into interesting and pleasing artistic visions.
Figure 7: The cat’s Paradox. Source: Wikipedia
The use of paradoxical constructions taken from science does not end here. Another case comes from quantic physic. Erwin Schrödinger described the nature of some quantic events by the metaphor of a cat in a box. Let say there is a cat and some poison inside a box. There is as well a 50% chances for the poison to be relieved and therefore for the cat to die. The way quantic physics works is that as long as the information about the cat has not been extract from the situation, the cat is in both states: alive and dead. Both states excludes each other and therefore it leads to a paradox that of ‘’ the living and the dead cat mixed or smeared out in equal parts.’’ (Schrödinger) What modern physics proposes as a solution the acceptation of both state for a certain period of time. This process is known as quantic bifurcation. Even if this is very difficult to accept as being true for neophytes, when transferred to fiction it leads to acceptable and interesting narrative constructions. An example of a movie using this type of multi-linear time frame is Source Code by Duncan Jones. In this movie, a soldier is sent multiple times in the past to prevent a terrorist attack. After failing multiple times, he achieves his goal and life continue normally in this new independent timeline. The use of quantic bifurcation appears in multiple science-fiction movies and communities of fans are sketching schemas to understand the structure behind the film. Movies like Primer by Shane Carruth and Looper by Rian Johnson have generate numbers of complicated charts using quantic bifurcation in order to explain these narrations. (Figure 8)
Figure 8: Looper movie chart by Rick Slusher. Source: Film.com
Another paradoxical time construction that has caused many problems is the time loop. Circular construction of time was accepted by many cultures around the world: Egyptians had circular time named Neheh (Assman, 137), the tzolkin, the Mayan calendar based on cycles of 260 days (Falcón, 19-21) and Hinduism has constructions of multiple intricate circles (Eliade, 134-136). This vision does not conciliate with the European linear construction of time, but it still easily apply to fiction.
Time being both in the future and the past appears in various fictional cases. First of all, whenever there is a time loop a cyclic time has to be accepted. Movies like Terminator (Cameron, 1984), Before the Rain (Manchevsky, 1994) or Chin Chin el Teporocho (Retes, 1976) all present this cyclic time construction. Indeed, time loops can be multiple and quantic bifurcation might again apply.
In the quantic bifurcation case as in the circular time construction, the paradoxical construction induces multivalued time states, discrete moment can be different but at a same time distance from a specific moment, the bifurcation point or those previous, or they can stand both in the future and in the past of a referential moment. It can also be seen as a specific case of multiple ontological states, as previously described, but with the specificity that the ontological state is defined by a time value.
It can be presented in a more mystic way as in the movie Voyage in Time by Tarkovsky and Tonino Guerra. In this case, the movie shows the directors talking about the film they will make about a trip they once had. The anecdotes supposedly in the past appears as well in front of the camera and therefore the time of the movie is triple, it stands for the past, the present and the future as in Three Sundays in a week, but without the logical explanation.
Perhaps the most well-known results about paradoxes is Gödel’s incompleteness theorem. After Cantor and Russell discoveries, logicians have tried to build a perfect and complete system for logic. The project happened to be more problematic than expected and new set of axioms surfaced. The outstanding result obtained by the mathematician Kurt Gödel changed radically the conception of logic and left the community in crisis. The incompleteness theorem states that no matter how many axioms we add to a logic system, there will always appear some statements that will be undecidable, meaning it will be impossible to prove them right or wrong (Nagel, 19-20). This is a perfect example of Paz’s perspective of grasping dual objects as such instead of considering them as problematic undefined concepts to reach a better understanding of it. In this case, the conclusion obtained by paradox is that paradoxes are inherent part of complex logical systems.
Paz’s consideration encompasses a big range of logical instance and, as seen previously, they apply to a wide variety of paradoxical objects: from optical illusions to narrative charts passing through quantum physics. It still does not hold for a type of undecidable statements. Some facts are not necessarily true or false; they stand somewhere in between as a result of incomplete definitions. They work as ambigrams but instead of offering mainly a finite amount of elements, they offer a continuous range of possible information. Such problems are common in everyday life since more situations are not clearly defined. For instance, we can pretend the sky is blue but it can’t be proven without adding precisions to the statement offered; at night the statement does not hold for instance. A relatively new branch of mathematics dedicates itself to such logical system. The idea behind this fuzzy logic, as it is coined, is to attribute truth values that varies continuously between the usual zero and one (Kandell). Therefore allowing any probability of truth ranging from zero to 100%. Such logical system coincides with perspective of quantum physics allowing diverse states with various probabilities. It is the case for instance for electrons in the atomic model were they navigate through a probabilistic area instead of following a precise trajectory.
