(Re-) Inverting the Architectural Perspective   

January 26, 2001

Plato and Euclid had an upside-down architectural perspective relative to nature in designing their 'ideal city', a perspective that we have inherited along with a deep entrenchment of the upside-down tool of Euclidian space and absolute time reference framing.

The honey bee does not focus on designing the 'ideal storage cell' and thus come up with its amazingly well optimized vaulted hexagonal cell design.

The honey bee is tuned to the curved space-time dynamic and the hexagonal cell is the result of 'container-constituent-coresonance' and its longer term transformational effect i.e. 'container-constituent-coevolution.   With a community oriented vision of opening up opportunity space for the young (larvae), the honey bee lets his 'assertiveness as an individual constituent of space'  fall into the service of 'inductive-assertive shaping of his containing space' ('inductive-assertive' in that as the bees 'assert', their codynamic reciprocally induces transformation in the shape of their containing space) putting the co-creation of 'opportunity space' for the larvae into the primacy over their 'independent action management'.  It was this 'curved space' architectural mode which led to the hexagonal cell (i.e. bees in earlier phases of evolution start off by making individual spherical larval containers which imply unnecessary duplication of cell walls and lead to almost forty percent space wastage between the spheres.  The wall-sharing practices have evolved naturally into the highly material and storage-space optimized hexagonal cell pattern)..   The architecture thus emerged from a collaboration based 'coresonance' between the inductive shaping of the enveloping space relative to the assertive behaviours of the bees; i.e. the bees let their assertive actions 'float' and thus be pulled naturally into coresonance with the reciprocally forming 'shape of containing space'.  Western man's architecture on the other hand, as Emerson says, (and Plato and Euclid particularly exemplify) inverted the architectural perspective, letting 'the tools run away with the workman' and seeing architecture in a purely assertive or constructivist light, out of the context of the reciprocal inductive-assertive resonant relationship between constituent and containing space, ... a relationship which leads to a 'container-constituent-coresonance' spectrum ranging from 'dissonant' to 'harmonious', ... the dissonant end associating with the constituency's lack of sensitivity to the innate 'container-constituent-coresonance' in nature.

Marshall McLuhan is saying the same thing when he observes "In terms of the ways in which the machine altered our relations to one another and to ourselves, it mattered not in the least whether it turned out cornflakes or Cadillacs."  That is, 'resonance' or 'dissonance' is the overriding factor.

Do we want to orient our architecture to managing 'our relations to one another and to ourselves' ('opportunity space management') or do we want to orient our architecture to managing 'what is there' ('action management')?  As Lao Tsu said, it is the 'opportunity space' which gives 'usefulness' to the material structure;

"Thirty spokes share the wheel's hub;

It is the center hole that makes it useful.

Shape clay into a vessel;

It is the space within that makes it useful.

Cut doors and windows for a room;

It is the holes which make it useful.

Therefore profit comes from what is there;

Usefulness from what is not there."

Whether we want to put 'opportunity space management' into the primacy over 'action management' or vice versa determines our space-time geometry requirements.   As McLuhan was implying (see his interview with Bruce Powers entitled 'Angels to Robots: From Euclidian Space to Einsteinian Space' in 'The Global Village'), the Euclidian view is "not a complete way to visualize the totality of the world" because it cannot deal with issues of 'resonance';

 * * *

Bruce Powers: We are constantly suppressing the awareness that the material universe is comprised of resonances; that no straight lines exist.

Marshall McLuhan: Exactly.  Because the Euclidian construct is controllable.  The 'center' of acoustic space is everywhere and, therefore, seemingly chaotic.

[the apparently 'chaotic' information in 'acoustic space' is space-time phase information which the 'listener' can 'image' by bringing the potential coherencies into connection in his mind, as in holography.]

 * * *

As Johannes Kepler also pointed out, in order to perceive 'resonant space' ('container-constituent-coresonance'), one has to 'immerse oneself' in the architecture and 'let go' of the notion of 'control'.  This 'inverted' mode of perspective is described by Einstein in his essay 'Geometry and Experience' in the following terms;

 * * *

Can we picture to ourselves a three-dimensional universe which is finite, yet unbounded?

The usual answer to this question is ``No,'' but that is not the right answer. The purpose of the following remarks is to show that the answer should be ``Yes.'' I want to show that without any extraordinary difficulty we can illustrate the theory of a finite universe by means of a mental image to which, with some practice, we shall soon grow accustomed.

First of all, an observation of epistemological nature. A geometrical-physical theory as such is incapable of being directly pictured, being merely a system of concepts. But these concepts serve the purpose of bringing a multiplicity of real or imaginary sensory experiences into connection in the mind. To ‘visualise’ a theory, or bring it home to one's mind, therefore means to give a representation to that abundance of experiences for which the theory supplies the schematic arrangement.

 * * *

What McLuhan, Kepler and Einstein are all saying is that if we want to understand and design 'resonant architectures', ... architectures such as our solar system in which there is a 'coresonance' between the assertive actions of the constituents and the shape of the opportunity space which simultaneously, reciprocally opens up these actions, then one will have to let go of the 'control' which is bundled into the Euclidian space assumption and embrace a curved space geometry.

Curved space perception is our primary and natural 'pre-cultural' mode of perception.  We didn't start off life by calculating our actions in Euclidian space and globally synchronous time coordinates; we started off by 'tuning in' to 'acoustic space' (as McLuhan calls it) and it remains our perceptive mode of choice when we are in our childlike friendly, open and accepting mode.  For example, when we are driving on a busy freeway and find ourselves amongst a group of friendly drivers (an increasing rarity?), since we are all conscious that our assertive actions simultaneously reciprocally transform the shape of dynamic opportunity space (the continually transforming inter-vehicular space), we can 'let go of our rule-based assertivity', putting our actions into the service of co-creating opportunity space for all of our fellow constituents.  In this mode, our perceptual focus is on the shape of space as it swells up here and shrinks down there to allow continuing resonant flow-through.

Curved space perception and opportunity space management, which enable a community to move into and sustain a state of 'container-constituent-coresonance' are relativistic, in the sense that the constituents key NOT to an artificial abstract Euclidian reference frame, but instead, to the geometry of space in which they are immersed constituent participants.  In this relativistic mode, as in the freeway driving example, there is no 'middle man space geometry' to reference to, ... one is keying directly to the 'shape of dynamic opportunity space' by the 'holographic' technique described by einstein and by Kepler in 'Harmonies of the World', ... bringing a multiplicity of space-time phase impressions into connection in the mind.  In McLuhan's 'cornflakes and Cadillacs' example, it is clear that if one simply encourages 'investment' in a community by the building of a factory, the induced effect on 'relationships' (the shape of opportunity space) is left to flap in the breeze, in the manner that the poor pool player focuses on 'making shot's' (action management) out of the context of the simultaneous, reciprocal transformation of opportunity they induce in the 'shape of opportunity space', ... a 'shape' which gates and modulates their continuing possibilities for 'making shots'.

The 'curved space' architectural mode is the architectural mode of ecologies and it is the architectural mode of exceptional teams (mini-communities) that I have studied, ... it is an architecture capable of inducing 'container-constituent-coresonance', something which is impossible in Euclidian space 'control mode'.

This notion of 're-inverting' our 'architectural perspective' back from the rigidity and control of the Euclidian space and absolute time referencing view to the resonance of the relativistic curved space-time referencing view can be somewhat bewildering because of long and deep cultural conditioning to the Euclidian view.  The following discussion seeks to revisit the typically unrevisited basement of our 'tools of inquiry', a cultural legacy which we tend to use without question and, when challenged, defend in the same manner that we tend to defend the religion and nationality that we happened to be born into.


