Article in Science / Math
We show how a radical involution in conventional mathematical and physical perceptions of reality can be brought about by regarding natural form as comprising energetically surfaced cavities of space, instead of local points of mass surrounded by space

Place-Time: The Flow-Geometry of Space

By Alan Rayner, Ben Sidebottom, David Peleshok and Philip Tattersall

No where is an Island, entire of it self


Here we show how a radical involution in conventional mathematical and physical perceptions of reality can be brought about by regarding natural form as comprising energetically surfaced cavities of space, instead of local points of mass surrounded by space. This involution provides a new way of understanding gravitational influence ‘at a distance’ and reconciling particle and field theories of matter. It enables evolutionary processes of all kinds to be understood in terms of ‘natural inclusion’ – the co-creative, fluid-dynamic transformation of all through all in receptive spatial context. All form is understood as ‘flow-form’, an energetic inclusion of omnipresent intangible space within, throughout and beyond dynamic figural boundaries.

Energy is mobile, superficial, formative presence while space is still, permissive presence, ultimately beyond energetic reach in its infinite inward and outward depth. Each presence in the other, distinct but mutually inclusive, is essential to natural energy flow as ‘place-time’, which comprises 100% intangible space, plus tangible energy in circulation and animated suspension respectively around and within inward and outward depth. Space, in this view, is a limitless pool of implicit induction; an intangible hollowness of local centres and axes, around which energetic form flows rather than residing within a fixed tangible ‘frame’ or ‘shape’ of explicit force. All energetic form circulates and reconfigures around and within an infinite scale of inductive influence, as an evolutionary dynamic of continual, cumulative transformation.

This understanding only becomes possible when the infinite depth of space as an intangible presence ultimately beyond energetic reach is acknowledged. The frictionless quality of this inaccessible intangible presence inside, outside, throughout and beyond distinctive energetic localities needs especially to be appreciated. Space can neither be abstracted from nor confined by structure. The apparent hardness of solid structure actually arises from the inaccessible hollowness within the depth of close-packed energetic configurations of space. In other words, space cannot be cut, not even with a diamond knife.

With this acknowledgment comes a paradigmatic transformation from perceptions based upon the definitive boundary logic of ‘abstract rationality’ to the fluid boundary logic of ‘natural inclusionality’. The fundamental natural geometrical ‘shape’ that manifests this fluid boundary logic is what we describe here as a ‘zero fractal’.


Throughout history, atomistic theories have regarded physical form as being composed of local points of mass surrounded and distanced by empty space. This perception has been brought into being by closed-systems of abstract logic, employed solely to approximate ‘observation points’ or ‘evolutionary points’ within an otherwise infinite continuum of cumulative energetic transformations. The resulting reductions of reality are embedded as absolute truths in the foundations of both classical and modern mathematics (Rayner, 2011a).

These abstract systems impose discontinuity between one object and another by treating boundaries as discrete limits and space as an isolating absence of tangible presence (cf. Tattersall, 2011). By treating a zero-dimensional point as a point of mass in space instead of a point of space enveloped by mass, they begin with a physical impossibility – the notion of a mass that has no shape or size. The resulting paradoxical inconsistencies render the agency of change - energy - into an extrinsic force instead of an intrinsic inductive influence.

Here we intend to show how the inconsistencies of abstract perception can be obviated by perceiving physical form as local cavities of receptive space enveloped by energetic surfaces. Space, in our view, consists naturally of an intangible, frictionless and thereby receptive presence that co-creatively induces tangible energetic presence into shape. It corresponds with what the ancients thought of as ‘aether’ and modern quantum field theorists speak of as ‘Higgs Field’. It is, in effect, the zero viscosity universal ‘medium’ that enables the non-zero viscosity informational ‘message’ of tangible energetic form to configure and reconfigure into myriad shapes and sizes – local ‘figures’ that cannot be isolated or cut away from their spatial ‘ground’ as independent singular entities (Rayner, 2011a). All figural shapes are hence inclusions of a cumulative energetic flux whereby any distinct locality can only be perceived as ‘one discrete object’ when abstracted or ‘singled out’ from larger and smaller scales within an infinite depth.

The following view of infinity was held by Blaise Pascal and many of his contemporaries, in the 17th Century:

“The actual infinity in material things as much as in the increasingly large as in the vanishingly small, that is, the actual division of each part of matter to infinity and at the same time the infinite vastness of matter, has been supported by Mr Pascal.” Leibniz (1695)

This, literally particular, ‘dual-infinity’ view suggests that it is not possible to observe either a largest scale or a smallest scale of material organization in the infinite depth of reality; that all material things are infinitely divisible, with infinite surface area; whilst also being additive to an infinitely larger, greater scale.

“...all matter is organic everywhere, and that however small a portion one takes, it contains representatively, by virtue of the actual decreasing to infinity that it encloses, the actual increasing to infinity which is outside it in the universe.” (Leibniz 1695)

In modern day ‘holism’, the universe is regarded paradoxically as a seamless, infinite whole, and infinity itself as an indefinable numerical quantity of matter. There is no perception of inaccessible vastness beyond material form, or of inaccessible non-material presence within and throughout material form. Space, in other words, is rendered ultimately co-extensive with material form or structure; rather than a distinctive, intangible and hence energetically inaccessible and indivisible presence that in its truly infinite and indefinable depth, exceeds the limitations of local tangible presence both inwardly and outwardly. Infinity is rendered a property of matter, both in terms of attribute and ownership, not an inextricable quality of space within, throughout and beyond material boundaries. Division is conflated with diminution and multiplication with enlargement.

By contrast, atomistic reductionism does envisage an inner limit to the divisibility and diminution of material form, by way of a ‘fundamental particle’ or unitary figure as a definitive ‘absolute, independent singleness’ that can be summed with others to limitless amount. This ‘single-infinity’ view is ultimately accomplished through the imaginary exclusion of space (and hence infinity) from figural form rather than conflation of space with form. The resultant figure is a paradoxical ‘point-mass’ or ‘infinitesimal’ that in being dimensionless lacks shape and size and can hence only be ‘nowhere’ – a ‘something’ that arises from and can only in itself be ‘nothing’. This is the abstract ‘point’ from which both classical Euclidean and conventional non-Euclidean geometries as well as all conventional arithmetic are derived. It cannot, in actuality exist – it can only exist as an imaginary abstract conception. It is a ‘zero point’ represented as a ‘unity’.

