Kozyrev’s Flow of Time, Hadronic Mechanics, & Conch Shell Morphology

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Kozyrev’s Flow of Time, Hadronic Mechanics, & Conch Shell Morphology


Chris Illert is the world leading expert in conchology, and succeeded in specialist studies during the 1980’s and early 1990’s to fi nd a universal algorithm to explain the growth pattern of all known sea shells (Illert 1983, 1987, 1989, 1990a,b, 1992, 1993, 1995, 1995b). This modeling of sea shell growth was only possible by a primary description of the growth trajectory in a certain supra-Euclidean space, projected through geometric deformation into sea shell growth as it appears for human perception. The supra-Euclidean description of the growth trajectory required was not possible with traditional supra-Euclidean geometry, such as Riemannian or Minkowskian, but required a more general geometry, which it is appropriate to name hadronic geometry. The mathematical physicist Ruggero Maria Santilli (IBR Ia) initiated the development of huge new classes of number fi elds with corresponding geometries and mathematical techniques, named hadronic mathematics, a scientifi c enterprise with revolutionary and already well established implications for physics as well as other disciplines. This development has now gone on for four decades and with a rising numbers of contributions from professional mathematicians. Hadronic mathematics encompasses, in progressive complexity, the new and more general fields of isonumbers, genonumbers and Santilli hypernumbers, with corresponding liftings of the totality of preceding mathematics, and with corresponding development of iso- , geno- and hypergeometries.

Stein Johansen  –



For a certain class of sea shells, namely sea shells with bifurcation, Illert proved that sea shell growth could only be understood by the acknowledgement of certain NON-TRIVIAL time categories presupposing hadronic mathematics and mechanics for a precise comprehension. For this class of shells, such nontrivial information flows in supraEuclidean space is projected from isospacetime (and its asymmetric isodual spacetime) through deformation into the ordinary Euclidean time line, where these information flows manifest as forward and backward LEAPS in time.

For this discovery and break-through in theoretical biology, Illert in 1995 was honored by winning the IBR International Prize for Biology. In the announcement of the nomination it is stated that Illert established «the inapplicability of conventional geometries (such as the Euclidean, Minkowskian or Riemannian geometries) for quantitative representations of sea shells growth, thus providing the foundations for potentially historical advances in biology» (IBR Ib). Illert has also made far-reaching contributions to nuclear physics, chemistry and linguistics. Due to inertia and inoptimal information flows in the global science ecology, these contributions are still not much known. However, in the SPIE Milestone Series, volume 15, «a reprint collection of outstanding papers from the world literature on optical and optoelectronic science, engineering, and technology» from last century (1900–1990), section one («Chirality and optical activity»), Illert was the only scientist honored by being represented with more than one paper of the eight papers picked. Both of Illert’s papers (1987, 1989) were dedicated to «formulation and solution of the classical sea shell problem» (Lakhtakia 1990).
Due to the importance of Illert’s results in conchology for supporting basic notions of Kozyrev, especially with regard to the nature of Time, we will present Illert’s extensive sea shell research in some detail. Illert’s contributions to conchology consist of many publications, but the most important and extensive is Illert and Santilli 1995 where Illert has written the fi rst part (p. 1–112) named Mathematical Representations of Sea Shells from Self-similiarity in Non-conservative Mechanics (i.e. a mechanics more extended than quantum mechanics). Illert’s representation reveals a UNIVERSAL algorithm (cf. eqs. 3.1 p. 72 and 3.2 p. 73, and equation 5 in Illert 1989:768) for sea shell growth, «from a solid empirical base encompassing 100.000 or so (living or extinct) molluscan shell varieties» (p. 4), more specifically «a unique second-order coupled differential equation (3.2) describing all of the several major categories of shell geometries found in the real world» (p. 101). The universal algorithm was tested against the most intricate and complex sea shell structures (among them Nipponites mirabilis — cf. p. 91) through extensive computer simulations, and with impressing empirical matching.

