Fractal time

There are a number of definitions of the notion of fractal time. The term was first mentioned by Benoît Mandelbrot [Mandelbrot, B. B. (1982): "The Fractal Geometry of Nature". W.H. Freeman, San Francisco.] , who used it to refer to statistically self-similar time series. A more general definition, based on primary experiences of time, was provided by S. Vrobel [Vrobel, S. (1998): "Fractal Time". Houston: The Institute for Advanced Interdisciplinary Research, Houston] . The notion of fractal time is implicit in concepts such as fractal space-time and recursive systems. Laurent Nottale's fractal space-time [Nottale, L. (2001): Scale Relativity, Fractal Space-Time and Morphogenesis of Structures. In: "Sciences of the Interface". Edited by H.H. Diebner, T. Druckrey, P. Weibel. Genista, Tübingen, pp. 38-51.] and El Naschie's Cantorian space-time [El Naschie, M.S.: Young Double-Slit Experiment, Heisenberg Uncertainty Principle and Correlation in Cantorian Space-Time. In: "Quantum Mechanics, Diffusion and Chaotic Fractals". Edited by Mohammed S. el Naschie, Otto E. Rössler & Ilya Prigogine. Pergamon, Elsevier Science, 1995, pp. 93-100.] are generalizations of Einstein's general theory of relativity. Daniel Dubois' hyperincursive systems [Dubois, Daniel (2001): "Incursive and Hyperincursive Systems, Fractal Machine and Anticipatory Logic". CASYS 2002. AIP Conference Proceedings 573, pp. 437-451, 2001.] are systems which contain a model of themselves and thus also portray a fractal temporal structure.

Benoit B. Mandelbrot

In his pioneering work, Mandelbrot coined the terms fractal and fractal time and introduced a way of differentiating by taking into account nested levels of description (hereafter referred to as LODs). These are temporal intervals within a time series as defined by their degree of resolution. He thereby provided a way of measuring and comparing time series with respect to their temporal density. This he achieved by assigning a fractal dimension to statistically self-similar (also referred to as scale-invariant) spatio-temporal structures.

For instance, as early as 1963, he found that the scale-invariance in the rate of change of cotton prices suggested that there is a correlation between daily fluctuations and long-term changes. This was surprising, because these two types of change were usually attributed to different causes, namely, random fluctuations and macro-economic influences (such as wars or recessions), respectively. This scaling principle of price change discovered by Mandelbrot was the first example of a fractal time structure, as the time series displayed statistical self-similarity. [Mandelbrot 1982, p. 337]

The arbitrary choice of the measuring rod applied by an observer, i.e., the time scales taken into account simultaneously, determines the outcome of the measurement, as it depends on how much detail is taken into account. Mandelbrot made explicit that the role of the observer is crucial to the extent that he defines the measuring rod and the LODs taken into account when observing or measuring a time series. However, his observer’s perspective is an external one, as the time series was treated as a closed system, with the analyst is taking an external vantage point.

When we want to describe our experience of time, by contrast, we need to do this from the perspective of an internal observer, since we are part of the system we wish to describe. Such an internal observer is subjected to a number of perceptual and cognitive constraints. As Pöppel [Pöppel, E. (2000): Grenzen des Bewußtseins – Wie kommen wir zur Zeit, und wie entsteht die Wirklichkeit? Insel Taschenbuch, Frankfurt] pointed out, our primary experiences of time (succession, simultaneity, duration and the Now) act as such constraints to an internal observer. This aspect did not play a part in Mandelbrot’s concept of fractal time.

Susie Vrobel

Vrobel’s Theory of Fractal Time, by contrast, does take account of an internal observer’s primary experience of time. [Vrobel, S. (1998): Fractal Time. Houston: The Institute for Advanced Interdisciplinary Research, Houston.] This is necessary because the assumptions and observations of any theory about the world are secondary constructs based on our primary experience of simultaneity, succession, the Now and duration. Vrobel’s fractal notion of time is based on our experience of duration, simultaneity and succession, which shape the structure of our Now. These primary experiences of time are defined as a prerequisite for modelling reality. We generate simultaneity by nesting, i.e., integrating events which are connected by during-relations. Such events are temporally compatible from an internal observer’s perspective. However, what is simultaneous in one observer’s Now need not be perceived as simultaneous by another observer. Succession is generated by integrating temporally incompatible perceptual gestalts, i.e., events which cannot be connected by during-relations, on specific LODs. Our experience and description of duration is based on the arrangement of nested content and contexts within our Now.

Our Now is structured by nestings of temporally compatible intervals and arrangements of temporally incompatible intervals. The former generate simultaneity, the latter succession. Vrobel’s theory provides a means of quantifying the internal structure of the observer’s temporal perspective, his Now, by differentiating between the length of time, Δtlength, the depth of time, Δtdepth, and the density of time, Δtdensity.