The scientific thinking of the western mainstream is oriented solely to the 'assertive behaviours of independent causal agents' (all inductive effects are secondary as Euclidian space is 'empty space' and has no innate inductive capability as curved space has.).

The important aspect here is that, in the western mainstream scientific mode, we are describing the physical phenomena we are inquiring into and trying to understand it, solely in terms of 'things' and their 'assertive actions'.

We take it for granted that we split apart space and time when we inquire in our 'scientific way' (or have long forgotten that we are choosing and imposing our space geometry on our experience), and use the notion of 'things in their own right' which occupy some coordinates in three dimensional Euclidian space, to describe things in the terms of 'kinetic trajectories and transactions', the displacement and interactions of 'things in their own right' as a function of absolute time. 'Absolute time' implies 'globally synchronous time' in that we want to use the same 'clock' for all of the 'things' in the physical phenomenon we are studying.

So, normally when we speak of 'four dimensional space' (x, y, z, t), we are speaking about split apart space and time, ... the rectangular space of Euclid complemented by (i.e. 'bundled in with) the notion of globally synchronous time, and it is this mathematical space which most of our scientific conceptualization is based in.

Because we live in a scientific culture which is 'swimming dependently in' technologies which have emerged from scientific knowledge, this mathematical space has become the de facto standard for our inquiry and we use it all the time without really thinking about.  In fact it has been part of our de facto assumptions about the nature of space and time since the era of Parmenides (500 B.C.) and his 'binary' ontology of 'things' and 'empty space', though it was Euclid (c. 330-275 BCE) who mathematically formalized this space.

The space of Parmenides-Euclid which features 'things' which are absolute and 'in their own right' contained in 'empty space' was not, in that early era, the sole alternative for a space-matter conceptualization, as the following historical observations illustrate;

 * * *

"By insisting that only permanent things could have real existence, the philosopher Parmenides (5th century BC) called into question the most basic claims about knowledge itself. In contrast, Heraclitus (c. 500 BC) maintained that all permanence is an illusion, for the things that are perceived arise through a subtle balance of opposing tensions. What is meant by "knowledge" and "proof" thus came into debate."

http://www.britannica.com/bcom/eb/article/printable/5/0,5722,118175,00.html (History of Mathematics, Encyclopedia Brittanica)

 * * *

[Clearly, Heraclitus was an 'acoustic space' man, like McLuhan, Kepler and Einstein, ... believing that nature's architecture was a 'container-constituent-coresonant' architecture rather than a solely 'bottom-up' assertive construction oriented architecture.]

Burkert observes that the Eleatic ontology of Parmenides and his successors, on the other hand, presents a much more likely source for the formal deductive logic which is essential to the success of the "elemental" geometries of Hippocrates and Euclid. . . .Parmenides' innovation, which turns out to be monumental, is to embrace an ontological first principle whose evidence is a direct consequence of its logical necessity. This is that being (to eon) is:

It is necessary to say and to conceive (noein) that being is, for there is being, but Nothing is not.
From this premise Parmenides deduces a number of necessary conclusions:

...that being is ungenerated and imperishable, entire, unique, unmoved, and perfect; it never was nor will be since now it is all together, one, indivisible.

The truth of these conclusions does not derive from empirical intuition, for indeed they contradict everyday experiences of change in the world, but instead rests upon the impossibility of their negation: That there be generation, destruction, divisibility, multiplicity, motion, or imperfection ultimately implies in one way or another that not-being is, which contradicts the necessary first principle, and hence leads to a logical impossibility. This is an explicit presentation of formal deduction in utter abstraction, and a new development in Greek philosophy.

http://ccat.sas.upenn.edu/~awiesner/gkdem.html (The Return of Odysseus and the Elements of Euclid' by Andrew Wiesner)

 * * *

Meanwhile Euclid's formalization of the notion of a rectangular empty space inhabited by 'absolute material entities' was based both on axioms, such as that provided by Parmenides as the absolute existence of 'things in their own right', and on geometric 'constructions' developed in diagram form (on a two-dimensional surface).   Euclid's faith in 'plane geometry' and diagrammatic construction had a lot to do with classical Greek studies of 'the ideal city', and aesthetic ratios as is pointed out by Abraham Akkerman in his essay 'Place and Thought', from which the following observations have been excerpted;

 * * *

During the winter of 332-331 BCE a plan of Alexandria was prepared by Dinocrates of Rhodes on a strictly orthogonal grid. Alexandria became the most magnificent of cities in the Hellenistic world, home of the Syracusan mathematician, physicist and inventor Archimedes (c. 287-212 BCE), a place where he is believed to have discovered his most significant laws of mechanics. It was also Alexandria that gave rise to the three geometricians, Euclid (c. 330-275 BCE), Erathostenes (276-194 BCE), and Ptolemy (Claudius Ptolemaus, Second century BCE). Ptolemy, also an astronomer and a geographer, expounded the comprehensive geocentric notion of our solar system, putting firmly in place an astronomical concept that would last for almost two millennia, till the 16th century. 

. . .

[Euclid seemed to follow on in the tradition of designing things diagrammatically in the two-dimensional plane, and looking for aesthetics in this non-volumetric 'ratio-cinative' mode, as the following notes from his 'Elements' suggest]

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less [...].
( Book VI, definition 3.)

If a straight line be cut in extreme and mean ratio, the square on the greater segment added to half of the whole is five-times the square on the half,
 (Book XIII, 1.)

i.e., a line of length 1 cut in extreme and mean ratio, has its longer segment 1/2( 5 - 1), its shorter segment 1/2(3 - 5), and their ratio 1/2( 5 - 1):1, approximating 0.61803398...45  [i.e. the 'golden mean']

http://duke.usask.ca/~akkerman/edition2/Q4.htm ('Place and Thought', by Abraham Akkerman)

 * * *

So the concept of split-apart rectangular space and absolute time has some rather specific historo-cultural 'foundations' (Parmenidian 'binary' ontology of 'something' or 'nothing', and the need for a 'proof' based on geometrical diagrammatic constructions on a two-dimensional surface).

Yet it is such a 'convincing' notion of space and time that it elicited the following statements from Emmanuel Kant (1724 - 1804);

" Euclidean geometry is the inevitable necessity of thought", ... and time is "A category allowing one to order events in a before-after-relationship".  

Meanwhile, back even to Euclid's own reflections, there was something uncomfortable about Euclid's proof of his three dimensional rectangular space geometry, and that discomfort resided in the fifth of the following five postulates, which implicitly tied back to the notion of parallel lines on a flat diagram (as opposed to straight lines on the surface of a sphere such as the earth).

  1. A straight line may be drawn between any two points.
  2. A piece of straight line may be extended indefinitely.
  3. A circle may be drawn with any given radius and an arbitrary center.
  4. All right angles are equal.
  5. If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.

The 5th postulate is also called the Parallel Postulate. The following, called Playfair's Postulate is equivalent: Given a line l and a point P not on that line, there exists in the plane of P and l and through P one and only one line m, which does not meet the given line l.

The problem with the fifth postulate was that you could not 'construct it' on a diagram, and the strength of proofs in Greek mathematics drew from 'diagram based constructions'.  It was this non-constructable aspect which had troubled Euclid and he had included the fifth postulate only reluctantly; i.e. he needed it to make the overall proof complete but could find no better postulate.  Many mathematicians from Euclid's time on, believing that the geometry of space was Euclidian, attempted to prove Euclid's fifth postulate, but failed.   Finally, in 1829,  Lobachevsky published his work on non-Euclidian geometry which 'disproved' Euclid's fifth postulate or rather proved that geometric space was not constrained to be conceived in terms of 'Euclidian space' (simultaneously, Karl Gauss and Janos Bolyai had come to similar conclusions as is discussed with diagrams in 'Euclid's Fifth Postulate and the Advent of Non-Euclidian Geometries' by Lawrence J. Crosswell at http://www2.ispa.fsu.edu/~lcrosswe/paper/paper.html ).  By other accounts, Gauss did not publish his results because it was highly controversial, particularly in the wake of Kant's statements, to suggest that the Euclidian space geometry was arbitrary.

