Natural Inclusion/Boundaries

Abstract and Natural Boundaries - Limits and Limitations

In conventional language, logic, mathematics, and decision-making, we generally regard boundaries as discrete or definitive limits or borders, which permanently and absolutely divide one thing or locality apart from other things or localities. For such definitive limits to exist, however, they would have to be so sharp as to have no thickness. By contrast, even when viewed from afar and over short durations, natural boundaries often appear diffuse, mobile and impermanent, defying such precise, abstract definition.

The apparent boundaries of a rainbow are only an illusion.

The naturalistic poet, William Wordsworth, recognized this difference between abstract and natural perceptions of boundaries when he claimed that: ‘in Nature, everything is distinct, yet nothing defined into absolute, independent singleness’. Natural boundaries, in other words, enable different things or localities to be distinguishable, but not to be entirely isolated or cut off from one another. Hence any attempt to treat natural boundaries as abstract, definitive limits will contradict how they naturally are and so give rise to misconceptions and paradoxes.

A survey of various types of naturally-occurring boundaries helps illustrate this point.

Some of the sharpest, smoothest boundaries found in nature are at edges of crystalline materials. Rough diamonds found in nature have distinctive straight edges, as do samples of many crystalline solids. It is important to understand, however, that this apparent straightness or linearity is only evident from a distance, not at high magnification, and so is not a primary condition. The regular close-packing of intrinsically spherical atomic localities establishes the crystal structure and this structure is evident through a broad range of scales over which the crystal is examined. Crystal patterns and lattices form as the material is drawn toward low-energy configurations. Crystals hence exhibit long-range order and symmetry to retain their low-energy configurations. Grain boundaries are interfaces where crystals of different orientations meet and interrupt the regular structure of the overall specimen.

Other sharp boundaries exist where hard solid materials meet other substances, especially fluids, but here also the appearance of a surface depends on the scale at which it is observed. The edges of a water-borne pebble may seem smooth and rounded to the naked eye and to touch, but will be seen to be indented under high magnification.

Boundaries In Fluids and Material TransitionsEdit

Surface tension causes the surface of liquids to contract to attain a low energy configuration. Surface tension causes beading of rain water on a waxy surface, formation of drops, separation of oil and water, and tears of wine. This is an example of an energy effect that causes a distinct boundary to form between a liquid and its surroundings.

Boundaries between different fluids are spatially distinct, yet often visibly dynamic, even in the short term. Consider drops of oil floating on water. The edges of each drop can be seen, but the shape will fluctuate, perhaps rapidly. If the oil drops are sufficiently small and well dispersed, the mixture may become an emulsion. Mayonnaise and milk are examples of emulsions. Distinguishing the boundaries between the butterfat globules and water that constitute milk only becomes possible at a microscopic scale.

The dynamic nature of the fluid boundary between colored wax and the surrounding liquid is apparent in Lava lamps. Because the fluids have different densities their boundary is unstable, and the shapes move continuously.

Examining the fusion zone of a weld shows a diffuse boundary formed by two solid metals that flowed together during the welding process. [Get reference and image]

Boundaries appear as materials undergo phase transitions. The edges of an ice cube are apparent as it floats in a glass of water. The boundary between water and ice is distinct as the water freezes. The boundary between water and steam can be discerned even at a rolling boil. But these boundaries lose their distinction and no phase boundaries exist at critical points of temperature and pressure.

Boundaries between high and low temperature regions form gradients rather than distinct transition points. Similarly, transitions between high and low concentrations of solute in solution and density or pressure in gases are better characterized as gradients rather than abrupt demarcations. The concept of a gradual transition is extended to include color gradients—gradual transitions of color, and a grade—a gradual change in terrain elevation. In each case the absence of a distinct boundary is acknowledged by using the term gradient. But again, scale of observation is important, as with the milk example, above: what appears as a gradual transition at low magnification may arise from the mixing of ingredients that remain distinctive at higher magnification.

Temperature gradients cause convection which helps propel weather events and ocean currents. Boundaries of high and low pressure regions, storms, and ocean currents are better characterized as gradients or transition zones rather than abrupt demarcations.

The boundaries of a cloud depend on the scale of the observations.

Clouds, mist, fog, haze, and smoke are all characterized by their inherently diffuse boundaries, which nonetheless arise from the mixing or interspersing of distinctive ingredients, i.e. water droplets or smoke solids and air.

The Paradox of the heap, also known as Sorites paradox, challenges our notion of defined boundaries by frustrating our attempts to identify a boundary that separates our definitions of a heap, a pile, and a single grain of sand.

The paradox goes as follows: consider a heap of sand from which grains are individually removed. The argument can be construed as follows:

  • 1,000,000 grains of sand is a heap of sand (Premise 1)
  • A heap of sand minus one grain is still a heap. (Premise 2)

Repeated applications of Premise 2 (each time starting with one fewer grain) eventually forces one to accept the conclusion that a heap may be composed of just one grain of sand, or even none!

