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)

Figure 1: Images from Zorns Lemma. Source: http://www.cia.edu/cinematheque/film-schedule/2013/02/zorns-lemma

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[1]– 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

[1] It was also called Fan-Tan at the beginning of the 20^{th} century (Bouton, 1902)

Where can I get a copy of the “Last Year at Marienbad” time map similar to the one you’ve used here?

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If I remember well, got it from Cahier du Cinema. I’m sure you could find it in books about the director as well.

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