Lobachevsky's reasoning was simple and extended the innate logic of the space from 'binary' to 'ternary'; i.e. he reasoned that as you increased the interior angles formed by a line crossing two indefinitely extending lines, the extensions of the lines will cross until you get them in the parallel position and then they will no longer cross.   In lieu of looking at a diagram one can think of a sagging letter 'H' or football goal whose uprights, if extended cross high above the playing field but as one straightens them up to the vertical, their upwards extensions will eventually no longer cross (nor will their downward extensions). Lobachevsky reasoned that there must be some bounding alignment wherein they go from crossing to 'not crossing', so we can count this boundary line also as a line which doesn't cross, therefore we have at least two lines m1 and m2  "in the plane of P and l which do not meet the given line l which contradicts Euclid's fifth postulate.

Now that may look like a pure mathematical abstraction with no 'real world' implications, but when one looks deeply into the whole issue of 'the geometry of space', as Henri Poincaré did, one finds that;

"Space is another framework we impose upon the world" . . . " . . . here the mind may affirm because it lays down its own laws; but let us clearly understand that while these laws are imposed on our science, which otherwise could not exist, they are not imposed on Nature." . . . "Euclidian geometry is . . . the simplest, . . . just as the polynomial of the first degree is simpler than a polynomial of the second degree." . . . "the space revealed to us by our senses is absolutely different from the space of geometry." . . . Henri Poincaré

Lobachevsky's non-Euclidian geometry can be visualized in terms of the earth sphere.  Lines, such as the lines formed by the streets and properties in Plato's and Euclid's 'ideal city' which are indefinitely extended on the spherical surface of the earth now become circles.  Imagine the line 'l' representing 5 degrees north latitude which encircles the earth, and imagine a point 'P' at 5 degrees south latitude at some particular but arbitrary longitude.  How many lines can we put through that point 'P' which do not cross the line representing 5 degrees north latitude?   You can visualize putting a bit of a tilt on the line, so that the circle it extends into doesn't quite correspond to a 'parallel of latitude' but strays up to, say, 2 degrees south at its most northerly point and back down to 8 degrees south at its most southerly point; i.e. even with the tilt away from the 'absolutely parallel', it it still doesn't cross the line 'l', the parallel of latitude at 5 degrees north.  In fact the mathematics show that there are an infinity of lines through the point 'P' whose extensions do not cross the line 'l'.

Ok, but what's that got to do with the space 'we live in'?  

It relates to the fact that the 'independent causal agent' assumption is a crude approximation and the 'assertive actions' of the constituents of space are not 'the whole story'; i.e. curved space geometry suggests that one's assertive behaviours have a reciprocal effect on the containing space which will in general reverberate back on the 'assertor'; i.e. imagine erecting a sequence of large stone 'domino's', each about 10ft high, 6ft wide and 1ft thick, and spacing them 5 ft apart, in a long straight line (a line which happened to go around the earth-sphere).  If one got in between two of these stone domino's and managed to topple one of them, ... one could lean back on the still upright stone domino at one's back, relax, and watch the ribbon-like toppling-dynamic snaking off over the hills and through the valleys on its straight line trip to infinity, right?

While this effect may seem 'sequential' because of its 'linear geometry', the general case is better seen by imagining playing the game of pool on the surface of a sphere (the pool table, because of its reflective banks, emulates spherical space; i.e. you can shoot a ball into and across one corner so that it makes two 90 degree banks, comes back out and across to the opposite diagonal corner, makes two more 90 degree banks and comes back through the point it started out, ... traversing 360 degrees as if it circumnavigated a spherical surface.).  Each ball on the spherical surface is, at the same time, 'an assertive agent' and 'an inductive agent'.  The unique geometrical shape forming from its 'inclusion' participates in the 'shaping' of opportunity space which gates and modulates the assertive behaviours of its fellows and itself.  When it moves, the shape of space simultaneously, reciprocally transforms, influencing the evolving patterns of motion.  This demonstrates the relativistic space-matter interdependence of curved space-time.   And, one can do the four bank shot described above, to come up from behind and hit the ball which was initially right behind, and knock it forward into the desired pocket (in theory).  This demonstrates the 'finite and unbounded' aspect of curved space time.

As Einstein pointed out in his essay 'Geometry and Experience', ... in the curved space of relativity, you have to concern yourself with 'reciprocal disposition', the shape of space which is reciprocal to the shape of the 'constituents' within the space, something we don't bother with when we think in terms of  Euclidian space, because since Euclidian space is infinite, the shape of space reciprocal to the shape of the things within it, is not definable, whereas it is definable in non-Euclidian space.   When one takes into consideration that the surface we live in (or 'within' if one thinks of the street level portion of the biosphere)  is a spherical surface-shell less than one millimeter thick if we scaled the earth to have a radius of one meter, and that we are now packing six billion people into that spherical space,... 'reciprocal disposition', the shape of the space between things can become rather important.  Einstein describes the 'finite unboundedness' of spherical space relative to the 'infinite unboundedness' of Euclidian space as follows;

 * * *

"What do we wish to express when we say that our space is infinite? Nothing more than that we might lay any number whatever of bodies of equal sizes side by side without ever filling space. Suppose that we are provided with a great many wooden cubes all of the same size. In accordance with Euclidean geometry we can place them above, beside, and behind one another so as to fill a part of space of any dimensions; but this construction would never be finished; we could go on adding more and more cubes without ever finding that there was no more room. That is what we wish to express when we say that space is infinite. It would be better to say that space is infinite in relation to practically-rigid bodies, assuming that the laws of disposition for these bodies are given by Euclidean geometry.

... Now we take an example of a two-dimensional continuum which is finite, but unbounded. We imagine the surface of a large globe and a quantity of small paper discs, all of the same size. We place one of the discs anywhere on the surface of the globe. If we move the disc about, anywhere we like, on the surface of the globe, we do not come upon a limit or boundary anywhere on the journey. Therefore we say that the spherical surface of the globe is an unbounded continuum. Moreover, the spherical surface is a finite continuum. For if we stick the paper discs on the globe, so that no disc overlaps another, the surface of the globe will finally become so full that there is no room for another disc. This simply means that the spherical surface of the globe is finite in relation to the paper discs. Further, the spherical surface is a non-Euclidean continuum of two directions, that is to say, the laws of disposition for the rigid figures lying in it do not agree with those of the Euclidean plane. This can be shown in the following way. Place a paper disc on the spherical surface, and around it in a circle place six more discs, each of which is to be surrounded in turn by six discs, and so on. If this construction is made on a plane surface, we have an uninterrupted disposition in which there are six discs touching every disc except those which lie on the outside. 

On the spherical surface the construction also seems to promise success at the outset, and the smaller the radius of the disc in proportion to that of the sphere, the more promising it seems. But as the construction progresses it becomes more and more patent that the disposition of the discs in the manner indicated, without interruption, is not possible, as it should be possible by Euclidean geometry of the Plane surface. In this way creatures which cannot leave the spherical surface, and cannot even peep out from the spherical surface into three-dimensional space, might discover, merely by experimenting with discs, that their two-dimensional ``space'' is not Euclidean, but spherical space. 