The abstract conflation of division with diminution, and multiplication with enlargement, which is evident in both reductive (single infinity) and holistic (dual infinity) views of material form, hence leads to a dual diminution of resolution in interpretation of actual perceptions gained far from and close to a fixed locality. The effect is either to merge or to divide the mutual relation between distinct but not discrete formative tangible (= finite) and spatial intangible (= infinite) presences into ‘one’ or ‘many’ complete ‘wholes’ and ‘parts’, which in themselves can only be rigidly inert because movement necessitates the distinctive, permissive presence of space. Holistic abstraction leads to the perception of an infinitely divisible whole that is more than the sum of its individual parts as infinitely divisible wholes, whereas atomistic reductionism leads to the perception of a ‘finite whole that is equal to the sum of its finite parts, that are ultimately divisible into dimensionless point masses’.

This dual diminution of resolution leads to paradoxical contradiction and ambiguity, which afflict both dialectic (holistic ‘both-and in mutual contradiction’) and propositional (reductionistic ‘either-or’) logics and their incorporation into mathematical physics (Rayner, 2011a). A fixed ‘observation point’ or ‘'evolutionary point’ is created by an observer in the process of taking a finite measurement – a snapshot of the infinite depth and dynamic fluidity of Nature. This point defines the scale of infinity (i.e. ‘proximity’ in space/time) at which a material form, for example an apple, is perceived as a singular object and hence is ascribed the number one; one apple. This is in contrast to other observation points at which for example, the apple and its internal organization is seen as a collection of cells; a multitude of molecular, atomic or quantum interactions; or just one small appendage of a tree in an orchard in a world in a universe. Each of these would herald vastly different mathematical interpretations, ratios and relationships, based on the same locality of ‘somewhere in everywhere’. At some proximity the apple and its constituents could be perceived as an inert ‘particle’ in space/time, at others as a dynamic ripple or wave of space/time. But neither of these definitively partial views could, in themselves, yield the imaginatively combined (Rayner, 2011b) view of all form as flow-form – infinite, intangible, immoveable space throughout and beyond local dynamic tangible figure, as a vital inclusion of each in the other in the natural energy flow of the cosmos that we describe below as ‘place-time’.

All energetic matter therefore appears to comprise local particles when viewed at some scale of this dual-infinite depth, but as being in flux when viewed at another scale. Even the most apparently inert objects are intrinsically in dynamic relationship with their neighbourhood. In quantum field theory this flux is essential to maintaining the shape of particles interpreted as ripples of a quantum field.

Natural Inclusionality – an involution of abstract perceptions of space and boundaries

As natural dynamic inclusions of space, all forms are variably fluid flow-forms. Their boundaries are energetic configurations of space, not exclusions from space. When they move, space remains still yet infinitely permeating and inducing flow. Space, in this view, is the frictionless, zero viscosity ‘context’ that co-creatively induces the resistive, non-zero viscosity ‘content’ into form.

The first serious attempt by a physicist to introduce an involution in the usual perception of space and matter was that of Osborne Reynolds (1903). Reynolds described space as an hexagonal array of minute hard but weightless granules of aether, similar in diameter to that currently proposed for ‘quarks’, with matter flowing in and through its interstices, similar in width to the ‘Planck length’ proposed for boson ‘strings’ or ‘branes’.

Reynolds’ reasons for doing this are as relevant today as they were in his time: he wanted to account for ten manifest physical qualities of nature as follows:

(1) The non-resistance of space to movement of tangible material form, such that the orbital velocity of planets and moons in the solar system, for example, remains virtually constant for thousands of years.

(2) The ability of space to allow the transmission of light from source to receiver.

(3) The gravitational attraction of material bodies for each other as an inverse function of distance.

(4) The cohesiveness of matter after breaking and fusing.

(5) The elasticity of matter as in a coiled spring.

(6) The variable frictional resistance of matter to the sliding movement of other matter, which can be overcome by sufficient pulling or pushing influence.

(7) The viscosity of matter, which allows a constant rate of flow down a gradient.

(8) The electromagnetic properties of matter, as revealed by the effects of static electricity.

(9) The capacity for a volume of fluid to equilibrate to uniform pressure.

(10) The chemical reactivity and decomposability of matter, as revealed in combustion and electrolysis.

This combination of qualities, Reynolds recognized, could not adequately be explained in terms of the conventional representation of the universe as passive empty space populated by mobile material objects driven by extrinsic force. But he thought they could be explained if the universe could be likened to a mixture of perfectly spherical sand grains and water – with the grains arranged as an hexagonal lattice of ‘space’ and ‘matter’ flowing freely in the interstices between the grains. Such a biphasic system would, he contended, account purely mechanically for all the known properties of light, electricity and gravitation. In particular, he attributed light waves to the ability of such a system to lock into a rigidly close-packed configuration under pressure due to its volumetric dilatancy (Reynolds, 1885). A similar phenomenon, where interstitial spacing is critical, is nowadays recognized to play a role in the spread of a ‘shock wave’ through a high speed traffic flow that piles up in response to local impedance (Sugiyama et al., 2008) . Years before de Broglie (1924), Reynolds (1903) declared the motion of matter to ‘have all the character of a wave in the medium; and that is what the singular surfaces, which we call matter, are – waves. We are all waves.’

Such a complete inversion of standard perceptions of the relationship between matter and space, in which space as an intangible, zero viscosity presence is vitally involved in, whilst not resisting the flow of form, anticipates in some ways, the evolutionary concepts of natural inclusionality developed by Rayner (2011a). It may go too far, however, in its use of definitive logic and choice of metaphor to render space and matter respectively as mutually excluding ‘particles’ and ‘waves’.