Finally, paradoxes can appear within humoristic or philosophical functions. The twist are often used in usually called intellectual humour such as Woody Allen’s work. In Allen’s quote from Annie Hall ‘’ The food here is terrible and the portions are too small’’, the double statement stands in the contradiction that, in fact, if the food is terrible there is no reason to ask for more, but complaining about small portions implies asking for more food. This kind of construction can be found as well in Annie Hall: ‘’ Life is full of mystery, loneliness, and suffering –and it’s all over much too soon’’. The role of the paradox is then, in this case, to release a tension constructed around the paradoxical statement. In this situation, the contradiction, or double truth value, stands as a sign that the joke has reached its climax. The contradictory aspect of the logic involves is to be read as a sign to character does not make sense anymore, therefore the humoristic relief. The humour can follow as a comment on a paradox: way before Russell, Lewis Carroll underlines that no one can contain himself because of excitement because nothing can contain itself (Benayoun, 84). These considerations follow the seriousness of Ambrose Bierce’s definition of logic as the art of thinking within human capacities (Benayoun, 113), the presence of this limit is in itself both humoristic and a relief.
Figure 9: Paradox humour
In a broader perspective, the same applies to koans, small stories or statement present in the zen tradition. The sentences serve to increase doubt and questioning. The simple logic behind the koan ‘’What is the sound of one hand clapping’’ is similar; it states the possibility and impossibility of the referred sound. Possible since it is stated there is a clap sound and the impossibility by the uniqueness of the hand producing it. In this case, the paradoxical information serves again to release tension. The same holds for the koan: If you have a stick, I’ll give one to you, if you don’t I’ll steal it from you. The tension is released with acceptation to work outside a strict logical frame, to accept our humanity as proposed by Bierce.
This work outside logic may serve as well, paradoxically, for theological arguments. To understand we have to go back to the unliftable rock paradox. God, being almighty, should be able to create an unliftable rock, but then if he can lift the rock he is not almighty. An easy solution to this problem is to state that God’s work beyond human based logic.
As we have shown, the contradictory dialectic raising from paradoxes only cause problem within its own paradigm of binary logical values as being true or false. When grasped as specific concepts gathering both values, or, even infinitely many values ranging from absolutely true to absolutely false, many applications can be found. Accepting such condition standing in between these poles is what allows us to enjoy narratives in different ways; first to enhance the emotive effect of a diegetic world by accepting the ontological quality of fiction as being an image of reality that is itself included in and presented as a simulacra of reality, secondly as to define science fiction or fantastic narratives as legitimate by extending the accepted diegetic world. Logical statements sharing both truth and false value are integral parts of human scientific and cultural knowledge. It is hoped that more research to consider paradoxes in their social appearances will be provided as to understand better their functions as a fundamental part of human thinking rather than solely as odd mythological thoughts gravitating in the abstract spheres of philosophy and logic.
First version September 2015
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We can classify visual paradoxes into two categories: the simply self-contradictory ones and the ambigrams. In looking at most of Maurelius C. Escher’s optical illusions, the spatial design seems to be both true and false. As pointed out by Biederman, since the shapes preserve adequate angle constructions, the objects seems to be credible. That is a direct result of the importance of line intersection in object recognition. Nevertheless, the overall conceptualisation of the object seems not acceptable as being true (Biedermann, 135-140). In this case, these visual paradoxes are simply self-contradictory because the macroscopic veracity statement contradicts the microscopic ones (as in the two sentence: The next sentence is true. The last sentence is false (Hofstadter, 19)) Many artists have found multiple interesting ways to produce these types of paradoxes. Escher indeed is well known for such constructions, but we can name as well graffiti artist Damien Gilley, Dutch artist Ramon Bruin and Istvan Orosz (Figure 3) Having met quite a wide popularity, it is normal that these simply self-contradictory visual objects have appeared in various situations. For instance, Penrose’s triangles has appear on post stamps, tattoos and many everyday objects.
