In 1967 Xenakis becomes teacher of music at the Indiana University, in Bloomington, U.S. He' not completely keen on teaching, but he accepts since he wants to realize a centre of research devoted to the relationships between music and mathematics. Unluckily almost all of the funds are redirected for the Vietnam war. In 1966 the composer resigns, and he comes back to Paris where he found the CEMAMu (Centre d'Etudes Mathématiques et Musicales, a non-profit organization dedicated to the study of the application of information technology to music). During these years he will realize the Polytopes, architectural spaces that today would be defined 'multimedia centre' and 'site-specific', dedicated to performances involving compositions made of light and sound.
The Polytopes (from the Greek 'poli'=multi and 'tòpos'=space) represent an idea of art that integrates sound, light and space. In that sense, polytopes are not only ancestors of the soundart, since the sound sculptors of our times are always working in a given space, and they sculpt it after choosing it, not before projecting it, and this is Xenakis' most complete realization, today not overtaken. The architecture of specific spaces created to enjoy live music and the creation of its own tools so to realize electronic music come together in Xenakis' mind and activity. When reading his writings of the 1980s, it is clear that what pushes the composer towards this form of total art is the same anxiety he has since the days of stochastic compositions: "Can we do tabula rasa of all the known compositional rules?", and "what a rule is?".
Since 1952 to 1956 Xenakis elaborates in first person in Fortran language a program in order to obtain scores that realize on cartesian axes analitic geometry, composing the pieces of the ST Series. The use of the computer as a compositional tool helps him to overtake 'the art of the fugue', which is transforming a theme following the rules of transposition, augmentation, temporal decrease, etc. inventing, instead, his own musical forms. The ST Series comes from applying to the stochastics the 'Markov chains' (responsible of the developing of computer science and of linguistics, at least until Noam Chomsky proved they were useless in that field).
Not only probability, but also repetition, determinism, so to measure the symmetry of a composition (Nomos Alpha is an example, again): 'repetition' is the definition of one of the smallest conditions to have a 'rule', following Newton. Nomos Alpha is also the best example of 'symbolic music' realized by the composer. Unsatisfied by the structures of Western music, for its limits in polyphonic development, as far as the time logic, Xenakis points at the popular music, in particular the Bizantine popular music, for his use of the pedal in vocal polyphony so that it creates a new tension, the same that will occurr in the music of Debussy or Schoenberg. The composition is divided in 24 sections, and it is composed by two layers: the first is composed by a group of 24 elements, while the second, in contrast, is no more determined by the group theory but it follows a continuous, evolving movement. Commissioned in 1965 by Radio Bremen for cellist Sigfried Palm, and dedicated to the mathematicians Aristossenus, Evariste Galois and Felix Klein, Nomos Alpha represents perfectly the dialectic of transformation of time between en temps and hors temps from which the idea of 'amnesia' comes ('to leave at the entrance the emotional and qualitative burdens passed on by musical traditions' taking under exam only 'the abstract relationships in every event' [Iannis Xenakis, Musiques Formelles, 1963]) and that represents the attempt to recover in music the 'ubiquity' typical of subatomic physics.
The Polytopes are the places in which to present music not only composed, but also played by computers. The Polytope of Montreal has been realized in 1967. Commissioned for that year Expo, it has been realized with wide concave and convex mirrors suspended to electric cables reproducing 'visual melodies' through lightning sources. The music created for the polytope will lead to Kraanerg, a piece of 75 minute of lenght, without any inner subdivision, including instead 20 moment of silence of different lenghts integrals to the development of the piece itself. The first section of Kraanerg contains equal portions of live orchestra and recorded tape, the second section is mostly live orchestra, and the third is almost entirely recorded tape. Performed for the first time at the National Arts Center in Ottawa in 1969, and conceived as accompaniment for Roland Petit's ballet company, Kraanerg has been performed until 1972 and then forgotten until 1988, when a new version coreographed by Graeme Murphy took it at a new peak, with a performance almost exclusively instrumental. The title of the piece comes from Greek, and it means 'accomplished act', referring to the youth movement of those years and the socio-political changes wished for.
The Polytopes diverge the one from the other for the spatial-temporal disposition of music and listeners. The Polytopes of Persepolis and Micenae are the only ones with a fixed stationing for the listeners while sound and light sources are dispersed. Works as Terretektorh and Nomos Gamma presuppose, on the contrary, the listeners to be dispersed between the musicians. The Polytope of Montreal sees the audience at the center of the architecture, with the light and sound sources around the audience.
During the 1970s Xenakis teaches composition and gives public lectures. He creates an atelier as the IMAMu at the University of Indiana, he teaches at the Sorbonne since 1973 to 1989, at the Gresham College in London since 1975 to 1978, and he can see his works performed even in Iran. His last work, O-mega, is accomplished before Alzheimer prevents him to compose. He falls into a coma at the beginning of February, 2001 and he dies in Paris few days later.