The notion of ‘space’ as having a limitless pre-existing structure as a crystalline array of hard aether ‘grains’ and ‘matter’ as flowing ‘space’ amongst but not included in the grains is difficult to reconcile with human sensory experience (i.e. ‘evidence’) or consistent reason. But it does help to draw attention to the nature of the problem of understanding the dynamic reciprocity of figural ‘message’ and contextual ‘medium’ as distinct but mutually inclusive and co-creative presences, and the radical involution in common perceptions that is needed to clarify this. To exclude one or other of these presences from consideration or to conflate them together within a seamless ‘whole’ or ‘unity’ cannot make sense of the manifest physical qualities of nature identified by Reynolds. In this paper we explore how these qualities could arise not through space alone or even as ‘space-time’ having an explicit, pre-existing structure, but through having an implicit, receptive presence and influence, that enables variably rigid and fluid tangible form to emerge co-creatively through its mutually inclusive relationship with energy in what we call ‘place-time’ (see also Rayner, 2008, 2011c, d). We suggest that space therefore has a far more important role as a co-creative, zero-viscosity contextual medium in evolutionary processes than has so far been recognized in current theories of relativity and quantum mechanics (see below)

‘Natural inclusionality’ is a new philosophy and fluid boundary logic of self-identity and ecological and evolutionary diversity and sustainability (Rayner, 2011a). It is intended to supersede the abstract rationality that has dominated human thought for millennia, based on definitive logic that can only apply to inert material systems that are unknown to exist anywhere in Nature. Whereas abstract rationality treats space as empty distance between, occupied by or outside completely definable tangible material structures or objects with discrete boundary limits, natural inclusionality recognizes space as a limitless, indivisible, receptive (non-resistive) ‘intangible presence’ vital for movement and communication. This recognition of space as a natural presence, instead of an abstract geometrical construction, allows all form to be understood as flow-form, distinctive but dynamically continuous, not singularly discrete. It enables the simple move from regarding intangible space and tangible boundaries as mutually exclusive sources of discontinuity and discrete definition to mutually inclusive sources of continuity and dynamic distinction. Natural, intangible space is included throughout and beyond all tangible figural forms as configurations of energy, whether as massy bodies or mass-less electromagnetic radiation.

From Abstract Artifact to Natural Inclusion: Bringing ‘Schrödinger’s Dead Cat’ Back to Entropic Life

In limitless space, as the proximity of an observation point to a definitively measured locality expands or contracts, so the measurement begins to show ‘dual-diminishing resolution’. At lengthening range, the locality progressively recedes ‘out of sight’ to a fixed ‘vanishing point’ at which its boundary is irresolvable and indistinguishable from its surroundings. At shortening range, the locality progressively looms large enough to become perceived as boundless and to surround the observation point. Moreover, what appears motionless at long range, over short duration or from within its depth may be revealed to be dynamic at short range, over long duration and from outside. This corresponds with our common experience of visual perspective, whether through our naked eye or through the magnifying lenses of our telescopes and microscopes and depending on whether we are situated within or outside a moving container. Although the locality does not actually increase or decrease in size, mobility or occurrence, our perception of its size, mobility and occurrence does change. Moreover, it changes in a way that is only explicable if space is neither a tangible presence (some kind of pre-existing ‘fabric’) nor an absence of presence (some kind of void nothingness or blankness that nonetheless comes between things), but an intangible presence (frictionless, omnipresent ‘no-thingness’ as 100 % of everywhere, without limit).

In conventional Quantum Physics, the ‘Copenhagen Interpretation’ of the ‘Schrödinger’s Cat’ thought experiment (in which a cat is placed in a sealed container with a vial of cyanide that may or may not get broken by a random event) considers that the quantum constituents of matter do not assume a quantifiable value, unless measured by a conscious observer. Here, the cat is confined in ‘suspended animation’ between life (wave function) and death (inert particle) but cannot be defined as alive or dead without opening the container. The constituents exist in a transient energetic flow-form, and not as a ‘unit’, discrete from the surrounding space and its neighbours – its context; unless perceived to be so. This fluidity of form, and the acknowledgement of the importance of the observer, already sits at the core of quantum field and particle physics theories (Wimmel, 1992).

The ‘animated suspension’ of natural, co-creative flow-forms is readily observable in nature, from subatomic to galactic scales and not least in embryonic, adult and decomposing cats (Rayner 2009, 2011b). Flow-forms are implicit in notions of Dual Infinity (Leibniz, 1695), relativity (Einstein, 1954) and Quantum Field Theory (Wimmel, 1992), but unlike these acknowledged abstract, reduced representations, find their actual ‘place-time’ in the continuity of space and energetic distinctions of dynamic boundaries/interfacings. They are dynamically distinct but not definitively discrete.

The fact that from a suitable observation point, flow-form(s) can, for short or long durations, appear discrete and stable, has led them to be ascribed definitive boundaries, relationships using the system of exact numbers and rigid geometrical form adhered to by conventional mathematics. But the fact that they can also be viewed in another way, in which they lose local definition has led to the notion of Wave Particle Duality, which suggests that all particles also have the characteristics of waves, including atoms and molecules, as well as photons and elementary particles (Eisberg & Resnick, 1985). Hence, if a particle or a wave is recorded, the necessarily hidden reality of the infinite depth is tacitly ‘cut’ as baseline assumptions are made, boundaries are imposed, and an approximated ‘closed’ system is created. The same tacit cutting of space by abstract definition is evident in Einstein’s formulation of relativity theory, as evident in the following statement:

“When a smaller box s is situated, relatively at rest, inside the hollow space of a

larger box S, then the hollow space of s is a part of the hollow space of S, and the

same “space”, which contains both of them, belongs to each of the boxes. When s is

in motion with respect to S, however, the concept is less simple. One is then inclined

to think that s encloses always the same space, but a variable part of the space S. It

then becomes necessary to apportion to each box its particular space, not thought of

as bounded, and to assume that these two spaces are in motion with respect to each

other.” (Einstein, 1954)

How, one might ask, can you move (‘cut and paste’) a box of space without dynamically bounding it? How can you dynamically bound a box (or, more aptly, sphere – see below) of space, without a gravitational centre? How can you distinguish the space inside a box from the space outside and still transmit light from an external source through the interior of the box at a constant ‘speed’, regardless of the movement of the box relative to its surroundings? How can you sub-divide a continuous, frictionless, intangible presence? The reality is that you can’t. This is because a presence that has no resistance can neither be cut nor resisted by a tangible frame. It is inescapably present throughout and beyond the boundaries of tangible figures. A tangible frame is an inclusion of and is included in

space but the frame is not the space. The tangible frame can move (or be moved) and be cut, but not the space. When the frame moves the space stays where it is: in relative terms by remaining still space permeates freely through the frame, the frame does not cut through the space. Moreover, if the frame is to move without being forced to do so by a force situated somewhere outside of it, it must have the capacity for movement within itself, i.e. the frame is itself a manifestation of energy, not inert structure—it is a variably fluid ‘framing’, not a permanent, absolutely rigid ‘framework’. This tangible ‘framing’, or ‘dynamic interfacing’, has to be present for form to be distinguishable in a feature-full cosmos, but it can neither ‘occupy’ nor ‘exclude’ the space that it includes and is included in (Rayner, 2011a).