Δtlength is the number of incompatible events in a time series, i.e., events which cannot be expressed in terms of during-relations (simultaneity). Δtlength defines the temporal dimension of succession for individual LODs.Δtdepth is the number of compatible events in a time series, i.e., events which can be expressed in terms of during-relations. Δt depth defines the temporal dimension of simultaneity and provides the framework time which allows us to structure events in Δtlength on individual LODs.Δtdensity is the fractal dimension of a time series. It describes the relation between compatible and incompatible, i.e., successive and simultaneous events, the density of time.N.B.: Δtdepth logically precedes Δtlength, as there is no succession without simultaneity.

The Newtonian metric may be defined as a special case of fractal time, namely, as Δtlength of the nesting level 0, i.e., Δtdepth = ∞.

(to be continued ... :

Laurent Nottale

Mohamed El Naschie

Daniel M. Dubois

Serge Timashev)

References

Further reading

*Barnsley, Michael: Fractals Everywhere. Academic Press, 1988.
*Fauvel, J. et al: Music and Mathematics – From Pythagoras to Fractals. Oxford University Press, 2003.
*Dubois, Daniel (2001): Incursive and Hyperincursive Systems, Fractal Machine and Anticipatory Logic. CASYS 2002. AIP Conference Proceedings 573, pp. 437-451, 2001.
*Nottale, L. (2002): Relativité, être et ne pas être. In: Penser les limites. Ecrits en l’honneur d’André Green. Delachiaux et Niestlé, Paris, p.157.
*Nottale, L. et al (2002): Développement Humain et Loi Log-Périodique (by Roland Cash, Jean Chaline, Laurent Nottale, Pierre Grou), In: C.R. Biologies 325. Académie des Sciences / Editions Scientifiques et Médicales Elsevier SAS, pp. 585-590.
*Ord, G.N. (1983): Fractal Space-Time: A Geometric Analogue of Relativistic Quantum Mechanics. In: Journal of Physics A.: Mathematical and General, Vol 16, The Institute of Physics, pp. 1869-1884.
*Rössler, O.E. (1996): Relative-State Theory: Four New Aspects. In: Chaos, Solitons and Fractals, Vol. 7, No. 6, Elsevier Science, pp. 845-852.
*Van Orden, G.C. et al (2005): Human Cognition and 1/f Scaling. In: Journal of Experimental Psychology: General 134, American Psychological Association, pp. 117-123.
*Vrobel, S. (1995): Fraktale Zeit. Draft dissertation, unpublished.
*Vrobel, S. (1996): Ice Cubes And Hot Water Bottles. In Fractals. An Interdisciplinary Journal on the Complex Geometry of Nature. Vol. 5 No. 1. World Scientific, Singapore, pp. 145-151
*Vrobel, S. (1998): Fractal Time. Houston: The Institute for Advanced Interdisciplinary Research, Houston.
*Vrobel, S. (1999): Fractal Time and the Gift of Natural Constraints. Tempos in Science and Nature: Structures, Relations, Complexity. Annals of the New York Academy of Sciences, Volume 879, pp. 172-179.
*Vrobel, S. (2000): How to Make Nature Blush: On the Construction of a Fractal Temporal Interface. In: Stochastics and Chaotic Dynamics in the Lakes: STOCHAOS. Edited by D.S. Broomhead, E.A. Luchinskaya, P.V.E. McClintock and T. Mullin. New York: AIP (American Institute of Physics), pp. 557-561.
*Vrobel, S. (2004): Fractal Time and Nested Detectors. In Proceedings of the First IMA Conference on Fractal Geometry: Mathematical Techniques, Algorithms and Applications. DeMontfort University, Leicester, pp. 173-188.
*Vrobel, S. (2005a): Reality Generation. In: Complexity in the living – a problem-oriented approach. Edited by R. Benigni, A. Colosimo, A Giuliani, P. Sirabella, J.P. Zbilut. Rome: Rapporti Istisan, pp. 60-77.
*Vrobel, S. (2006a): A Nested Detector with a Fractal Temporal Interface. In CASYS ’05 – Seventh International Conference on Computing Anticipatory Systems. Conference Proceedings, HEC-Lug, Liège.
*Vrobel, S. (2006b): A Description of Entropy as a Level-Bound Quantity. In CASYS ’05 – Seventh International Conference on Computing Anticipatory Systems. Conference Proceedings, HEC-Lug, Liège.
*Vrobel, S. (2006c): Nesting Performances Generate Simultaneity: Towards a Definition of Interface Complexity. In: Cybernetics and Systems Vol 2 (Edited by Robert Trappel). Austrian Society for Cybernetic Studies, Vienna, pp. 375-380.
*Vrobel, S. (2006e): Temporal Observer Perspectives. In: SCTPLS Newsletter, Vol. 14, No. 1 Society for Chaos Theory in Psychology & Life Sciences, October 2006.