From the latest results of the theory of relativity it is probable that our three-dimensional space is also approximately spherical, that is, that the laws of disposition of rigid bodies in it are not given by Euclidean geometry, but approximately by spherical geometry, if only we consider parts of space which are sufficiently great. Now this is the place where the reader's imagination boggles. ``Nobody can imagine this thing,'' he cries indignantly. ``It can be said, but cannot be thought. I can represent to myself a spherical surface well enough, but nothing analogous to it in three dimensions.''

...We must try to surmount this barrier in the mind, and the patient reader will see that it is by no means a particularly difficult task. "

 * * *

So, there are a lot of practical insights in non-Euclidian space, ... it focuses our attention on the importance of the shape of 'inclusor space', the shape of the space which wraps around the 'things' seen as 'inclusions' within the space.  Inclusor space corresponds to the 'cheese' ('containing space') in Swiss cheese which envelopes the 'holes' which are the 'inclusions' ('constituents') in the cheese.

Now, when we start thinking of the importance of 'reciprocal disposition', we are moving towards eastern thoughts of space, such as the importance of space to the philosophy of Lao Tsu (a contemporary of Heraclitus and Buddha) as mentioned above re his 'Thirty spokes share the wheel's hub' poem.

Lao Tsu's view of space is like Heraclitus's, ... it is 'dynamic' rather than static.   Lao Tsu makes the point that space has two attributes, its constituents, and the opportunity for movement which is reciprocal to the constituent.  The above poem by Lao Tsu could perhaps make him a candidate for the 'father of architecture'.  Euclid was also very much concerned with architecture, but in this case the 'assertive architecture' of 'the ideal city' which seemed to follow on from 'Plato's Republic' and which was based purely on the notion of 'assertive agents' in the form of control hierarchies, neglecting the 'shape of opportunity space aspects'.  The control based architectures of Plato's Republic lend themselves to 'Euclidian space and absolute time' and the ignoring of the reciprocal 'shape of opportunity space'.

In this regard, Donald Kunze (professor of architecture at Penn State Univ.) observes that medieval societies were more 'biological' and 'inclusionally nested' (more 'container-constituent-codynamical') but have reverted in modern times towards the mechanical 'Plato's Republic' format due to the application of technology in extending 'the lines of power' for greater centralization of controls.

 "In Medieval society, there were many LAYERS. Each social class had its internal ruling structure which paralleled the royal rule of the king and the theological rule of heaven. The layers moved inside each other, like a system of concentric circles. Because layers were mirrors of each other, the WHOLE was present in each PART. All members of society, from the highest to the lowest, participated directly in a more or less unified view of the world... The NEW SYSTEM, based on continuous movement, easily took over the OLD SYSTEM based on conditional passage, because it was able to convert the old boundaries into abstract conditions. The city wall, for example, was originally a political, military, and religious entity. In the new system, the wall was made superfluous by political homogenization and useless by artillery... The NEW SYSTEM worked along lines of power that radiated from a core that controlled the periphery. This new idea helped promote centralized, representative government, the development of nations, and the streamlining of transportation. In PERCEPTION, these changes were reflected in the way INSTRUMENTS such as telescopes, microscopes, theodolites, and other instruments extended the power of the eye; and also in the ways that REPRESENTATIONS such as drawings, maps, and -- later -- photographs were accepted as reliable substitutes for the visible. The LINE OF SIGHT was treated as a line of CONTROL and potential POWER. The representation was a means of control and power. It facilitated the inward flow of knowledge and the outward flow of power." (See Donald Kunze's essay 'Representation' at http://wgn111.ce.psu.edu/representation/representation.html  )

These ideas on the 'architecture' of community are very much tied to our conceptions of space.   In centralized control-based systems (systems which consider only 'assertive behaviours of independent causal agents') one ignores the role of the 'shape of dynamic opportunity space' even though it is of essential importance.   One can work these issues in terms of 'plans' or diagrammatic constructs as in the case of Euclid's work on 'the ideal city'; i.e. one does not bother to make a volumetric model to find out how 'resonantly' or 'dissonantly' things flow from the point of view of the 'immersed' rather than 'voyeur' observer.  This ignorance can be likened to the poor pool player who manages only on the basis of 'shots' and fails to consider how the shots transform the opportunity for further shots; i.e. he shoots each ball as well as he can, as if each ball were independent and as if the 'game' was determined by the sum of its kinetic transactions.   If the balls could speak from their 'immersed viewpoint', however, they might say;... "doesn't that idiot realize that every time he moves one of us, he transforms the shape of the opportunity space in different ways for each and every one of us, ... and how our opportunities evolve is the overriding influence on how the game evolves.  To make good shot opportunities, we must resonantly co-create our corridors of opportunity."

In any case, from an architectural viewpoint, one wants to understand, from an 'immersed observer's perspective', how the shape of space facilitates the codynamics of people and goods (constituents) in the edifice and in the community, and then it is not just a question of the 'assertive behaviours of independent causal agents', but of 'container-constituent-codynamics'.   Some 'inclusor spaces' facilitate good flow-through and some do not. The friendly freeway driver knows this as well as the skilled pool player.   They also know that it is possible to co-create the shape of 'dynamic opportunity space' in a manner which facilitates the continuing opportunized assertion of the dynamical agents into this shape; i.e. the asserting agent simultaneously, reciprocally transforms the shape of dynamic opportunity space and can do so in a 'container-constituent-coresonant' or 'container-constituent-dissonant' fashion.  The crowds in the 'city', on the basis of their immersed experience walking across a public terrace know that they can 'co-create' a harmonious flow (container-constituent-coresonant flow) if they 'reference to' and let their assertive movements be guided by the 'shape of dynamic opportunity space', rather than by simply 'barging across' assertively in a straight line.  (See Hasan Isawi's site for an example of the move towards the immersed volumetric view).

Now, we can return to Heraclitus' above-cited notion that "all permanence is an illusion, for the things that are perceived arise through a subtle balance of opposing tensions".

While, in one way, this sounds a bit odd, ... if we think in geologic timeframes, it makes a lot of sense.  And if we think of medieval notions of community, a la Camelot and King Arthur, ...the architecture is decidely non-hierarchical, and is instead curved space-ish.  That is, the purpose of the 'round table' was to do away with hierarchy (the 'King' was seen as a respected 'elder' in the indigenous tradition who is not necessarily 'old') and the notion of 'community' was in the sense of the 'geometry of community' or its 'container-constituent-codynamical' geometry which continued to evolve in spite of the 'recycling' of its consituent buildings and citizens.  In other words, the architecture for this relativistic 'space-over-matter' community had no innate dependencies on 'things in their own right' as did the Platonic ideal city with its Euclidian geometric space representation, but was based on the coevolutionary resonance between container-opportunity and constituent-purpose.   In the film 'First Knight' with Sean Connery, the dialogue between Arthur and Lancelot includes the following; 

 * * *

King Arthur (to Lancelot as he shows him around Camelot) ...if you must die, die serving something greater than yourself... ... (Regarding the round table and his style of politics Arthur says) ...No head, no foot. Everyone equal. Even the King.

 Lancelot: (reading the words around the center of the Round Table) IN SERVING EACH OTHER WE BECOME FREE.

Arthur: That is the very heart of Camelot. Not these stones, timber, towers, palaces, burn them all! and Camelot lives on because it lives in us. It is a belief that we hold in our hearts.