The most general assumption in Illert’s systematic presentation — as in most theoretical mechanics — is the concept of energy (p. 3) and the principle of least action for energy fl to «dissipate stresses» during sea shell growth to resemble optimal tensile clocksprings» (p. 9). To reveal the hidden universal growth algorithm, Illert uses the principle of self-similarity (including scale-invariance) of growth — elaborated from Aristotle’s notion of gnomon. (p. 27–64) from which Illert derives and explains «in a natural way» the self-similarity differential equations with two specified constraints (eqs. 2.41 and 2.42, p. 67), this leaving only two arbitrary constants which values Illert groups in different classes leading to various classes of clock-spring trajectories (p. 1 and p. 9) corresponding with the empirical variations of sea shell forms (p. 72–105).
In developing the equation for the universal growth algorithm, Illert discovered the necessity of moving — technically speaking — from a real to a complex Langrangian which requires a LIFTING from Euclidean space to what is called ISO-EUCLIDEAN space in the modern isomathematic branch of mathematics (cf. p. 101). This was necessary because the two mentioned «critical constants, associated with trajectory “curvature” and “torsion” often have to be complex numbers» (p. 2). Iso-Euclidean space is a certain multidimensional complex space, in Illert’s case basically with SIX dimensions. The concept of such space was NOT known before the initiation of iso-mathematics (Santilli 1988), and is not to be confused with trivial multidimensional modeling or with hyperdimensional geometry in general, dating back to Riemann in 1854. Iso-mathematics is a new and more extensive landscape of mathematics where ALL earlier known mathematical operations, supposing the number of 1 as the basic unit, is GENERALIZED and LIFTED to encompass ANY other unit which COINCIDES with the original basic unit, and at the same time has an ARBITRARY functional dependence on other variables. Hence, iso-mathematics rose from detrivializing and generalizing the conventional unit of mathematics.
This means that Illert’s systematic examination revealed a highly non-trivial general result: that the hidden universal algorithm for sea shell growth could ONLY be discovered with the extension of 3D space to «at least five space-like and one time-like dimensions» (p. 2). This has far-reaching implications for an adequate understanding of the ontological architecture of space itself, degrading the ordinary 3D perception of space to a MANIFESTATION of a higher order (in the sense of David Bohm) of space organization. Some quotes from Illert in this regard: (In this article comments of mine are in brackets, and emphasizes of mine are in boldface.) the growth-trajectory that we see (hereafter called a CLOCKSPRING) is only the real part of a more general (–) curve through a multi-dimensional space. Even the underlying physical principles (such as HOOKE’S LAW) only emerge coherently, and seem to make sense, within our full complex-space formalism (–). Real space <Euclidean 3D> just doesn’t seem adequate. So are seashell geometries profound enough to tell us that we live in a world that doesn’t quite make sense unless we assume that it has at least five spacelike and one time-like dimensions? (–) Certainly, if we do take shell geometries seriously, our insights are all the more powerful because they emerge from totally classical, non-quantum, reasoning (p. 2) forms that are different in normal Euclidean space may be unifi ed in this more general geometry <i.e. isospace>. (–) We already know that shell growth trajectories are iso-euclidean, but, if we tried to force them into purely Euclidean space, they would wrinkle and the shells would crack or explode. (–) the iso-euclidean trajectory of Nipponites mirabilis starts out in a regular planar spiral before eventually becoming serpentine. But if we force it to exist in a more «Euclideanish» space (–) the whole curve meanders grossly from beginning to end, it is just like stuffi ng elastic piano-wire into a smaller box thereby forcing it to wrinkle more severely (p. 101–102).

Illert classifies clocksprings in first and second kind, depending on if their representation requires first or second order discrete mathematics. Even quite simple sea shells, classified as clocksprings of the first kind, can have a growth trajectory where the imagined «wire» may pass through itself. Illert argues this to not represent any crucial difficulty since the «wire» is imagined as INFINITELY thin in his approach (cf. p. 82). (However, there exists ONE topological structure, the diagonal woven Klein-bottle discovered by Morgan (and further discussed by Purcell (2006), where the wire passes through itself in 3D WITHOUT being infinitely thin.) While sea shells with self-intersection as such may not be too big a deal in Illert’s theory, there is a certain subclass of such shells that poses a huge and highly interesting challenge for the scientifi c understanding, namely the so-called BRANCHING clocksprings. I prefer to quote Illert at length here, because this may be a discovery in the history of science of uttermost importance for a more profound and extended understanding of the nature of time: shells such as Yochelcionella, Rhaphaulus, Rhiostoma and Spiraculum all utilize self-intersecting clockspring trajectories; actually BRANCHING at points of trajectory-intersection, there after growing simultaneously along two separate branches of the clockspring! Some shells branch during the earliest developmental stages (as in Yochelcionella daleki, a self-intersecting clockspring of the First Kind (–)), whilst others (such as Janospira nodus, a self-intersecting clockspring of the Second Kind) wait almost till the end of ontogeny before branching. The palaeontologists who fi rst studied these branching clockspring geometries described the shells as «curious», «ridiculous» «absurdities» but we can now see them as the same optimale tensile spirals which other non-branching shells also utilize. And as trajectory-branching seems to occur widely, in unrelated species, the usual «once-off» biological explanations won’t suffi ce… there is a deeper geometrical principle at work! (–) how can the trajectory at the branchpoint (–) be causally linked to the FUTURE ongoing pathway (–)? It seems as if Janospira, at the instant of branching, «knew» (ahead of time) about the existence and location of a future portion of the clockspring trajectory even though the outermost whorl had not, at the time of branching, actually looped about to (and indeed, never ultimately would) physically create the future intersection-point. We are talking here, about action with foreknowledge, action outside the expected linear Newtonian sequence, rather as if an impending future event acted BACKWARD THROUGH (future) TIME to influence the present (p. 93–94). Illert illustrates the issue with the following vector-spiral diagram from his vector-equation for the clockspring trajectory (p. 95):