Tacit ‘space-cutting’ is also implicit in the inclusion of ‘entropy’ in calculations by a variety of disciplines. In Thermodynamics, for example, entropy is essential to predicting the occurrence of dynamic equilibrium, a balance created as a result of energetic exchange. Rudolf Clausius originally described entropy as an increase in disordered energy, tending to a maximum and never depleting within an isolated system (Clausius, 1865).

Since, in space as a continuous, intangible presence, a system can never truly be isolated (cut free from the space it includes and is included by, see Rayner 2011a), what is abstractly perceived as ‘the external environment’, the universe, must be regarded as a dissipative sink. Correspondingly, entropy has more recently defined as the energy that is no longer available for work in a system, a reservoir of ‘useless’ energy accumulating as the ‘useful’ energetic work is exhausted and equilibrium is reached (Moore, Stanistski, Jurs, 2005). William Thompson in 1851 hence proposed that eventually, in a finite universe, with a smallest and a largest scale, the ever-increasing entropy (energy that is ‘no longer available for work’) will herald a descent into total amorphous disorder, the ‘heat death’ of the universe. This, and variations thereof are still commonly held views in modern science (Adams, Laughlin, 1997).

Nonetheless, in Thermodynamics, a reversible process is acknowledged never truly to reach equilibrium and therefore become completely inert. A point of ‘no net change’ is attained, but this remains dynamic. The ‘useful’ energy can hence never be fully depleted; it is more that the potential for change is reduced locally, as the energy flow becomes more stable, maintaining localized shape. This local stability of shape is however always vulnerable to external stimuli, that may create new potential for change, thereby introducing more ‘useful energy’.

In an infinite system, entropy could more simply be acknowledged as energy transferred to the infinite depth beyond the perceptually imposed boundaries of the observed system. While it is no longer perceived as useable by the observed system, it is still an inclusion of infinite flow; rather than a descent into either chaos or uselessness. Molecular, atomic and sub-atomic relationships remain dynamic, as demonstrated by their detectable bond resonance, and as implied by Quantum Field Theory; so do planetary and galactic orbits. This could also imply that the inevitable production of entropy, as a result of continual change is not finite either – constrained to the locality described by the mathematics, but even when dynamic equilibrium has been established.

Entropy, when viewed in this infinite context, would seem to be more in accordance with the flow-forms that can be observed in nature (Rayner 2009, 2011b), as theoretically required by Quantum Field Theory (Wimmell, 1992), and even the bond-resonances that we can readily observe in a dynamic equilibrium. It takes on a role in the continual reconfiguration of energetic flow, rather than contributor to the ultimate ‘heat death’ of the universe. ‘Schrodinger’s Cat’ ultimately reconfigures into life, not ultimate corpse.

Fractal Geometry – A ‘stepping stone’ to natural inclusional flow geometry, or a ‘stumbling block’ stuck in abstract definition?

‘Fractal geometry’ has widely been heralded and taken up by varied fields of research in recent decades as the ‘geometry of nature’ and ‘path to infinity’ (Mandelbrot, 1982; Stewart et al., 2004). In some respects this may be true, and there is no doubt that the study of fractals – as figures created by the extrapolation of recursive equations, which exhibit infinite self-replication – has shown both the limitations of integral Euclidean geometry and importance of iterative processes in understanding the generation of natural flow-form. There is at least a superficial similarity to natural branching and nested structures. Some examples are shown below.

Fig. 1. Pythagoras Tree. (Wikipedia, 2005)

Fig. 2. Menger Sponge. (Baserinia, 2006).

Although claimed to transcend Euclidean geometry (Mandelbrot, 1982) fractals are actually constrained by their conventional arithmetical formulation, to the surface (Fig. 1), or within rectilinear sets (Fig. 2) of Euclidean planes. They explore the effect of infinite subdivision into mutually exclusive spatial and material localities within a Euclidean container as a whole – and so effectively cut space by boundary definition. But they do not and cannot take into account the continuous space throughout, beyond and inaccessible to figural boundaries. They do not and cannot explore curvature, except, as in calculus, through approximation to infinitesimal scales. They are, quite literally, fractionations of, not expansions from Euclidean dimensions. As implied by the above ‘Menger Sponge’ example, to become truly natural inclusional flow form, they need to include somewhere vital, and this somewhere is everywhere - space.

Natural Inclusional Flow-geometry: the dynamic spatial relationship between curved and linear form in the origins of variably resistive and mobile boundary configurations: crystals, channels, pulses and circulations

With the recognition of natural, energetically inaccessible intangible space as a vital inductive influence, which tangible energetic form both has an affinity for and cannot be isolated from, emerges the possibility of a natural flow-geometry that both transcends and transforms the abstractions of conventional mathematics. ‘Zero’, as a local point of space is brought from outside to inside tangible form as a receptive centre for energy to circulate around, but not reach. This bringing from outside to inside is what constitutes the involution in perception of natural form as ‘place-time’, which is implicit in natural inclusionality. It offers a simple new way of understanding observable geometric configurations in Nature that arise from the dynamic spatial relationship – not irreconcilable disjunction – between curved and linear form, which allows a non-paradoxical synthesis of the findings underlying relativity and quantum mechanics to be made.

All things being equal, the natural form of fluid droplets and bubbles is spherical. The sphere is that configuration of energy/matter in dynamic relationship with space in which the ratio between surface area and internal volume is minimal. Any departure from the infinite radial symmetry of spherical form increases the surface area to volume ratio, and hence the capacity to absorb or dissipate energy from or to outside. Also, as the internal volume of a sphere decreases, the ratio increases towards infinity at ‘zero’ volume. The energy/matter invested in the surfaces of a set of small diameter spheres is correspondingly much larger than that in the surface of a larger sphere of the same volume as that of the small diameter spheres combined.

