Easily mistaken for the infinity sign, a circle or any number of more complex pretzels and knots, the Lissajous Figure is a picture of compound harmonic motion named for French physicist and mathematician Jules Antoine Lissajous (1822-1880). The shape is drawn by plotting a two-variable parametric equation as it iterates itself over time – the resulting figure is the picture of two systems falling into and out of phase.

In 1855 Lissajous constructed his "beautiful machine," devised to draw a picture of two systems superimposed and constructed in his Paris workshop of a pair of tuning forks placed facing at right angles, each with a mirror attached. The light source is focused through a lens, bouncing off the first onto the second and projecting to a large screen a few feet away. As the tuning forks are struck and tones are produced, simple vibrations begin to move the mirrors in a regular oscillating pattern. The projected image begins to form the strange and beautiful curves of a Lissajous Figure.

For his machine Lissajous was awarded the Lacaze Prize in 1873 and was exhibited at the Paris Universal Exhibition in1867. He did not otherwise distinguish himself as a scientist or mathematician. In fact, almost fifty years earlier American Nathaniel Bowditch had already produced similar figures with his harmonograph.

The simple harmonic motion which Lissajous was measuring is easily described by the motion of a clock's swinging pendulum. As the pendulum swings its speed isn't constant, but rather it accelerates and decelerates following a precisely predictable curve. If plotted over time, as the clock ticks the motion of its pendulum draws a sine wave – the so-called "pure wave" or zero-picture of a simple moving system. Ocean waves, sound waves, light waves, even average daily temperatures all produce this same oscillating sine wave pattern.

Compound harmonic motion, then, is simply the superimposition of two sine waves as they register, interfere and produce a series of overlapping waves. When juxtaposed at right angles, two sine waves recording simple harmonic motion produce the surprisingly complex figures that Lissajous identified.

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Sound Pattern 03

Sound and harmony made visible. Lissajous patterns made with 2 audio oscillators, a loud amplifier, plastic wrap, a bowl, and a laser pointer.

Lissajous Figures can be easily found today in computer graphics, in science museums, in laser light shows and, perhaps most precisely, burned into the green phosphor screen of a cathode-ray oscilloscope. A standard piece of electronic test equipment, the oscilloscope allows signal voltages to be viewed as a two-dimensional graph of potential differences, plotted as a function of time. When testing an electronic system, the phase differences between two signals form opposing sine waves on the screen of the oscilloscope connected together, constantly drawing and redrawing themselves in a precise and regular pattern.

These two varying signals produce a perpetual infinity (figuratively and literally as it will actually construct itself in the shape of the infinity sign given the right initial values). The Lissajous Figure becomes a picture of timing and sequence, registration and resonance, sound and music.

Specific shapes are produced corresponding to the resonating harmonic intervals familiar from western music (major fifth, minor third, major sixth, etc.) Any figure may be transformed into any figure and an infinite number of in-betweens as the oscillating sine waves pass in and out of harmonic resonance.

Jules Antoine Lissajous created a way to see sound (using mirrors, light and vibrating tuning forks.) But the most radical possibility of his mathematics might be in the commitment it asks of its audience. The image that Lissajous produces forms slowly right in front of your eyes — imperceptibly changing, forming, adjusting and re-aligning over time.

Is there any feeling, besides happiness, that surpasses the experience of adventure? Yes, the pleasant surprise of the new. This can happen even when well into old age and pockmarked by years bygone, if you remain open to the experience of receiving these novel encounters.

André Thijssen, Aksacoka, Turkey

As a maker of images, with the camera as my tool, I’ve done the necessary travelling in search of adventure. Awakening in a strange bed, in a new location, and entering a new world after breakfast, full of anticipation, is always pleasantly charged.

André Thijssen, Amsterdam, Netherlands 2011

One can avoid an expensive plane ticket by veering from your usual routine. Preferentially by walking, so that you’ll be able to catch details that may seem unimportant

André Thijssen, Amsterdam, Netherlands 2011

I find it an extraordinary sensation to glimpse an intriguing mystery in the corner of your eye that makes you slow your pace.

When the two dimensional reproduction of such a mystery still conjures that same enjoyable feeling of the inscrutable in both yourself and in others, one could say that you have a successful work before you.

André Thijssen, Amsterdam, Netherlands 2012

The images in my archive are not always of equal obscurity; some are of a more anecdotal nature. These photos are more about situations, sometimes puzzling as to why someone would leave a situation a certain way.

André Thijssen, Aksacoka, Turkey, 2011

For editorial commissions I often delve into this part of my archive first. An image placed next to text demands a different approach: an autonomous image applied associatively results in a more exciting interaction that a servile illustration spelling all out. Editorial images don’t always need to clarify but can, as I prefer, to evoke discussion.

