Using a mathematical model, we show that a Triple Helix (TH) system contains self-interaction, and therefore self-organization of innovations can be expected in waves, whereas a Double Helix (DH) remains determined by its linear constituents. (The mathematical model is fully elaborated in the Appendices.) The ensuing innovation systems can be expected to have a fractal structure: innovation systems at different scales can be considered as spanned in a Cartesian space with the dimensions of (S)cience, (B)usiness, and (G)overnment. A national system, for example, contains sectorial and regional systems, and is a constituent part in technological and supra-national systems of innovation. The mathematical modeling enables us to clarify the mechanisms, and provides new possibilities for the prediction. Emerging technologies can be expected to be more diversified and their life cycles will become shorter than before. In terms of policy implications, the model suggests a shift from the production of material objects to the production of innovative technologies.

The model of "Open Innovations" (OI) can be compared with the "Triple Helix of University-Industry-Government Relations" (TH) as attempts to find surplus value in bringing industrial innovation closer to public R&D. Whereas the firm is central in the model of OI, the TH adds multi-centeredness: in addition to firms, universities and (e. g., regional) governments can take leading roles in innovation eco-systems. In addition to the (transversal) technology transfer at each moment of time, one can focus on the dynamics in the feedback loops. Under specifiable conditions, feedback loops can be turned into feedforward ones that drive innovation eco-systems towards self-organization and the auto-catalytic generation of new options. The generation of options can be more important than historical realizations ("best practices") for the longer-term viability of knowledge-based innovation systems. A system without sufficient options, for example, is locked-in. The generation of redundancy-the Triple Helix indicator-can be used as a measure of unrealized but technologically feasible options given a historical configuration. Different coordination mechanisms (markets, policies, knowledge) provide different perspectives on the same information and thus generate redundancy. Increased redundancy not only stimulates innovation in an ecosystem by reducing the prevailing uncertainty; it also enhances the synergy in and innovativeness of an innovation system.

The study of interhuman communication requires a more complex framework than Claude E. Shannon's (1948) mathematical theory of communication because "information" is defined in the latter case as meaningless uncertainty. Assuming that meaning cannot be communicated, we extend Shannon's theory by defining mutual redundancy as a positional counterpart of the relational communication of information. Mutual redundancy indicates the surplus of meanings that can be provided to the exchanges in reflexive communications. The information is redundant because it is based on "pure sets" (i.e., without subtraction of mutual information in the overlaps). We show that in the three-dimensional case (e.g., of a triple helix of university-industry-government relations), mutual redundancy is equal to mutual information (Rxyz = Txyz); but when the dimensionality is even, the sign is different. We generalize to the measurement in N dimensions and proceed to the interpretation. Using Niklas Luhmann's (1984-1995) social systems theory and/or Anthony Giddens's (1979, 1984) structuration theory, mutual redundancy can be provided with an interpretation in the sociological case: Different meaningprocessing structures code and decode with other algorithms. A surplus of ("absent") options can then be generated that add to the redundancy. Luhmann's "functional (sub)systems" of expectations or Giddens's "ruleresource sets" are positioned mutually, but coupled operationally in events or "instantiated" in actions. Shannon-type information is generated by the mediation, but the "structures" are (re-)positioned toward one another as sets of (potentially counterfactual) expectations. The structural differences among the coding and decoding algorithms provide a source of additional options in reflexive and anticipatory communications.

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