Figure 4: Inception.s stairs scene by Christopher Nolan
These self-contradictory visual objects can contribute solving other artistic problems. In Christopher Nolan’s movie Inception the infinite stairs illusion is used and as real instance within the diegetic world to trap the enemy. In this case, it is a use of a visual paradox to solve of narrative problem. (Figure 4) On his hand, Cameron Browne have found interesting ways to merge the optical illusion construction with another old problem; the paving of the plane (Browne, 2007). Browne has constructed infinite patterns of self-contradictory visual objects that can be used to fulfill the entire plane. He worked as well with contradictory fractal structures (Figure 5)
Ambigrams are figures that show two incompatible information at the same time, inasmuch a paradox, they work on a scale more nuanced than the dichotomic paradoxes. The figure of a young-old lady is a popular example of ambigram due to H.H. Hill (Delahaye, 91). In this image we can actually perceive two different portraits. One pictures a young lady and one offers the profile view of an old lady. It is a paradox since it contradicts itself, not in the previous case straightforward manner, but by ricochet. If it is a young lady, then it is impossible to be the old lady at the same time and vice versa. In the other hand, we can interpret the image as containing two informations, two different images. Ambigrams, working on a larger scale; they can contain more information. For instance, in figure 5 one can find six apparitions of the word palindrome, all put upside-down and to be read in both directions.
The advantage of using a word like ambigram is that it underlines an interesting property of paradoxes, that fact of being containing multiple statement that would usually not appear in general in a coherent manner. To go further in this sense, we have to go back to an analysis made by Mexican sociologist and writer Octavio Paz. In an analysis of complementary and dual concepts existing in various societies, Paz stresses the importance of considering such pairs as a whole by focusing on the relation between them. For instance, body and no-body are not to be considered as specific meanings except to express contraries (Paz, 55). This perception of duals as a whole can lead to interesting results when applied to paradoxes.
Figure 6: Ambigram
 We mean by this that the information is not straight opposite, like true and false, up and down, etc.
Russell’s paradox is an example that shows how thinking about paradoxes can be fruitful. Bertrand Russell studied set theory at a time when it was still in a naïve simple form at a time the vary basics of logic were studied in philosophy and mathematics. He realised the idea of set was really powerful but was still problematic since it involved the following paradox equivalent to an extension of Grelling’s paradox: separate all sets into two categories, the sets including themselves and the sets not including themselves (Reinhart, 23). As in in Grelling’s case, the paradox appears when we try to classify the set of all sets not including themselves. The paradox works in the exact same fashion as Grelling’s one. Finally, Russell added some axioms to the naïve set theory and included levels of inclusions for a theory called type theory (Vidal-Rosset, 17). A set defined at a certain level of inclusion cannot be taken as a set of a lower level. Therefore, the question of including the set of all sets not including themselves within itself or its dual is not a legitimate question since it becomes a prohibited operation.
Although this paradox has been solved in a logical point of view, it still holds in many fictional construction. In constructions that Brian McHale, based on Gerard Genette’s theorical work on narratology, defines as being on different ontological levels. McHale provides with many examples of novels and short stories based on this idea. Many science-fiction movies are also based on this principle of multi-stage inclusions: it is the case for EXistenZ by David Cronenberg or Avalon by Mamoru Oshii. In both cases, as underlined by McHale for one of Robbe-Grillet’s short stories Project for a Revolution in New-York, the reader cannot identify the ontological level on which the action is happening (MacHale, 117). This narratological strategy helps staging a Daedalus that extends to many ontological levels and reinforces the impression of being lost. This state of delocalisation of the protagonist has a double impact: the loss of referent implies by the fact that they can’t identify on which ontological level they are, and by secondly by erasing the value of the initial referent as the real initial and authentic state. This construction of infinite ladder of diegesis is constructed from confounding a set with a higher (or lower) level set.
The section of mathematics dealing with set, their properties and their axiomatic use as a starting point for mathematics is called set theory. Set theory also had to deal with other conceptual problems at the time and this had to do with bijective functions: correspondence between the elements of two sets where an object is related to one and only one object on the second set (Grimaldi, 279). As an example, we can set such a function between the set of positive integers and the set of negative integers by linking every number to its negative equivalent. No matter how many numbers there are, an infinity as a matter of fact, we will always be able to construct the function. The problem arises when we put in relation sets that seem to clearly have a different number of elements. This time, we can create a bijective function between every positive integers and every positive even integers. In this way, one is linked to two, two is linked to four, and three is linked to six and so forth. Since we have an infinite number of integers we will always have enough to construct the relation. This is counterintuitive since all the elements of the set of even numbers are present in the set of integers but the converse is false. We can set other kinds of strange relations between finite length segments and infinite length segments. The two-dimensional stereographic projection provides such an example. We proceed as follow: we set a circle on an infinite line and from the top of the circle we trace rays that cross the circle at a point and then continues until it hits the line. By proceeding as such and scanning the rays on 180 degrees, every point on the circle will be linked to a point on the line and this time the reverse holds. Even more surprising, it has been shown that the line segment can be put in relation with the square which at the time seemed very curious since both geometrical objects did not even share the same number of dimensions: the line is one dimensional whereas the square is two dimensional (Sagan, 115).