 * * *

As was discussed in 'Including the Tools of Inquiry in the Inquiry', the mathematical physics descriptions of nature, which are typically based in Euclidian space and absolute time, do not consider the reciprocal transformation of the shape of dynamic opportunity space which occurs simultaneously with assertive action (since the reciprocal space is not definable due to Euclidian space being infinite).  So the notion of a 'community' being a thing which has a life and evolution of its own which transcends the materiality of the constituents, is also not definable in the terms of Euclidian space and absolute time.  The 'simultaneity' involved in the curved space-time continuum comes from the substitution of the 'little story' notion of 'kinetic trajectories and transactions' (things which can be proven by diagrammatic construction) by the 'bigger story' notion of 'space-time transformation'; i.e. the eye of the hurricane 'looks like' a 'thing in its own right', but it is really only an implicit center of 'whole-and-part' coherency.  We can say that 'it' moves, but since 'it' is, at the same time, its containing atmosphere, what is 'really' going on (the 'bigger story') is space-time transformation.

So who ever said that 'material things' were in the primacy over space?  One person who did, who had a huge influence on our cultural mode of perception and inquiry was Parmenides, and Euclidian space goes hand in hand with Parmenides 'binary' or 'discretist' worldview.

How come we persist in embracing this 'upside-down' 'matter-over-space' worldview in our 'scientific' culture?

We don't really persist in embracing it, we only persist in 'succumbing to it', particularly in our 'architecture' of management and governance.  The local group or 'community of freeway drivers', if they are friendly and collaborative, opt for the curved space architecture of 'container-constituent-coresonance', wherein the geometry of space is placed in the primacy over the kinetics of the constituents (rather than kinetic assertiveness being placed in the primacy).  Dynamic opportunity space is, essentially, the 'living entity', and it is sustained by the co-creative kinetics of its material constituents who, themselves, come and go while the dynamic opportunity space 'lives on'.

Another example of the shape of space being in the primacy over the assertive behaviour of the constituents is the airforce aerobatics team performing a routine in which three aircraft flying fairly high and parallel to the ground fly towards their intersection point (collision point) while a fourth, starting at a lower altitude flies vertically up on an orthogonal through the virtual collision point and 'threads the needle' just before the shrinking triangle formed by the other three 'inverts' from converging mode to diverging mode (naturally, the three planes plan to slip by each other rather than literally hitting the collision point).  In fact, they are co-creating a volumetric triangular prism which inverts, and which looks something like the molecular structure of a quartz clock.

In any case, this 'community of four' continually reference to the evolving geometry of their 'community' which, like the storm in the atmosphere, is self-referential and whose motion can only fully be described in terms of 'space-time transformation' rather than in terms of 'kinetic behaviours' (trajectory based constructions are incapable, dimensionally, to aspire to volumetric shape-of-space descriptions).  True, we can describe it 'after the fact' in terms of kinetic behaviours, in the same manner that the poor pool player can describe a game of pool in terms of the 'shots' without ever addressing how the shots simultaneously, reciprocally transformed the shape of dynamic opportunity space.  Such a description is simply an incomplete 'miming' of what happened which is void of any understanding of how the geometry evolved the way it did.    The problem is that 'the co-created shape of space' is what is in control (the pilots are 'referencing to it' or making themselves 'subservient to it') and it is not possible to mathematize this arrangement in Euclidian space and absolute time, as mainstream science is wont to attempt to do.  Euclidian space and absolute time based models cannot speak to 'container-constituent-coresonant' situations (relativistic situations) wherein 'space is a participant', a 'greater than the sum of the parts' situation which the constituents reference to and 'serve'.  This shape of space forms from the collective conscious of the constituents who realize that their actions reciprocally transform opportunity space and further realize that the can co-create a resonant situation where everyone's purpose can be served up with opportunity for assertive action.  When the community of freeway drivers, aerobatics pilots, or 'billiard balls' goes into the 'co-creation of container-constituent-coresonance state, it is beyond the domain of standard 'mathematical physics' (pre-relativity) and mainstream science models, yet it exemplifies community behaviours which are pervasive in nature from the scale of the solar system down to the scale of atoms in crystalline minerals and including flora and fauna of all types.

Douglas Caldwell et al, in 'Do Bacterial Communities Transcend Darwinism' (Advances in Microbial Ecology, Vol. 15, Plenum Press, 1997) have shown through sophisticated experiments on multi-species bacterial communities how this nested 'container-constituent-coresonant' geometry applies in nature and how the Euclidian geometry based 'natural selection' of Darwinian evolution theory is akin to the 'poor pool player's 'assertive action only' interpretation of the game of pool.  That is, Darwinian 'selection theory' is to the 'nested proliferation theory' of Caldwell et al as Newtonian theory is relativity.  Caldwell, Wolfaardt, Korber and Lawrence comment;

"Until the development of fluorescent molecular probes and confocal laser microscopy, there were few alternatives to isolating microorganisms from their communities prior to laboratory study. Isolation was necessary to obtain a sufficient amoont of homogeneous cell material for chemical analyses, yet it constrained most laboratory work to the molecular, cellular, or organismic level. However, fluorescent probes and other molecular techniques now allow the analysis of individual microorganisms without isolation. ... This affords the opportunity to perform community-level laboratory experiments that are not possible with plants and animals due to their large size. However, inconsistencies between evolutionary ecology, ecosystem ecology, microbial ecology, germ theory and information theory make it difficult to formulate testable hypotheses that are relevant in understanding ecology at the community level. Consideration of communities as units of proliferation (and hence as units of evolution) requires a more generalized theory of life, amenable to the formulation of community-level hypotheses and tests. . . . As an alternative [to the evolution of explicit entities] we suggest a proliferation hypothesis that offers a simpler and more comprehensive explanation of ecology and evolution by recognizing the possibility of propagation and reproductive success at many different levels of biological organization simultaneously (genes, plasmids, cells, organisms, communities, ecosystems, etc.) rather than solely at the level of individual organisms (species populations). We also suggest that laboratory communities of bacteria may provide one of the few experimental systems readily amenable to the testing of this hypothesis."

The degree to which the culture at large has an entrenched investment in 'selection theory' is the degree to which the culture at large has an entrenched investment in Euclidian space and absolute time.

 The point is that Euclidian space and absolute time are the simplest approximations for describing the geometry of space and, as Poincaré says ( the authoritative depth of Poincaré's on non-Euclidian space and 'The Relativity of Space' is unmatched by modern discussions on space geometry); "the space revealed to us by our senses is absolutely different from the space of geometry."

Einstein recognized the tremendous cultural lock-in to Euclidian space (the equivalent to the lock-in to 'perspective' noted by architects such as Don Kunze) and was very vocal on the fact that space is not Euclidian. In 'Ether and Relativity', Einstein goes to great lengths to point out that space, ... "not only conditions the behaviour of inert masses, but is also conditioned in its state by them.", ... "the recognition of the fact that 'empty space' in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials g(mu-nu), has, I think finally disposed of the view that space is physically empty."  And together with Leopold Infeld in 'The Evolution of Physics' says; "Our world is not Euclidian.  The geometrical nature of our world is [reciprocally] shaped by masses and their velocities.  The gravitational equations of the general theory of relativity theory try to disclose the geometrical properties of our world. . . . It [relativity] forces us to analzye the role played by geometry in the physical description of the world."