FIG. 1. (from Illert in Illert and Santilli 1995:95)

The universal algorithm with the adequate value of the two critical constants gives the growth trajectory for this sea shell INCLUDING the dotted part of the trajectory. The dotted trajectory is NOT manifested in the physical structure, but the PROLONGED trajectory (m) from the branching point CONTINUING this dotted and 3D-VIRTUAL trajectory is. Hence, the prolonged trajectory (m) can ONLY be discovered from assuming that the dotted part has a crucial HIDDEN reality, obviously because the universal algorithm has an even HIGHER reality. Also, for this to be the case, the hidden algorithm has to include a determination of the LENGTH in space (both in hyperspace and 3D space) and time (cf. later) of the hidden part, and by this also the exact LOCATION in space and time of the branching point.

Illert’s interpretation in and of fig. 1 is to view the growth trajectory as a combined result of three different trajectory parts with three corresponding different categories of time:
1) Interval [-infi nity, n] with ordinary time fl ow or «action from past to present». We may name it «PLAY» for a convenient video analogy.
2) Interval [n+1, m] with «action forward through future time», by Illert coined Sheldrake propagator after Rupert Sheldrake’s notion of such a time category (1981). We may name it «FORWARD» for short.
3) Interval [n+1, m–1] with «action back through future time», by Illert coined Gatlin propagator after Lisa Gatlin’s notion of time- reversed information fl ow (1980). We may name it «BACKWARD» for short.
The combined result is established in the succession from 1) to 2) to 3). 2) represents an addition to interval 1), while 3) represents a subtraction or deletion of and from interval 2) with the remaining exception of the «head» of 2): the new branch m anchored in the branching point.
2) and 3) represent highly non-trivial categories of time, and if Illert’s theory is adequate, this of course must have crucial implications for ALL sciences. With regard to the non-triviality Illert writes: “The main thing to realize is that branching clocksprings arize naturally from the same theory that describes all other known shell geometries, and that examples such as Janospira occur in Nature. To be predicted by theory and observed in practice is a powerful metaphysical position: how one mentally reconciles the causal implications is a psychological problem (p. 96)”

The discovery of the universal growth algorithm was only possible by looking for it and formulating it in ISO-Euclidean space. However, the nature of isospace also has DIRECT and UNIVERSAL implications for the understanding of TIME, consistent with Illert’s highly non-trivial results in the case of the sea shell branching phenomenon.

It is interesting to note that Illert himself argues that non-trivial Sheldrake and Gatlin propagators also are highly relevant for understanding of PARTICLES, and presents a case inside physics itself: charged lepton decay and neutrino production (p. 96–100), with the possibility of description from Illert’s Langrangian antiparticles as well as particles, and with the possibility of charged particles to travel in clockspring trajectories which sometimes branch. Also, Illert argues the possibility of the muon-antineutrino existing OUTSIDE the normal time flow as a time-reversed electron-neutrino, and the muon as NOT pointlike Newtonian, but smeared over a region of space-time as a TEMPORAL version of Young’s famous double-slit experiment.