The sphere can correspondingly be regarded as the primary fluid form from which all other configurations of energy in natural flow geometry can be derived by elongation and/or close-packing. These include linear configurations in curve-faced cylindrical form and plane-faced, close-packed hexagonal and tetrahedral form of the kind familiar in rigid, crystalline and frozen structures. The latter structures correspond with the ‘locked’ configurations of dilatant fluids under pressure recognized by Reynolds (1885), and their resistive quality sets up the potential for oscillations between stalled and flowing movement of the kind illustrated by the spread of a ‘Mexican wave’ around a sports stadium produced by people alternately standing up and sitting down.

Loss of spherical symmetry, associated with uptake of free energy, allows elongated forms – ellipses, tubes, channels, branches and spirals to emerge and grow, as in the roots and shoots of plants and the shells of snails. Uptake of heat allows expansion of surfaces in less condensed, more energetically relaxed form that are less liable to break symmetry. The spores of some fungi illustrate this point well. At high temperatures they germinate by swelling isotropically to form ‘giant cells (Hanlin 1994). At moderate or low temperatures they germinate to produce a protoplasm-filled tube, called a hypha, which elongates and produces branches and anastomoses from parabolic apical growing points to form a collective organization known as a mycelium. As shown in Figs 3 and 4, both in the way that it explores unpredictable, heterogeneous environments and in the way it can fuse with or interfere with others, this collective organization graphically illustrates the interdependent relationship between polarized (linear) and circular flow-form in a living organism.

Fig. 3. ‘Fungal Foraging’. A fungus finds an ‘oasis in a desert’, by fluid-dynamically spreading and narrowing its energetic focus. The wood-decaying fungus, Hypholoma fasciculare, has been inoculated into a tray full of soil on a block of wood (‘starter’food source), with an uncolonized wood block (‘bait’ food source) placed some distance away from it. Distinct stages are shown in the radial spreading of the fungal mycelium from the inoculated wood block, followed by the redistribution and focusing of its energy in one direction following upon contact with the bait. Similar fluid dynamic patterns of gathering in, conservation of, exploration for and redistribution of energy supplies are found throughout the living world, from subcellular to ecosystem scales of organization (From Dowson et al., 1986; see also Rayner, 1997).

Fig. 4. Early (left) and late (right) stages in compatible (upper) and incompatible (lower) pairings between mycelia of the wood-decaying fungus, Phanerochaete velutina grown from wood blocks inoculated into soil, showing the formation of persistent and degenerating channels across the zone of overlap. (From Dowson et al., 1988).

Continuously curved spherical surfaces cannot, on the other hand, be derived from discrete linear components – not even contiguous spheres of equal diameter - without reducing these to infinitesimal sizes – and even then, only as a convenient approximation, useful in calculus. This is clear from the fact that the surface area of a sphere is 4πr², where π is a well-known, so-called ‘irrational’ number. A truly continuous curved surface is possible only if it is itself an inert amorphous fudge – begging the question of how and why this could come into being – or its ingredients are fluid, capable of flowing into and out from others. In either case space too must be continuous (incapable of being cut or defined). Here, it can be recognized that the boundary of a fluid sphere is not a discrete limit that isolates inner space from outer space, but a dynamic interfacing between inner and outer realms across which space, as intangible presence, is continuous.

The spheres of flow-geometry are therefore not self-contained, locally discrete entities, as in conventional Euclidian and non-Euclidian geometries, but energetic configurations of local space within ‘somewhere’ distinct as an inclusion of non-local space everywhere (Shakunle & Rayner, 2008, 2009). Here it may be recalled that Euclidean geometry is the abstract geometry of zero-dimensional (size-less) points, one-dimensional (breadth-less) lines, two-dimensional (depthless) planes and three-dimensional solids (self-contained volumes). Its figures are used to represent definitive tangible structure and yet can only actually represent the intangible presence in the core of tangible form because it is impossible to reach zero size, breadth or depth without removing the tangible presence. The same applies to the so-called ‘non-Euclidean’, Riemannian and Lobachevskian parabolic and hyperbolic geometries of curved surfaces.

The reality is correspondingly that abstract Euclidian and non-Euclidean points, lines and planes/curved surfaces can consist only of intangible presence, not tangible presence! By the same token, it is impossible to drive or rotate a solid body from or around a solid fixed centre. The central ‘still’ point, axis or plane of symmetry of any bodily form can only consist of intangible presence, with correspondingly zero viscosity. In effect, conventional mathematics and its discontinuous underpinning logic thereby treat ‘1’, as a ‘unit of tangible presence’, as if it is ‘0’, a vanishing point of intangible presence. They literally attempt to construct ‘one thing from nothing’ and then to sum an infinite number of these one things up into an infinite ‘whole’ as a ‘one’ that is also ‘many’, whilst discounting the very presence that truly is infinite and indivisible, at all scales (space).

This difficulty can only be resolved realistically by accepting that in Nature, tangible and intangible presences are distinct but mutually inclusive. This is the point recognized by the flow geometry of natural inclusionality. Here, space and boundaries are regarded as mutually inclusive sources of continuity and dynamic distinction with variable connectivity, not mutually exclusive sources of discontinuity and discrete definition, as in standard Euclidean and non-Euclidean geometries. So far, the only mathematical formulation explicitly to accept and incorporate this natural inclusion of omnipresent space in and throughout local figural form is the ‘transfigural mathematics’ introduced in 1985 by Lere Shakunle (see, e.g. Shakunle, 1994; Shakunle & Rayner, 2007, 2008, 2009).

Natural inclusionality effectively transforms the fixed structural frameworks and boundaries of standard Euclidean and non-Euclidean geometries into fluid framings of omnipresent, non-local intangible space everywhere, within (intra-), throughout (trans-), between (inter-) and beyond (extra-) local tangible energetic form (cf. Shakunle & Rayner, 2009). This opens the possibility of a dynamic, co-creative, mutually inclusive relationship between internally and externally situated non-resistive (and hence receptive) intangible spatial presence and locally situated, tangible energetic presence. Of central significance to this interplay is the fact that tangible energetic presence can only circulate around, it cannot occupy the intangible core of spherical flow-form. Moreover, the more closely it circulates around this core, the greater will be the intensity of energy invested in its surface area relative to diameter and the stronger its internal coherence needed to restrain its expansion. As in a vortex, such increased coherence can be provided by enhancing the rate of circulation in response to a steepening pressure gradient from outside to inside. Alternatively, it may be possible by reducing the heat content of the system.