While our life experience (e.g. the freeway driving etc. examples) tell us that 'space conditions our behaviours' and that 'we are also conditioned by space', and while this is taken for granted by indigenous cultures, it is hard for us of the western culture to assimilate because the notion of the world being describable in terms of 'the assertive behaviours of independent causal agents' (out of the context of any reciprocal relationship with our containing space) is the underpinning of almost everything we do, ... hierarchical control based management, win/lose competition, the primacy of action management over opportunity management etc.  It may be worth re-visiting what Euclid might have been thinking about, to better understand how and why our minds have been conditioned to the degree they are.  As Abraham Akkerman says in 'Place and Thought';

 * * *

"The adherence to the square or to rectangular landscaping patterns appears to manifest, once more, a city ideal whose different aspects meant to correspond to the universe or to the human soul.   . . . By the time Greece reached its Classical period, c. 4th century BCE, much of the estate in both town and country throughout was already subdivided into uniform rectangles, thus ensuring equitable land distribution.41 In so far as landscape permitted, orthogonal layout of new settlements or rebuilt old settlements was, in fact, the norm in much of classical Greece.42" . . ."During the winter of 332-331 BCE a plan of Alexandria was prepared by Dinocrates of Rhodes on a strictly orthogonal grid. Alexandria became the most magnificent of cities in the Hellenistic world, home of the Syracusan mathematician, physicist and inventor Archimedes (c. 287-212 BCE), a place where he is believed to have discovered his most significant laws of mechanics. It was also Alexandria that gave rise to the three geometricians, Euclid (c. 330-275 BCE), Erathostenes (276-194 BCE), and Ptolemy (Claudius Ptolemaus, Second century BCE). Ptolemy, also an astronomer and a geographer, expounded the comprehensive geocentric notion of our solar system, putting firmly in place an astronomical concept that would last for almost two millennia, till the 16th century."

 * * *

Had the radius of the earth been smaller, the reciprocality between space and matter might have come more strongly into the cultural consciousness (for a 'fairy-tale' version of how that might have played out, see 'King Ruhtra of Tolemac'.)

So, in spite of having technology-amplified our faculties and thus shrunk earth down into a 'global village' where we are very aware of the reciprocity between 'action' and 'opportunity', our mainstream science and culture are still 'hung up' in Euclidian space and absolute time frames.  What are its attractive features which keep it in position as 'the preferred space geometry' of the western culture?

There are a few reasons that I can think of, as follows;

1. We manage and govern via centralized control hierarchy architecture which emanates from Euclidian space.

It is not a trivial matter (in terms of the psychology of those entrenched in it, though youth has far less a problem) to revert from a Euclidian control hierarchy to the curved space-management system of indigenous tradition or the 'round table' of Celtic tradition (ancient-medieval) where leaders do not 'run for election' but are appointed by the people on the basis of their 'wisdom' and the harmonious influence they induce in the community. 

As Descartes says in prefacing his 'down-and-back-up-again' analytical model in 'Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences', for handling all manner of 'problems';

"Et comme la multitude des lois fournit souvent des excuses aux vices, en sorte qu'un État est bien mieux réglé lorsque, n'en ayant que fort peu, elles y sont fortement observées, ainsi, au lieu d'avoir ce grand nombre de préceptes dont la logique est composée, je crus que j'aurais assez des quatres suivants, pourvu que je prisse une ferme et constante résolution de ne manquer pas une seule fois a les observer.

 (And as a multitude of laws often furnishes excuses for evil-doing, and as a State is much better ruled when, having but very few laws, these are most strictly observed; so, instead of the great number of precepts of which logic is composed, I believed that I should find the four which I shall state, quite sufficient, provided that I adhered to a firm and constant resolve never on any single occasion to fail in their observance.") [the four being; --- never accept what is not evidently true, --- fragment the problem into many small parts, --- solve from the bottom-up and, --- be totally thorough.  It's worthwhile noting that Descartes published 'Discours de la méthode' in 1637, four years after he abandoned and destroyed most of his 'non-Aristotelian' work on 'Monde ou Traité de la lumiere' on learning of the condemnation of Galileo in Rome.]

There remains a prevailing advocacy in the modern culture for this stern approach to 'problem solving' in the 'modern ideal republic' as well as in 'science' at large. The curved space administrative tradition puts trust into the primacy and uses control as support, while the Euclidian administrative architecture does the reverse.  Another way to state it is that management and governance in the curved-space tradition is 'inductive-over-assertive' rather than 'assertive-over-inductive'; ... the friendly freeway drivers 'co-create opportunity space into which they can assert', while the 'drive-to-rule' freeway drivers put the quality of their independent assertive driving skills into the primacy, out of the context of 'the shape of dynamic opportunity space' (as do the poor pool players who, while they may make 'great shots' and please the gallery audience, are no match for the skilled 'shape-over-shots' managers. ).

2. Science is economically driven by the demand for technology.

There is no 'shortfall' in Euclidian science with respect to mechanical systems, systems which need only 'assert' out of the context of their 'inductive' effect.  The native constituents of space, from minerals to humans, as both Kepler and Poincaré note, do not guide their behaviours by the laws and equations of science, but respond directly to the dynamical shape of space in which they are immersed.   This opens the door to the curved space-time view which enables the 'container-constituent-coresonance' of ecologies (as in the case of the friendly freeway drivers).  The point is that scientific description based on Euclidian space concepts deals only with 'assertive behaviours' and if one wants to study 'container-constituent-coresonance' as found in the 'harmonic' aspects of the solar system and natural ecologies (including social ecologies), then one needs to consider the inductive-assertive reciprocal effects.

Kepler noted (in the terminology of his time) that the curved-space view was necessary to understand the 'container-constituent-coresonance' aspects of our solar system, and this meant avoiding dependency on the 'absolute trajectories' of the planets and keying to 'space-time phase information' to visualize the volumetric codynamics (the 'shape of the dynamic opportunity space').   While this is treated more extensively in 'Including the Tools of Inquiry in the Inquiry', a passage from Kepler's 'Harmonies of the World' makes the point;

 * * *

"For if the ratios of the journeys are harmonic, all the other effects which the planets have will be necessitated and bound up with the journeys, so that there is no room elsewhere for establishing harmonies.  But whose good will it be to have harmonies between the journeys, or who will perceive the harmonies?  For there are two things which disclose to us harmonies in natural things: either light or sound: light apprehended through the eyes or hidden senses proportioned to the eyes, and sound through the ears.  The mind seizes upon these forms and, whether by instinct (on which Book IV speaks profusely) or by astronomical or harmonic ratiocination, discerns the concordant from the discordant.  Now there are no sounds in the heavens, nor is the movement so turbulent that any noise is made by the rubbing against the ether.  Light remains.  If light has to teach these things about the planetary journeys, it will teach either the eyes or a sensorium analogous to the eyes and situated in definite place; and it seems that sense-perception must be present there in order that light of itself may immediately teach.  Therefore there will be sense-perception in the total world, namely in order that the movements of all the planets may be presented to sense-perceptions at the same time.  For that former route --- from observations through the longest detours of geometry and arithmetic, through the ratios of spheres and the other things which must be learned first, down to the journeys which have been exhibited --- is too long for any natural instinct, for the sake of moving which it seems reasonable that the harmonies have been introduced

 Therefore with everything reduced to one view, I concluded rightly [287] that the true journeys of the planets through the ether should be dismissed, and that we should turn our eyes to the apparent diurnal arcs, according as they are all apparent, from one definite and marked place in the world --- namely, from the solar body itself, the source of movement of all the planets: and we must see, not how far away from the sun any one of the planets is, nor how much space it traverses in one day (for that is something for ratiocination and astronomy, not for instinct), but how great an angle the diurnal movement of each planet subtends in the solar body, or how great an arc it seems to traverse in one common circle described around the sun, such as the ecliptic, in order that these appearances, which were conveyed to the solar body by virtue of light, may be able to flow, together with the light, in a straight line into creatures, which are partakers of this instinct, as in Book IV we said the figure of the heavens flowed into the foetus by virtue of the rays.