Illert’s conchology studies do not discuss any relation between mass changes and non-trivial time flows. However, others of his results are highly relevant with regard to Kozyrev’s theory of time, such as:
1) Time exists in classes and modes that are FAR FROM TRIVIAL, and not recognized — or recognizable — by most physics.
2) Hidden or supra-Euclidean time categories have PRIMACY compared to Euclidean time to describe and explain the overall pattern of time fl ows with connected observable phenomena in Euclidean spacetime.
3) Supra-Euclidean time fl ows projected to Euclidean time is necessary for the manifestation of certain PHYSICALLY observable phenomena, including in BIOLOGICAL nature, and such time fl ows can include LEAPS as measured along the ordinary time line. This is consistent with the results of the astronomical observations by Lavrentiev et al. (1990a,b, 1991, 1992) documenting non-electromagnetic and highly non-trivial effects from stars on PHYSICAL sensors from positions of the stars in the PAST (corresponding to the visual positions we observe from receiving their light), their real positions in the PRESENT (in the case of the sun also documented to effect BIOLOGICAL sensors), and their positions in the FUTURE (symmetrical to their past positions, measured across the axis of their present positions).
4) Supra-Euclidean time fl ows projected to Euclidean time are necessary for the manifestation of certain IRREVERSIBLE phenomena in biological nature, such as branching in sea shell growth. Notice that this case is an EMERGENT irreversible phenomenon, ADDING more complex order (branching compared to not-branching growth), contrary, or rather complementary, to the well documented Kozyrev irreversible «cause» deforming the «effect» to more entropy, sought explained by infl ow of additional «Time energy». This irreversible antientropy effect is consistent with the effect from the present, non-visible position of the sun on biological sensors (growth of microorganism colonies) as documented by Lavrentiev et al.
5) The ordinary notion of causality between physical objects and states needs DETRIVIALIZATION and COMPLEXIFICATION, in-
cluding comprehension of the infl uence of supra-Euclidean time fl ows on objects and events in Euclidean spacetime, to reconcile the paradoxes rising from the ordinary notion which considers time jumps impossible.
6) «TIME TRAVEL», backward as well as forward, is not any logical absurdity or any fanciful construct, but an undeniable and quite crucial aspect of the ordinary state of affairs in Nature’s dynamic architecture, as illustrated by even a quite simple biological system as branching sea shells. Hence, there is nothing surrealistic to the idea of imitating Nature’s time fl ow by means of adequate human technology, as illustrated by the time machine experiments already executed by Chernobrov (1996, 2001).
7) Conventional notions in physics concerning the topology of overall spacetime have restricted relevance due to shortcomings in ontological rigor and sophistication, while the topology of the KLEINBOTTLE may offer a crucial key.

Illert argued that the basic structure of space may be somewhat similar to the structure of a complex sea shell, which — when selfintersecting — is close to the suggestion of a Klein-bottle structure. Also, a certain class of sea shell analyzed by Illert was coined «Moebius conoids» because a set of allowable spire shapes is successively ordered as an unfolding Moebius strip (1987:fi g.9; 1990b: 1613, eq.1, fi gs. 1 and 3), and Illert has pointed out that «Moebius-ness is a telltale sign of Klein-bottle-ishness» (Illert 2007).

Profound signifi cance of the Klein-bottle for topology and physics, as well as for other disciplines, has been increasingly acknowledged by scientists during the last generation (cf. Rosen 1988, 1994, 1995, 1996; Morgan I, Purcell 1998, 2006; and also Brodey I and Johansen 2000, 2004, 2006).
Moebius Band magnetic monopole devices as described by Shakhparonov (I) were developed from theoretical insights in Klein-bottle projection into physical 3D space having highly non-trivial implications for energy creation, flow and density.

The fundamental tenet of the causal mechanics developed by Kozyrev can be formulated as follows: There are two types of energy in the Universe. The positive or «right» energy acts as a factor of entropy increase. The negative, or «left» energy tends to decrease the entropy, i.e. acts as a factor, which regulates the entropy increase. The «right» energy is transformed to the «left» one and this fact may be interpreted as a course of time from the past to the future. When the energy is transformed from the «left» to the «right» form, time is reversed. Kozyrev supposed (–) that through revolving of a body together with a coordinate system along a circumference the right coordinate system is transformed to the left one at the moment, when the body reaches the point situated at the opposite side of the diameter (Shakhparonov I:275–276).

Moebius Band technology exploits the effects of knitting these two opposite points together by a Klein-bottle projection, «gluing» the two coordinate systems in revolved superposition. From Illert’s results the success of this recent and unorthodox technology inspired by Kozyrev’s work does not appear so surprising.

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