These paradoxical constructions led Georg Cantor to the creation of the transfinite numbers theory. Cantor defined different types of infinity: the countable and uncountable (Cantor, 1976) Countable simply means they can be put in a bijective relation with the natural numbers. As mentioned previously, the set of even numbers is countable for that very reason. The two sets, natural numbers and even numbers are then of the same cardinality because they are both infinite countable. The same logic holds for the circle and the infinite line: both of them are infinite uncountable and therefore are of the same size. The expansion of the theoretical frame for infinite numbers explained as well the natural relation between the line and the square. In 1635, Bonaventura Cavalieri already proposed the idea that plane figure were made of infinitely many line segments (Alexander, 70). With the work of Peano (1890), the idea of filling the square with a single curve spread widely and many mathematicians proposed such curves. These curves where not bijections, they were in fact surjections; points of the square were actually covered many times by the same curve.
These curious objects also brought light on other concepts that were taken for granted like the idea of dimension. As a result, many definitions for dimensions have been proposed and objects called fractals have been found having non integer dimensions. For instance, the Koch curve, a well-known fractal, has dimension 1.2619 (Mandelbrot, 36). The Koch curve is famous as well for being paradoxical to the notion of continuity. When Cauchy developed the concept, he believed that continuity implied derivability, i.e. the existence of a tangent line (Wallace, p. 187). Bolzano and Weirstrass constructed such curves, but von Koch, unsatisfied by the too analytical model of these curves decided to construct geometrically his now famous curves with the property of being continuous but nowhere differentiable. (von Koch 1904-1905) Various other examples from that time shared similar paradoxical value over the canonical comprehension of continuity and dimensions. They participated in the birth of the notion of fractal geometry by French mathematician Benoît Mandelbrot in 1975. Again, redefining paradigm from arising paradoxes led to improvement of various theory in mathematics.
Another common way to use paradoxes appears in mathematical proofs. In a proof technique called reducto ad absurdum a statement that seems to be false is taken to be truth. The proof holds if a contradiction with the hypothesis is to be found. A common example of such proofs is Aristotle proof that √2 is not a rational numbers. It concedes the number a rational form and by dividing by all possible cases of appearance of even or odd occurrences for the numerator and denominator, contradictions appears in all cases. The hypothesis is therefore impossible (Boll, 31-32). Euclid showed in a similar way that there is an infinite amount of prime numbers. He started by setting the highest prime number on the theoretically finite list and then show he could construct in bigger number not divisible by any of the finite prime number list (Grimaldi, 222). The list here could be very long but the result would be the same: paradoxes can be use actively in search of knowledge. It maps this knowledge by defining areas of impossibility, therefore implying areas of certainty.
This strategy holds for other scientific area. In neuroscience, paradoxes often helps understanding the way the brain gather and compute information. A typical case is the Ames room, named after the American ophthalmologist Adelbert Ames Jr. In this particular construction, a room is deformed in such a way that, viewed from a specific perspective, it seems to be a normal room. In that sense, it is a three-dimensional anamorphosis (Baltrušaitis, 7). When looking at people in the room from that specific point of view, it seems that they are changing sizes like balloons. The Ames rooms creates a falsifical paradox: we clearly know that these people are not changing size, but our brain is unable to actively interpret the visual information as such. This experiment shows how our brain treats information and take some shortcuts for granted instead of computing them all the time. In this case, the granted fact is that rooms are rectangular prisms regulated by right angles. (Figure 2)
The importance of the intersection of lines in our perception has been proved again to be of prime importance by Biederman in 1987. In his study, he shows how the brain is faster in recognizing drawings from which no intersection segments have been erased compared to drawings where totally random segments have been withdrawn. By this mean, Biederman also explains how work optical illusions, objects that we can define as visual paradoxes.
Figure 2: Istvan Orosz Source: pour la science 2005, nu 330.
 We do not imply here that these narratives are a direct result of Russell’s work, we simply state that overall shape of its structure is similar to these narratives. More detailed historical researches could lighten this question of direct implication.