 * * *

Science today, as science historians have pointed out, is more about 'doing science' than doing 'scientific conceptualization'.   As Caldwell observes in 'Post-modern ecology - is the environment the organism?' in regard to why it is difficult for scientists to see beyond 'natural selection'; "... devising increasingly intricate explanations for altruistic behaviour in terms of either individual or group self-interest (kin selection, group selection, ecosystem selection etc.) is the generation of mythology rather than the advancement of scientific understanding.  It serves no useful purpose unless selection theory is an end in itself." ... "The primary difficulty is that, although technology has been strengthened during the past 50 years, scientific thought has weakened.  Scientific reasoning is sometimes referred to in the popular press as 'mind-numbing post-modern jargon (Cartmill, 1998), and it is often completely absent from technical journals, being regarded by technologists as philosophical rather than scientific.  Few scientists are required to take courses in the philosophy of science, and some do not realize that they have degrees of philosophy (PhD)."

The same complaint is being heard in medicine where the runaway Euclidian train is going even harder and faster with its 'epicycular activity' than in biology, fuelled by the huge market in symptom-suppressing drugs.  Those who would argue that the body involves 'container-constituent-coresonant' phenomena are treated as heretics (e.g. in the HIV - AIDS and Genetically Modified foods debates.) and deprived of funding, opportunity and 'access to the microphone'.

3. Euclidian space based scientific theory is supported by deceptively convincing experimental replication of theoretically predicted results..

Here again, Henri Poincaré has spoken well and clearly;

"It is not enough for a theory not to affirm false relations ; it must not conceal true relations"

Euclidian theory is not equipped to deal with simultaneous, reciprocal transformation of the shape of space (space-time transformation) as is the larger 'flip-side' view of 'kinetics', ... it is only equipped to deal with the 'assertive behaviours of independent causal agents'.  Thus the experimental predictions made with Euclidian science do not include (are innately precluded from including) 'container-constituent-coresonance' effects as are manifestly obvious in nature.  In effect, science which constrains itself to Euclidian space and absolute time conceals true relations , and that is the issue that Kepler, Poincaré and Einstein were 'going on about', about how our dependent use of Euclidian space neglects the effects of the participation of the geometry of space in physical phenomena.  When the little red light goes on that the parking lot is full, the traffic strangely stops turning in to the parking lot.  Does a powerfully assertive force-field issue forth from that little red light?, ... or does it 'induce' some alteration in the pattern of assertive behaviours?  Scientific theory which speaks only to 'assertive behaviours' and ignores simultaneous space-time transformation effects will tell you more than you need to know about how the vehicles assertively pass down the street, but such theory will not tell you why they stopped assertively turning in to the parking lot because that is a non-assertion, ... an 'induced effect' which operates on the assertive patterns, patterns which are fully defined by the 'shape of space' and cannot be fully defined in terms of 'assertive trajectories'.   The non-assertions are important data for understanding system evolution.  The cars who wanted to park in the lot were 'snookered' in pool terms and the geometric 'shape of space' patterns of snookering and opening of opportunity corridors are the relativistic reference frame which induces assertive behaviours.

The shape of opportunity space, the reciprocal to the form of the constituents is undefined in science based on Euclidian space and absolute time thus, the theoretical predictions of mainstream (Euclidian) science can neither include nor validate opportunity space transformation information, but are 'blind' to it.

Summary: --- How did 'architecture' get inverted wrt nature and why do we need to 're-invert' it?

Pronunciation: 'är-k&-"tek-ch&r
Function: noun
Date: 1555
1 : the art or science of building; specifically : the art or practice of designing and building structures and especially habitable ones
2 a : formation or construction as or as if as the result of conscious act <the architecture of the garden> b : a unifying or coherent form or structure <the novel lacks architecture>
3 : architectural product or work
4 : a method or style of building
5 : the manner in which the components of a computer or computer system are organized and integrated

Architecture, in its broadest sense, as 'the art of achieving coherent, unifying form' cannot 'start' from the notions of structure and construction, but must start from the relationships between 'things' and their containing space.

At first glance, Plato and Euclid's 'ideal city' architectures seem to fall perfectly well into that definition of architecture, ... but there is one important characteristic of nature's architecture which Plato and Euclid did not address, and that is 'container-constituent-coresonance'.  When you design something and then try to use it, ... fill it with people of different types, men, women, soldiers, legislators, ... 'interferential' effects emerge which cannot be predicted on the basis of the properties of the constituents, and if one wants to achieve 'coherent, unifying form' in a codynamical sense, as nature often does, ... then an 'architecture' must seek an understanding of coherency and unification in codynamical terms rather than designing by summing the dynamical properties of the included constituents.

A moment's reflection tells us that space-matter (container-constituent) codynamics gives rise to situations where the constituents 'take their cue' from the shape of the containing space which forms from their codynamics.    Easy to visualize examples include the codynamics of street crowds and traffic on a freeway, ... where what is important is to co-create the opportunity corridors for assertive movement which serves everyone's purpose.

What goes on here is that the constituent 'references to' the shape of his containing space and lets his assertive actions be guided by their influence on the co-created shape of space.   In other words, the reference frame for his movement is 'the shape of dynamic opportunity space', .. his reference frame IS NOT EUCLIDIAN SPACE AND ABSOLUTE TIME.

Further reflection informs us that this 'space' we are referencing to is 'unbounded' in space and time, ... it is a local co-created (co-formed) estuary of the space-time-continuum, a continuously transforming 'shape' otherwise known as the 'universe'..

Coordinating our movements by referencing to a local co-created estuary of the space-time continuum is manifestly possible, and its description is beyond the capabilities of finite systems of logic and mathematics.  This referencing, wherein the constituent keys to the shape of the codynamics of his fellows; i.e. the shape of the dynamic opportunity space which he can move into in such a way as to co-sustain a coherent and unifying codynamical form; i.e. 'container-constituent-coresonance', is not confined to humans, in fact the architecture we are immersed in seems to possess this same coresonant characteristic   As E. E. Richards says;

* * *

"Each cosmic body---a planet, a moon, or a star---utilizes the spherical shape as its energy containment. The sphere is an ideal shape for a resonant cavity. The very nature of the sphere means that it resonates over vast spectrums of frequency. For example, if we start by considering the earth circumference of approximately 7.5 Hz., as a fundamental, we may calculate and detect many higher harmonically-related frequencies. In addition, there are radius frequencies with higher harmonics present. Harmonic waves in a spherical solid set up a periodic distribution within the inner and outer spherical cavities. The Van Allen energy belts surrounding the earth also present a multitude of resonant harmonics at lower frequencies than the circumference 7.5 Hz. Figure (9) shows some of the earth related frequencies. …

 … "The Music of the Spheres", an ancient concept of the Universal Song, may be seen as a reality when considering the motions of the solar system. When the revolutions and rotations for the planets and their moons are converted to frequencies, there appear many harmonic relationships. For examples: the Moon's revolution is harmonically attuned to the three largest moons of Jupiter, which are themselves one octave separated from each other in their revolutions. Jupiter's rotation is a harmonic of the Earth, and Pasiphae, the outer moon of Jupiter, is in harmony with the Earth's revolution. There are many other similar solar system harmonics.


* * *

The summary point is that the codynamical aesthetics of architecture go beyond the 'golden mean' of Plato and Euclid, a static ratio, and find their deepest and truest home in the resonant harmony of space and matter, as in the coherent and unifying codynamics of the solar system.   The structural emulation of this codynamical architecture, where Plato and Euclid started off from to design their 'ideal city'  is a secondary precipitate and while one can reduce codynamical architecture to structural precipitates, as happens with the honey bees whos co-creative serving of their larvae precipitates exquisitely optimized hexagonal storage cells, one cannot start from the structures and ensure that codynamical harmonies will follow.

Something is missing in architecture that starts from rational considerations of the constituent structures, properties and behaviours, .. it is not possible to build the structures and wait for the 'container-constituent-coresonance'.   As Richard Feynman says in 'Cargo Cult Science' (adapted from a Caltech commencement address in 1974), we need to get honest about architectures which sound good in theory but which 'don't work';

 * * *

"Another example is how to treat criminals. We obviously have made no progress -- lots of theory, but no progress -- in decreasing the amount of crime by the method that we use to handle criminals.