The coastline paradox also challenges our notions of a well-defined border by demonstrating that the coastline of a landmass does not have a well-defined length. The measured length depends on the scale of the measuring device used to traverse the indentations, whether these comprise bays and headlands or atomic substructure.

Boundaries are shaped by energy balances. The superficially distinct shapes and sharp edges of solid crystals result from of the low energy configuration of the crystal structure. The compact shape of water drops is a result of reducing the energy at the liquid surface. Ocean currents flow as they chase lower energy configurations. Clouds form and fog rolls in as each responds to energy shifts in the environment.

A drainage basin and its containing watershed is a good example of a natural form produced by a system seeking a low energy configuration. Water from rain, or melting snow and ice soaks into rock and runs downhill under the influence of gravity. This forms various gullies, streams, and rivers as the water flows toward an ocean, inland sea, lake, or closed basin. Drainage basins are naturally occurring flow-form networks shaped to assimilate water by gravity and the water cycle.

Mycorrhizal fungi live in a symbiotic relationship with plant roots.

Boundaries in Natural SystemsEdit

All life forms are ultimately shaped by energy flows. Trees and many plants form leaves to gather in the energy of the sunlight needed to perform photosynthesis and transform water and carbon dioxide into the carbohydrates and lignin that comprise much of the tree structure. Much as in the formation of river tributaries, water is drawn into and through roots and their mycorrhizal fungal partners by osmosis and capillary action as roots extend into soil, forming a flow-form network with an assimilation structure that leads into the conductive xylem vessels and tracheids within the tree. Stomata in the leaves provide openings through which carbon dioxide can be drawn in and oxygen and water vapor released to sustain photosynthesis and enable more water to rise through the wood.

Mycelia are the nutrient and water gathering systems of most fungi, consisting of an indeterminate mass of branching, tubular hyphae. Hyphae grow from their parabolic, extensible tips through a combination of assembly and polymerization of cell wall components that coat their exterior, internal production of new cell membrane and hydraulic thrust arising from assimilation of water and nutrients.

Amoebae have no fixed shape. As they feed on microorganisms living in surrounding water or water films, they can form arm-like structures called pseudopodia, extending from any part of their bodies. When they detect food, they extend their pseudopodia in its direction, move towards and then engulf it within a food vacuole containing digestive enzymes. Any undigested food waste is then released through its body surface.

Just as the M.C. Escher woodcut print Sky and Water confounds figure and ground, natural boundaries confound inside and outside. Boundaries face both ways. Consider a soap bubble floating freely through the air. Does the thin film of soapy water isolate the interior space of the bubble from the spacious exterior or does it isolate the spacious exterior from the bubble interior? For another example, consider a swirl created from blue and red finger-paint. Does the boundary at the edge of the blue belong to the blue paint or the red paint? Any distinction is clearly arbitrary.

Osmosis is the net movement of solvent molecules through a partially permeable membrane into a region of higher solute concentration, to equalize the solute concentrations on the two sides. It is arbitrary to assign the boundary formed by membrane to either the higher or lower concentration regions; the membrane faces both ways. Also, although the membrane is distinct, solvent molecules move through it and are not contained by the membrane boundary.

The boundaries of the sea extend throughout the intertidal zone of a beach.

Because natural boundaries are not absolutely sharp, it is often useful to challenge boundary definitions without entirely dismissing them. Insisting a boundary is either well defined or non-existent is a false dilemma. Gary Davis challenged the entire construct of international borders when he created the World Passport. Fields of mathematics such as chaos theory, complex systems theory, fractals, and fuzzy logic rely less on the sharp boundaries and straight lines that form the basis of Euclidean geometry, even though they retain the unnatural discontinuity between figure and spatial ground embedded in their abstract foundations. Boundaries of land areas such as wetlands, beaches, littoral zones, estuaries, and even forests are inherently diffuse and dynamic. Land ownership and assignment systems that use fixed boundaries to represent these regions are illusions that occlude reality. New systems inherently based on flow-forms are needed.

The Great Barrier Reef is the world's largest coral reef system composed of over 2,900 individual reefs and 900 islands stretching longer than 2,600 kilometers over an area of approximately 344,400 square kilometers. The reef is located in the Coral Sea, off the coast of Queensland, Australia. Identifying individual reefs or the precise boundaries of the reef becomes arbitrary. It is composed of billions of tiny organisms, known as coral polyps. These grow and die as part of the greater ocean ecosystem they are part of. Attempts to identify a definitive boundary between the reef and the sea arbitrarily separate each from the other it depends on.

