Geometrical Model of Anticipatory Embedded Systems

I have defined (Garnier-Malet, 1997) the fundamental movement of doubling which transforms any initial system into anticipatory (Rosen, 1985) embedded (Dubois, 1996 and 1997) systems. I have demonstrated that six levels of embedding are necessary in the initial system which is the zero level during its transformation. Each level has its observer. With scalings of transformation' s-spaces and times, each level is a zero level. During the doubling transformation the initial observer cannot observe the other observers. But, at the end of the transformation which is always the beginning of another transformation, the initial and the third observers, then the third and the sixth observers, exchange their space's and time's perception.·These exchanges are the only way for the initial observer to know and anticipate the consequences of an experience of embedded systems before having time to realise it in the initial system and, above all, without modifying this initial space. The perception's exchange·of the observers must be the consequences of this necessity at the end of the transformation. These exchanges imply three speeds of doubling which I have calculated. They are necessary at the end to juxtapose the six embedded levels in the initial system which must be necessarily one ten-dimensional space. We shall see that this implication is as fundamental as the movement of doubling.