Yet these things are said to be scientific. We study them. And I think ordinary people with commonsense ideas are intimidated by this pseudoscience. A teacher who has some good idea of how to teach her children to read is forced by the school system to do it some other way -- or is even fooled by the school system into thinking that her method is not necessarily a good one. Or a parent of bad boys, after disciplining them in one way or another, feels guilty for the rest of her life because she didn't do "the right thing", according to the experts.

So we really ought to look into theories that don't work, and science that isn't science.

I think the educational and psychological studies I mentioned are examples of what I would like to call cargo cult science. In the South Seas there is a cargo cult of people. During the war they saw airplanes with lots of good materials, and they want the same thing to happen now. So they've arranged to make things like runways, to put fires along the sides of the runways, to make a wooden hut for a man to sit in, with two wooden pieces on his head to headphones and bars of bamboo sticking out like antennas -- he's the controller -- and they wait for the airplanes to land. They're doing everything right. The form is perfect. It looks exactly the way it looked before. But it doesn't work. No airplanes land. So I call these things cargo cult science, because they follow all the apparent precepts and forms of scientific investigation, but they're missing something essential, because the planes don't land."


 * * *

The planes are not landing on a lot of the modern architectures we have been preparing for them.  What is missing (Feynman goes on to talk about what is missing) in the architecture of Plato and Euclid is 'dimensionality'.   You can't get to an architecture which derives its coherency and unity from the shape of unbounded space-time if you design on the basis of the 'assertive behaviours of independent causal agents'.   'Cause' is laundered out of those situations where the actions of the individual are driven by the dynamic shape of volumetric space co-created by the 'community'.  There is no 'cause' in this situation in the conventional sense of the term, ... what is happening is 'emergent space-time transformation'.   The medical architectures for cancer and AIDS, where they do not consider interferential architectures and stick with 'Euclidian causal architectures' are, in effect, 'cargo-cult science'.  Unfortunately, this is where the mainstream of medicine 'is'.

When Lobachevsky squeezed more than one line in through a single point adjacent to a straight line, without having any of their extensions touch the extensions of the adjacent parallel line, .. Lobachevsky was essentially proving the primacy of the shape of space over any discrete bounding geometry.  A rectangle projected onto a curved landform, which has a hill on it, will give the farmer who gets it more than an equal share of arable land.  Thus the 'perfect' rectangles in Plato and Euclid's 'ideal city' were of a 'perfection' limited to its 'flatspace class'.   As A. Square says, in Edwin Abbot's 1880 classic 'Flatland' to the paragon of three-dimensional perfection, Lord Sphere; 

"My Lord, your own wisdom has taught me to aspire to One even more great, more beautiful, and more closely approximate to Perfection than yourself.  As you yourself, superior to all Flatland forms, combine many circles in One, so doubtless there is One above you who combines many Spheres in One Supreme Existence, surpassing even the Solids of Spaceland."

Had Plato and Euclid 'thought big' and laid out their 'ideal city' over the whole earth, perhaps they would have realized the primacy of the shape of space over the perfection of the discrete structural form, but then again, maybe not since Euclid's Elements stressed the innate ideality of the square and Plato's elements assigned fire to the tetrahedron, air to the octahedron, water to the icosahedron and earth to the cube (the edge properties would have likely been allotted to the lower classes).

In any case, this is supposed to be a 'summary' so I had better 'keep moving'.

The first summary point, just above, was that the codynamical aesthetics of architecture go beyond the 'golden mean' of Plato and Euclid, a static ratio, and find their deepest and truest home in the resonant harmony of space and matter.   That is, it is apparent that the legacy of logic and mathematics which is built into our western culture is of the 'flatspace' variety which does not recognize the natural primacy of 'the shape of space', as in relativity and curved space-time.  It is our flatspace mathematical tools (Euclidian space and its bundled partner absolute time) which have been the foundations of our disciplinary science architectures.

To paraphrase Poincaré, while we have imposed Euclidian space and absolute time on our disciplinary architectures, we cannot impose them on nature.  The constituents of nature do not in the natural course of affairs respond to their environment on the basis of the laws and equations we formulate using Euclidian space and absolute time.  What mainstream science gives us, based on the Euclidian frame, is a voyeur 'flatspace' description of physical phenomena which denies or ignores the primacy of the relational, dynamical form of space over the discrete constituent structures and their independent kinetics.

.If we want to address 'container-constituent-coresonance' and 'container-constituent-coevolution' effects in the architecture of our science, as in post-Darwinian evolutionary biology and in 'resonant' architectural designs, we need 'curved space' representations to do this.

That is, if we want not only to describe the assertive behaviours of the constituents of space, but also how their assertive behaviours induce transformation in the geometry of opportunity which modulates their patterns of assertive behaviour, then we need a curved space geometry.

If we want to consider the world in terms of 'dynamic opportunity space in the primacy over assertive behaviours of the constituents' as in the Heraclitean, Keplerian, Einsteinian and post-Darwinian evolutionary biology models, then we need an architecture based in curved space-time representation.

If we want to understand psychological and physiological illness in terms of dynamical equilibria amongst multiple interfering constituents (e.g. bacteria, cells etc.) then we need a psycho-somatic architecture based in curved space representation since Euclidian space and absolute time gives us only a 'causal view' without consideration of the predominating resonance and dissonance effects.  In fact the Euclidian paradigm clearly has us searching for 'phantom cause' while making us blind to the interferential ('container-constituent-dissonance') origins of social, psychological and physiological malady.

If we want to understand 'how' a group of collaborating freeway drivers co-creates the shape of dynamic opportunity space so as to sustain the opening of opportunity corridors for their continuing assertions (how they get into 'container-constituent-coresonance') and how one or two 'independent causal agent drivers' can infuse dissonance into the 'community', then we need an architecture based in curved space representation.

The entrenched use of Euclidian space dates back to our 'inverted-with-respect-to-nature' architectures for 'community' (the ideal city,... its assertive structures and its control based management) which have been based on 'flatspace' geometry.   This is, in effect, 'cargo-cult architecture' since all structures in nature are precipitated by curved space-time dynamics which cannot be represented by flatspace architectures.  Excluded middle logic puts assertive statements dealing with absolute 'in their own right' entities into the primacy over the space-time relationships in which they are immersed.  Gödel's Theorem, basically says that finite systems of logic and mathematics cannot deal with 'self-referentiality' (such as, for example;. 'this statement is false') and find themselves in unresolvable paradox.  That is, 'logic' cannot stand on its own head and see how its 'doing' with respect to its containing space, ... the electronic fuel system keeps pumping gas while the car is burning, ... one could patch this oversight, but the possible container-constituent-codynamical relationships are infinite, .. and thus flatspace architecture designs tend to be 'purely assertive', based on particular purpose out of the context of the shape of dynamic opportunity space in which the architecture is an 'inclusion'.  Yet we know that our assertive action simultaneously, reciprocally transforms the shape of dynamic opportunity space, e.g. asserting into a private and unique estuary of space-time in the act of making love (referencing one's actions directly to the shape of space-time), simultaneously, reciprocally closes down opportunity to make love and induces changes in local patterns of assertive behaviour.

The structures of our Platonic-Euclidian architectures, by failing to account for the simultaneous reciprocity between 'what we assert' and the enveloping 'opportunity to assert', are in effect 'holding a torch' for the re-emergence of 'container-constituent-coresonance', ... meanwhile our lighting bill is rising at a far faster rate than is coherency and unity in our social-environmental, inductive-assertive containing sphere. 

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