Approaching the subject from an unexpected angle can result in new, surprising meaning.

André Thijssen, Rincon de la Victoria, Spain 2013

http://theotherpicture.com

http://nl.blurb.com/books/4531276-fringe-phenomena-3

André Thijssen, Sania, Hainan, China 2007

André Thijssen, Haarlem, Netherlands 2012

André Thijssen, Haarlem, Netherlands 2012

During the fifties and sixties of the last century, the first pioneers in digital art used computers to realize their visual experiments to create algorithmic art. They would write computer programs, otherwise known as algorithms, to generate images, usually by using advanced programming language such as COBOL or Fortran, but also by using machine language. They would often work in the dead of night, whenever a university or research institute would grant them a few hours to make calculations on their expensive IBM-mainframes. These computers were built for computing punch cards, which meant that using them to make visual art became an abstract and mathematical procedure that called for the formulation of rules to determine the construction of an image. The computer carries out the algorithm after which the output is made visible on a plotter (a drawing machine) connected to the computer.

Cubic Limit, plotter drawing, ink on paper, 1977

Manfred Mohr

Artists in this field of visual computer art, such as Ben Lapofsky, Lilian Schwartz, Frieder Nake, Manfred Mohr, Edward Zajec and Vera Molna, were strongly influenced by cybernetics and closely linked to the informational aesthetics developing during that time. It became evident that composing algorithms that performed repeatedly to produce the same image was not particularly artistically interesting, however valuable this development would appear for the later advancement of computer graphics. Much more interesting are the algorithms that, when repeated, produce different results. Although the first generation of computer artists created an output of unique plotter drawings, their main artistic interest lies in fundamental research into composition. They also touch upon complex questions concerning the essence of the artistic practice, by often bluntly addressing the question of authorship with computer generated plotter drawings. The question, what is art, is approached conceptually through an algorithm that automatically spits out one unique drawing after another, as though produced on a factory assembly line. Generally, the question was answered by assigning authorship and artistry to the actual formulation of the algorithms. The construction of algorithms for artistic purposes was seen as a scientific form of visual research that opened up a new stage for the development of art.

P148, "inschrift", plotter drawing ink on paper, 50cm x 50cm, 1973

Manfred Mohr

The works made by these pioneering computer artists is closely linked to the avant-garde movements of the sixties (like GRAV, de Groupe de Recherche d'Art Visuel), and conceptual art. After all, Joseph Kosuth and Sol Lewitt likewise made works that consisted of formulations. Indeed, the end of the sixties saw a short-lived convergence between computer art and conceptual art. The computer artist's attitude towards technology was incomprehensible to those involved in conceptual art and classical art criticism, and was to some even suspect. Nevertheless, many computer artists (like Frieder Nake) were even more radical in their rejection of the bourgeois art system than the conceptual artists, who were operating within the galleries and museums.

P-133, "cluster phobia", plotter drawing ink on paper, 50cm x 50cm, 1972

Manfred Mohr

The innovations made by the first generation of computer artists are simple when compared to what’s possible now, fifty years later. But their experiments laid the foundation that makes the prefab box of tricks possible. What's more important - for art, that is - is that the computer artists also pioneered in the exploration of conceptual questions that still remain fundamental for computer art.

144 Trapèzes (16 variations), plotter drawing, ink on paper, 20x25 cm, 1974

Vera Molnar

P91, 1971, plotter drawing, ink on paper, 50 x 50 cm

Manfred Mohr

P-122, "scratch code", plotter drawing ink on paper, 50cm x 50cm, 1972

Manfred Mohr

P-71, "serielle zeichenreihung", plotter drawing ink on paper, 40cm x 50cm, 1970

Manfred Mohr

T.V.C. 20 68179 71, 1971.

Edward Zajec

Quadrate, 1969/1970.

Herbert W. Franke

Plotter drawing, ink on paper, 1965

Frieder Nake, Klee

51/80 Scratch Code, 1970-1975.

Manfred Mohr

P-050/R, "a formal language", Ink/paper/wood, 1970, 100cm x 100cm

Manfred Mohr

In 2002, I received a New Year’s greeting from the Amsterdam Archive in which they announced a grand exhibition about city maps from across the centuries. I imagined making the most current map of Amsterdam. A living map. Formed by use of the city, using the most modern technology. While traversing the streets, the users of the city would create a trail, as though the streets were formed by the pedestrians the way that roaming cows form paths in the mountains.

Visible trails. This could be made possible by giving a number of pedestrians a GPS system and sending their information to a central computer. This computer would, in real time, sketch their routes on a blank screen. The more movement and the more users, the more visible Amsterdam would become. Different types of users would be appointed different colours, so that one could see the difference in how the automobile driver, cyclist and pedestrian uses the city.