Pando is a colony of a single male Quaking Aspen determined to be a single living organism by identical genetic markers and one massive underground root system. The plant is estimated to weigh collectively 6,000,000 kg, making it the heaviest known organism. The root system of Pando, at an estimated 80,000 years old, is among the oldest known living organisms. Debating if this is one organism with thousands of tree stems rather than a collection of individual tree stems becomes an arbitrary exercise in semantics.

The Dynamic Nature of BoundariesEdit

Financial accounting systems that artificially draw a boundary around a single corporation, a few transactions, a short time frame, or a particular organization unnaturally exclude economic externalities. This causes many distortions and often leads to excess, abuse, and social and environmental impacts.

It is rarely only black or white.

Even the boundary between true and false is blurred. The study of certainty demonstrates, in the words of Voltaire: “Doubt is not a pleasant condition, but certainty is absurd.” This does not, however require us to drift aimlessly in an anything-goes world of relativism were all truths and values are relative and without any basis. Degrees of certainty and sensible concepts of common good allow us to navigate the real world unencumbered by arbitrary and often baseless artificial boundaries.

Recognizing the intrinsically dynamic nature of boundaries highlights the importance of including neighborhoods, context, and ecosystems along with each individual. Consider the problem of establishing the boundaries of even a single tree. Separating the leaves from the branches, branches from the trunk, or the trunk from the roots will quickly kill the tree. And separating the tree from the ground, the air, the water, or the sunlight will also quickly kill it. The tree cannot exist outside of the neighborhood and context in which it lives.

On his deathbed, Louis Pasteur, the pioneer of germ theory: “Bacteria are nothing, the terrain is everything” as he highlighted the vital role of context for any living system.

Consider this question: “Is the apple an array of atoms and space; a collection of chemical molecules; a lump of organic cells; or a piece of fruit? Or is it the tree and ecosystem that is waiting to grow within it?” Attempting to answer highlights the arbitrary nature of commonly defined boundaries, their dependence on the scale of the observations being made, and the importance of context and ecosystems.

Natural living systems are dynamic inclusions of their neighborhoods, and are vitally dependent on their context. There is no natural place to draw a boundary that isolates one living system from its natural neighborhood without stifling it. An ecosystem is sometimes spoken of as the smallest environmental unit that can be isolated as an autonomous whole, and the whole Earth is sometimes spoken of as ‘our ecosystem’ – but in reality no ecosystem, as a dynamically bounded, energetic configuration of space and in space can be isolated from anywhere. There are no isolated systems that we know of – or could know of – in the cosmos.

We are left to wonder how abstract perceptions and representations of boundaries could have come to dominate our thinking and ways of life in the way that they have done. All we have to do to appreciate the fundamental difference between abstract and natural boundaries is ask a simple question: what enables natural form to be and become distinguishable? It becomes apparent that the only way of answering this question is to acknowledge the occurrence of at least two kinds of natural presence: a receptive context or medium which provides freedom for local movement and/or expression, and local formative content, which informs or configures that context. The former is necessarily spacious, the latter necessarily cohesive. Moreover, for form to be and become distinguishable, each of these presences must naturally include the other. Spacious presence alone would be formless void, and formative presence alone would have no shape or size. They are necessarily distinct, but mutually inclusive presences. They can neither be abstracted from one another as independent entities, nor be homogenised into ‘Oneness’. The only way in which this necessity can be fulfilled is for one of these presences, natural space, ultimately to be everywhere, continuous, intangible (i.e. frictionless) and immobile, and for the other ultimately to be somewhere, distinctive, tangible and continually in motion. Natural space and figural boundaries are hence, respectively, continuous and dynamically distinct (i.e. dynamically continuous) energetic interfacings between the insides and outsides of all natural forms as flow-forms

In summary:Edit

  • Natural boundaries defy abstract definition.
  • Natural boundaries are energetic configurations of space, sustained in dynamic balance between adjacent localities.
  • Boundaries face both ways – as energetic interfacings between distinctive localities.
  • Challenge boundary definitions without entirely dismissing the presence of boundaries.
  • Individual identities include neighborhoods, and can only be understood in context, as inclusions of ecosystems as inclusions of space.

All form is manifest as flow-forms, energetic configurations of space; each in the otherness; obviation of paradox, contradiction, conflict, co-operation, reductionism, holism through recognition of natural co-creativity and continuity; organic life as a dynamic inclusion of natural energy flow; evolution as cumulative energetic transformation via 'natural inclusion' - the co-creative, fluid dynamic, transformation of all through all in receptive spatial context.[1]

Dr. Alan Rayner came to his understanding of boundaries as described here upon publication of his book Degrees of Freedom - living in dynamic boundaries.[2]

The nature of boundaries and space is explored more organically, poetically, and artistically in the following materials:


  1. The Vitality of Each in the Otherness, March 26, 2013, Alan Rayner, Best Thinking
  2. Rayner, A.D.M.(1997), Degrees of Freedom - Living in Dynamic Boundaries. London: Imperial College Press.