What emerges from these considerations is a picture of a fluidly bounded sphere as a ‘balancing point’ of inaccessible, intangible, non-resistive space configured by a tangible, resistive rotational flow (‘spin’ or ‘swirl’) of energy – in other words, an energetically surfaced cavity or ‘local sphere of spatial influence’. This spherical swirl around a local centre of zero pressure, is potentially both an acquisitive ‘sink’ and a generous ‘source’ of energy flow from and to others in its dynamic neighbourhood.

Of special interest are the possibilities that arise when the ‘local spheres of non-local influence’ of natural inclusional flow-geometry overlap with and dissociate from one another to yield potentially complex and varied patterns of flow, counter-flow and generative and degenerative interference. These possibilities can be visualized through a consideration of the figure, known as a ‘mandorla’ or ‘Vesica Piscis’ produced by two spheres each overlapping to the centre of the other, as depicted in the sonic interference pattern shown in Fig. 3, which shows a striking resemblance to the formation of bridging channels between fungal mycelia shown in Fig. 4.

Figure 5. Sonic interference pattern in the vesica piscis between two sets of annular ridges and troughs (from

This figure provides the foundation for deriving the configurations known as ‘breathing points’ in Lere Shakunle’s ‘transfigural mathematics’. Fully overlapping, the spheres coincide as ‘one’, but as each is drawn out to one side from the other a convex lens-shape forms, which balances the ‘tension’ set up as each reciprocally gains and loses energy from the other. This is illustrated in one plane using two hemi-circles in Figure 6.

Figure 6. Coil and recoil – how a sphere of spatial influence reaches to and draws in from receptive centres of zero potential (zeroids) in its neighbourhood. (From Shakunle & Rayner, 2009)

In this case, as the ‘upper’ hemi-circle moves ‘horizontally’ to encompass the ‘positive’ to its ‘right’ (using conventional mathematical representation), a ‘deficit tension’ (‘minus’) is set up to its left, resulting in a ‘clockwise’ spiral inflowing countercurrent towards the central focal point-influence, which has meanwhile been stretched into a ‘line’. Simultaneously, a ‘surplus tension’ (‘plus’) is set up to its right, which results in an ‘anti-clockwise’ spiral inflow. In Shakunle’s ‘transfigural mathematics’, the ‘horizontal S-shaped, or sigmoid, figure that results from this ‘left-to right flow’ and its countercurrents is called an ‘alpha fold’. By the same token, the reciprocal sigmoid figure produced as the ‘upper’ hemi-circle is drawn into the ‘negative’ to its ‘left’ is called an ‘omega fold’. The superimposition of alpha folds and omega folds as seen along a ‘horizontal line’ in one plane yields a two-fold figure, known as a ‘zeropline’, which is one of several kinds of zero spirals (Fig.7).

Figure 7. Dynamic configuration of a ‘breathing point’ along a horizontal axis to yield a ‘zeropline’ (shown here, for simplicity, omitting details of folding through the core, but depicting its intangible balance point-centre as a dark spot) (From Shakunle & Rayner, 2009).

If the same geometrical process is repeated along a ‘vertical line’, a four-fold, ‘breathing point’ is revealed, corresponding with a tetrahedral arrangement, of the kind familiar around a 4-valent carbon atom (Figure 8).

Figure 8. Dynamic configuration of a ‘breathing point’ along both horizontal and vertical axes (shown here with more elaborate folding through the core identity included).

(From Shakunle & Rayner, 2009)

Using the zeropline to represent the basic dynamic configuration of the breathing point (as both what can be thought of as a ‘point-channel’ – a point that expands into a channel – and a ‘line-channel’ – a channel that condenses into a point), the way a ‘flow-channel’ can form as a linear series and become organized into gravitational bodies is illustrated below by coupling one breathing point into another (Figure 9).

Figure 9. Coupling of neighbouring zeroplines to yield a linear series. (From Shakunle & Rayner, 2009)

Such a series can vary enormously in length, and this length, along with its central axis of internally located space, could either be stretched out into elongate form or coiled upon itself into a compact form, like a snake, caterpillar, worm or millipede. But ultimately, every ‘segment’ along this length will retain its local identity in dynamic relationship with its neighbours and only be capable of communication with members of other, non-connected flow-lines through the intervening space between them. Moreover, every distinct, though not spatially isolated, flow-channel will have a local-in-nonlocal centre of inductive spatial influence and hence manifest as ‘massy body’. In conventional physics such bodies may be alluded to as consisting either of ‘matter’ or ‘antimatter’, depending on which way around the flows and counter-flows are arranged.

A radically different arrangement of flow and counter-flow situation occurs in flow-circuits where the axis of internal space is de-localised into the configuration of a toroidal annulus or ‘ring’ through the formation of what has been called an annular ‘superchannel’ (Shakunle & Rayner, 2007, 2008, 2009). The way in which this possibility arises can be seen from Figure 10.

Figure 10. ‘Flow and Counterflow’ (From a painting by Alan Rayner, Oil on canvas, 2008). The continuous ‘superchannel’ of transfigural geometry spatially expands the discrete, one-dimensional, purely material line comprising contiguous but spatially discontinuous and dimensionless numerical point-masses upon which classical and modern mathematics are founded. Each discrete point is transfigured from a static, lifeless entity into a dynamic, breathing identity as a local informational (electromagnetic) sphere of non-local spatial influence, a ‘breathing point’. The breathing points reciprocally inspire from and expire to their immediate neighbours, creating a double helical energy flow through coupled numerical neighbourhoods of three.

The formation of such circuits enables ‘current’ to flow freely and reciprocally through a double helical arrangement of alpha and omega folds such that their inductive influences mutually balance one another, rather than being ‘weighted’ one way or the other (as in linear flow-channels). All kinds of natural and human-made flow-circuits may be based on this principle. Biological examples include the flow-networks of fungal mycelia, central nervous systems of animals and vascular systems of plants and animals. In organic chemistry, the stability of aromatic rings compared with aliphatic chains is attributed to the hybridization and delocalization and of s and p and p electron orbitals respectively into sigma bonds and pi clouds. Superchannels may also be the fundamental flow-form in which electromagnetic radiation, which has no detectable ‘mass’, occurs and emanates from or gives rise to ‘equal quantities’ of what are spoken of as ‘matter’ and ‘antimatter’ (Shakunle & Rayner, 2009).