I called the Municipal Archive and spoke with Ludger Smit. He made the great suggestion of attaching equipment to ducks so that canals too would become visualised. I understood that he understood where I was going.

Every Amsterdammer has, as I imagine, an invisible map of the city in their head. The way they move throughout the city and the choices that he makes while doing so are determined by this mental map.

When different users leave their traces in different colours, the viewer will be able to see how individual the map of Amsterdam can be. The favourite routes of a bike courier differ immensely from that of a taxi driver. The mode of transportation, the customer’s request and the person’s mental map will shape the trail that he leaves behind.

Amsterdam REALTIME

The visual result of 75 GPS-tracked volunteers their different routes in Amsterdam

I imagine that trails older than a few days will erase themselves. In this way, a constantly changing, very current, but also extremely subjective map of Amsterdam will be created.

The data will be stored and will result in a film at the end of the project. In the best case scenario, the film will also provide a time frame that will indirectly make visible large events and crowds, like a marathon or a royal wedding.

A completely different option is to use this installation to undermine the power of the cartographer. It would be possible to invite a number of people to use this drawing machine to create their own map. One could use the GPS system to trace the name of their loved one by cycling through certain streets. Another might attempt to draw the official border of the Vondelpark (regardless of having to brave fences and bushes in the process), Another might decide to walk across the same small path hundreds of time, so that it appears on the map as the most important route in the whole of Amsterdam. However, as more people make use of the GPS tracking system, it will start becoming more recognizable as an objective map.

To begin my project, I headed to the Stedelijk Museum’s library to research what other artists have done using maps and charts. What I found most stimulating was a work by Kim Dingle. She had asked teenagers to draw a map of America from their imagination. This resulted in bizarre splotches that she arranged in a rhythmic pattern on a white plane. It looked good as a reproduction. The teenagers were anonymous, yet I could see straight into their minds. It makes one think of the possibility of creating a psychological test based on the analysis of their internal map. You could ask someone: make a realistic map of the terrain that you see as your living environment. A domestic type will draw a map of their house; an adventurous type would draw the whole world. The discrepancies in proportion say a lot about someone’s personality, Reading a palm and reading a map in one.

I also found Morit Kung’s book Orbis Terrarum to be fantastic, it contains an extremely rich content. The combination of historical maps and contemporary art is, of course, very relevant to my own projects, and I have to admit that the contemporary works motivate me to study the old maps in more detail. Enjoying these objects is an acquired taste. There are, of course, errors in the maps, but they only make their further accuracy all the more impressive. God knows how they were capable of creating a reasonably accurate depiction of the world back then. Without satellite photos... was this all just done with the aid of a compass and the sun?

Orbis Terrarum - Roman Empire

I also attended an event organised by the hiking club Nemo; a lecture by John Eberhardt, head of the publishing company Buijten en Schipperheijn. Eberhardt creates inventories of routes, and so cartography has become his occupation. Cartography is a Dutch invention. A man named Jacob van Deventer was ordered by the king to draw 250 city maps in 30 years by securing a chain between his legs and counting his steps. The maps are still remarkably accurate.

City map of Amersfoort

Made by Jacob van Deventer commissioned by koning Filips II

Eberhardt admitted to mapping stretches of neglected paths as functioning cycle paths, in the hopes that they would end up actually becoming cycle paths. In some cases his plan worked: sometimes so many complaints were issued to the municipality that something really must be done about that bike path. As I heard the cartographer speak about his map, it became apparent that The Map, with its intrinsic air of objectivity and authority, is in fact determined by the subjective choices of the map’s maker and the constraints of technology, time, and money. What I had also never realised is that freedom of press is equally applicable to the publishers of maps. Eberhardt even spoke of the ‘dissident cartographer,’ which I find in itself a beautiful title. The world that is mapped is constantly changing. Each ‘layer’ of the map changes in its own tempo. The difference in the tempo of movement of continents and of migratory birds is immense, and the changes in buildings and plant growth are wedged in between there. His ideal map is one that is in constant movement, which is always changing. My idea of the most current map is very similar. The map itself draws this constant changing of the map.

Reading a map is a form of virtual reality. A model of reality can also change reality (think of the cycle path). By viewing a map through a new perspective, one can discover things that even the cartographer missed. A smart archaeologist, for example, found that all the dolmens of Drenthe are aligned in an exact straight line. The maker of the map had drawn it, but had overlooked it. A map is, then, a reality within itself, where new discoveries are always possible.

A map is also a Utopia. Each map and each user holds his own Utopia. A map for fishermen, hikers, drivers, or geologists each manifests another imagined world. Each map will allow you to enter a landscape with a different experience.