From Hierarchy to ‘Lowerarchy’: a natural inclusional interpretation of gravitationally inductive geometric and numerical organization and evolution

Based on the above considerations, natural inclusionality enables new interpretations to be made of cosmological organization and evolution. These interpretations bring together some insights from Reynolds’ granular aether, relativity, quantum field theory and fractal geometry, whilst crucially NOT imposing definitive boundary limits between, or attempting to unify formative energy and permissive gravitational space, but treating these tangible and intangible presences as distinct yet mutually inclusive.

An involution becomes possible from paradoxical and inconsistent perceptions of rigidly hierarchical organization, based on definitive exclusivity and extrinsic force, to an appreciation of the inductive coherence around central zero points and axes of infinite intangible presence. This coherence arises from the intrinsic depth of space and intrinsic dynamic of fluid boundaries as mutually inclusive presences. The need for physical ‘law enforcement’ within rigidly three-dimensional confines is removed. The inductive influence of spherical and cylindrical ‘holes’ subsumes the hierarchical power structure of cubical ‘wholes’. Receptive hollowness and formative fluidity subsume solidity and rigidity.

With hierarchical perception the question arises of why gravity has such a relatively weak influence, when compared to and when acting upon fundamental forces that act at smaller scales (as evident from the fact that you can use electromagnetism to lift an object, against gravitational influence). The issue has had many mathematical explanations applied to it, mainly involving the need for one, or a series of dimensions in time and space, extra to the four predicted by General Relativity. These extra dimensions have not yet been observed experimentally (Randall & Sundrum, 1999). It is one of the core questions for the Standard Model of Quantum Mechanics, as the model does not conclusively include for gravity or explain the origins of gravitational influences (Novaes, 2000).

We cannot assume uniform characteristics to the greater scale inductive influences that comprise ‘Gravity’, amongst the evidence of other, unexplained phenomena in Space; such as dark matter and black holes. These signify that there is as much variety and complexity to the greater scales of infinity, as we observe in the smaller scales.

The need to account for this infinite complexity / surface area in both spatial depth and energetic scale, may provide an ever-receding horizon for any reduced solution to the search for a mathematical ‘theory of everything’ or Unification Theory. Natural Inclusionality implies the organically co-creative, inclusive nature of each inductive influence within and between the other; infinitely varied influences, within the larger overall sphere of influence of a larger object and beyond to infinity; each distinct, but not discrete from an infinite flow / counter-flow context, permeated and induced by the receptive depth of space. As these influences would not be evenly spread or fixed across the heterogeneous Earth, this would imply an irregularly shaped, flow-form cumulative gravitational field, as has been experimentally observed by the GOCE Satellite (Fig. 11).

Figure 11. GOCE Satellite Mapping of Earth’s Gravitational Field. ESA/HPF/DLR, 2011.

A very simple metaphor can be used to describe how this co-creative accumulation or inclusion can occur. As a yacht travels through a sea-channel, there are a number of perceptual observation points, for example:

    The Propeller: As the hull displaces the local water and the propeller rotates, the molecules of steel and water interact in a perceptually ‘chaotic’ way. The local effects of the local inductive influences upon the molecular level are, at a relative scale, much greater than the local effect of the larger scale waves that the smaller influences none-the-less embody.

    On Deck: On deck, foam can be observed at the point of contact, between the boat and the water, but the perceptual ‘chaos’ at observation point 1 is much less relevant. The effect of this local chaos is not additive but ‘inclusionally cumulative’ observed as a steady wake, leading from the back of the boat.

    The Dolphin: A dolphin employs the wake of the boat to provide propulsion in the water. The perceptual chaos at observation point 1, has become an inductive influence, ‘force’ or even ‘orbit’ at a larger scale.

In this way, a dual-infinite, naturally inclusional reality allows the properties of gravity to be understood and interpreted more comfortably, in the limitless dimensionality of receptive space, unconfined conceptually by Euclidean three-dimensionality or the need for additional dimensions of time and space.

The relative ‘weakness’ of gravity when acting upon the smaller scale also provides a further insight into the property of ‘dual-diminishing resolution’ described earlier. This property is dependent upon both scale and proximity in space, as well as perception. Resolution is diminished in a relative sense, but the inductive influences are still mutually constituent. It may be that the differences in reconciling the relative strengths of the fundamental forces is purely an artifact of attempting to account for many different magnitudes of inclusive, space-permeated energetic flow within unacknowledged infinite scale/depth, using a scale/depth limited, finite system of definitive mathematics.

Beyond the horizon to zero fractals, and the 1/0 diminution

The transformation of the Menger Sponge into intricately nested natural inclusional spheres of influence is, in effect, simply accomplished by fluidizing the rigid 3-dimensional boundary-definition via the incorporation of receptive space within, throughout and beyond the figure. The resulting figural flow-form may be called a Zero Fractal; Figs 12 - 14 show examples from nature, which demonstrate the Zero Fractal in context. In all of the below images, the zero points / particles in resolution can be acknowledged to comprise, and co-create within, further ‘unseen’ detail / divisibility at both close and far depth – they exhibit dual diminution of resolution.

Fig. 12. Simulation of molecular dynamics of DNA Molecule. Singh et al. 2010

Fig. 13. The Triangulum Galaxy (NASA/Swift Science Team/Stefan Immler, 2011)

Fig. 14. These two images of a three-light-year-high pillar of star birth demonstrate how observations taken in visible and infrared light by the Hubble Space Telescope reveal different perceptions of an object. (NASA, ESA, and M. Livio and the Hubble 20th Anniversary Team, STScI, 2010)

Dependent upon both the observation point and the duration of observation, dual diminution of resolution allows us to perceive the energetic configurations as being divided into infinite zeroidal patterns – particles, or perceived as continual flow – waves, membranes. This applies to all the energetic particles / waves, whether perceived as matter, electromagnetism, light and/or other force transmitting particles described by quantum mechanics.

“When you detect a photon, you can say where, when, and with what frequency it arrived, but before the measurement, these parameters are undefined. The photon's existence is embodied in a wave function, which gives the probability of measuring the photon at any time, place, and frequency. The wave function for a single photon is usually a "wave packet"--nearly zero everywhere except in a narrow range of space and time. But as long as you don't detect the photon directly, you can manipulate its wave function into any complicated shape, in theory.” (Kolchin et al, 2008)

Fig. 15. Light shaping. Shown here are the measured oscillations in a multi-photon light pulse (Kolchin et al, 2008)

The Zero Fractal consists of non-linear, indefinitely divisible, inclusive, transfigural zeroids configured as energetic spheres of influence, but always fully and freely permeated by continuous space. They are not necessarily interconnected in a rigid matrix or a linear wave, though they may appear to be so at the Observation Point, yet still assume co-creative, dynamic flow-form through interplay with other, proximal zeroids. The infinitely deep, receptive nature of space plays a key role in inducing this flow of energy, as described previously.

Fig. 16: Scanning Tunneling Microscope image showing the individual atoms making up this gold (100) surface. Reconstruction causes the surface atoms to arrange in columns several atoms wide with pits between them (Image originally created by IBM Corporation)

Figure 16 shows a ‘snapshot’ of this flow-form at atomic level in Gold (100) under STM. At once, both the zero fractal shape can be observed, as can how a perceptually rigid structure can be perceived from the interrelation of proximal zeroids, closely packed, but always permeated by continuous space. It is apt to investigate and to engineer this instantaneously observed digital, rigid-lattice structure, but the presence of further space-permeated flow-form, inclusively, between the Cosmological and Quantum Observation Points and beyond, must also be acknowledged. This suggests that linear and digital relationships are perceptual and emergent at scale, rather than intrinsic properties throughout infinite space and energetic flow-form.

All flow-form begets inductive influence, all inductive influence begets flow-form; and each requires the permeation and receptivity of limitless space. Natural Inclusionality would suggest an infinitely deep, indefinitely surfaced, energetic reality, comprising simultaneously interrelating, diminishing spheres of influence – emergent particle / wave flow-forms, permeated throughout by the infinite depth of space. This depth and detail extends not only across the surface area of perceptual mass, but across energetic and inductive surface area, and beyond, ad-infinitum. No element within this infinite flow-depth need be or can be extracted from the infinite context.

At a particular Observation Point, a perceptual numerical scale can be ascribed, a smallest and largest scale selected, and therefore a ‘Horizon of Diminution’ created – the ‘edge’ of a perception or a mathematical / theoretical system. It is not fixed, being dependent upon both the Observation Point and the resolution available to the perceiver, but it is ever present within perception of infinite depth. Hence, the ‘Horizon of Diminution’ is a characteristic of perception, not of space, energy or matter. Each zero ‘point’ within the images above can be described as a ‘Horizon of Diminution’, as the depth of flow and space within is not perceptually obvious – they are perceived as ‘whole’ units, singularities, that can be described by the number 1.

As it is a ‘mobile’ characteristic of perception and understanding, rather than of energy, space or matter, a new horizon – ‘whole’ unit or number 1, and hence entire derivative, abstract numerical scale, can ‘appear’ at any scale of infinity ‘in theory’. We can approximately transcend these linear, Euclidean scales and relationships via our use of ‘irrational’ constants e.g. Pi, although this always entails losing resolution due to ‘rounding’ up or down and thereby ‘cutting’ the infinite depth of space.

The dual diminution of resolution provides the basis for the Euclidean perception of a ‘whole’ unit, discrete from the surrounding space. The number 1 could be described as the smallest perceivable ‘whole unit’ of energy/mass; and 0 as the absence of presence – space. In contrast, a naturally inclusional flow-form necessarily includes both 1 and 0, consisting of infinitely-deep, continual, receptive space, bounded by indefinitely-diminishing, energetic spheres of influence. And so, the smallest flow-form, or flow-length, perceivable as a ‘whole unit’ always incorporates both energy and space, both 1 and 0.

This is the ‘1/0 Diminution’ – the abstract exclusion of 0, and hence infinity, from 1 in perception only; a clear paradox, and yet the fundamental basis of our mathematical and scientific paradigm. The ‘smallest perceivable flow-length’ – 1 must in actuality include infinite ‘limitless space’ – 0, with infinite ‘potential of dissolution’. To define a number as ‘whole’ – 1, the non-number – 0, and hence infinity, must be ‘abstracted’ from the dynamic boundaries of the number, thus yielding only linear and linear-recurrent relationships. These linear numerical relationships and systems are therefore ultimately scale-limited by perceptual abstraction.

The number 1 as a singularity cannot be divided into further ‘whole’ parts, and therefore can have no depth or structure, no dynamic flow-form, and hence no mass. It is a paradox, a perceptual ‘horizon of diminution’ in the infinite depth and permeability of energetic flow and space, which can be given life and form by acknowledging the infinite flow-depth within.

Within Reach

Here, I am

Calling from within you

To all

Who call

From my heart’s desire

To be

Full filled

With nothing less

Than nothing more

Reaching everywhere


Beyond each lingering moment

Of transient life

Spinning around


Turning inside my dear


With nowhere further to go

Than somewhere deep inside

Without walls

Within walls

Beyond eye shot

Beneath ear shot


I cannot be pierced

Not even by the fiercest


Mounted from a place

Without my consent

By those Hell-bent

On reaching my infinite depth

Such a vain, hopeless venture

Not the spirit of adventure

That brings you close

Within my reach

Beyond your grip


Alan Rayner

2nd January 2011


We would like to thank Bruce L. Rosenberg for alerting us to the work of Osborne Reynolds, Peter Jackson for his comments about ‘Einstein’s Boxes’ and Paul Stiles for sharing his insights regarding the difference between depth and distance.


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Dowson, C.G., Springham, P., Rayner, A.D.M. & Boddy, L. (1989) Resource relationships of foraging mycelial systems of Phanerochaete velutina and Hypholoma fasciculare in soil. New Phytologist 111, 501-509.

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Rayner, A.D.M. (2011c). NaturesScope: unlocking our natural empathy and creativity – an inspiring new way of relating to our natural origins and one another through natural inclusion. O Books.

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Fig. 16. Gold (100) surface under STM Microscopy. Image originally created by IBM Corporation.

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Alan Rayner
Dr Alan Rayner is a naturalist who uses art, poetry, fluid mathematics and careful science to enquire and communicate about the evolutionary

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