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9th April 2006 | Uncompleted

Punctuated Evolution of Marks

Questions in the intelligent design of numbers and Zen ox-herding pictures

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This exploration is stimulated by the incompleteness implicit in a question and the manner in which that is signified by the design of a question mark -- "?" The graphical form of such a design may be understood in terms of:

The challenge to comprehension in its various stages is emphasized in this exploration from two contrasting but complementary perspectives:

An effort is made to point to a degree of correspondence between:


Principles of evolution in the light of intelligent design

  1. A point emerges from an implicit circle, more explicitly as the circle itself becomes more explicit
  2. An explicit point extends into a line, whether explicitly or in an implicit dimension
  3. An explicit line tends:
    • to curve, seeking closure into an explicit circle and, in doing so,
    • to extend implicitly as a plane (thus curving into a cylinder)
  4. An implicit cylinder tends to curve, seeking closure, such that the two curved ends become explicit as separate circles (possibly only partially closed)
  5. An explicit curve, as it closes into a circle, engenders a point (cf point 1)
  6. An implicit cylinder (continuing the process of point 4) tends to seek closure as an implicit torus in 3 dimensions and
  7. The torus in 3 dimensions (continuing the above progression), forms into an implicit toroidal hypersphere as a geometrical generalization of the circle (the 2-sphere), and 3D sphere (the 3-sphere), in more than 3 dimensions [more | more]

An emphasis is placed here on the "explicit" or visible features of design in contrast with "implicit", emergent features -- an "explicate order" versus an "implicate order" (cf David Bohm, Wholeness and the Implicate Order, 1980):

As explored in various classic fictionalized mathematical descriptions (cf Edwin Abbott Abbott, Flatland: A Romance of Many Dimensions, 1884; Dionys Burger, Sphereland, 1965: A. K. Dewdney, The Planiverse, 1984; Ian Stewart, Flatterland, 2001; Rudy Rucker, Spaceland, 2002):

Application to the design of numbers

These principles are applied to the sequence of numbers on the assumption that there is an element of "intelligent design" in the emergence of the graphical form of the numbers.

It is readily assumed that numbers are in some way simple conventional "answers" -- arbitrarily chosen in response to the need for a label for a static, well-defined category that is essentially "closed". The exploration here, however, assumes the possibility of a degree of "openness" to understanding associated with a given number. It is further assumed that there is some merit in understanding numbers as being in process of a certain kind of change or evolution. Two is "in process" of becoming three. There is, in this sense, a plasticity to numbers. They "flow" and "grow". This process is to some degree reflected in the graphic design of the form whereby a number is represented as the mark on a two-dimensional plane.

Graphically this emergent tendency is perhaps suggested by the possbility of a number being represented with a serif, rather than without. In metaphorical terms, the number form may then be understood as "putting out shoots".

In some sense therefore, it is assumed here that numbers are not static "answers". They are emerging "questions".

The transitory shapes associated with a question mark may then be understood as transition forms between numbers.

  Transform Point Line Curvature (2D) Cylindrical (3D) Toroidal (3D) Remark
1 extension extended extended point        
1   - idem - - idem - serif      
2   (implicit in curve) reoriented expanded, seeking closure partial developing implicitly    
3 flipped 2 extended extended point   partial implicit cylinder curved to make other end explicit effect of toroidal curvature in 3D apparent in 2D  
8             Möbius strip
9             Klein bottle

Zen ox-herding pictures

The classical Zen sequence of ox-herding images exemplifies the stages of progressive understanding [more | more | more].

Kubota Ji'un (Ten Ox-herding Pictures with the Verses Composed by Kakuan Zenji, 1996)

This process of individuation is frequently depicted in Zen Buddhism by a traditional sequence of 10 ox-herding pictures, each with a brief commentary (cf D T Suzuki). These are of special interest because of their indication of a person's progressive discovery and interplay with a shadowy element denoted by an ox. The following is an attempt to suggest how that classical sequence might be interpreted for clues to an unfolding relationship between humanity and its shadow (in the shape of the complex of world problems).

The phases in the sequence are (Integration of Perceived Problems; commentary in Encyclopedia of World Problems and Human Potential):

  1. Undisciplined exploration of the problematique: Humanity, having violated its own inmost nature, loses track of the problematique and its significance. It is then led astray by the delusions to which it succumbs, such as desire for gain and fear of loss, and is confused by a multiplicity of views of right and wrong, appropriateness and inappropriateness. Although distracted by this confusion, and exhausted by its efforts, humanity continues its search for a sustainable solution. At this time, it would appear that humanity, as represented by the international community, continues to be embroiled in the pre-systemic, single-factor perspectives of this first phase (ozone, acid rain, "health-for-all", substance abuse, illiteracy, terrorism, AIDS).
  2. Recognizing traces of the problematique as an integrated system: Repeated (and basically unsuccessful) attempts to locate and contain the problematique through uncoordinated initiatives provide humanity with occasional insights into its nature, especially when more integrated approaches are used. Although recognizing that the problematique, by whatever means, is in some sense engendered by humanity as a whole, there remains a basic confusion between truth and falsehood, especially when it seems obvious to some that another particular group can be usefully blamed for specific problems. Environmental and systems insights (tropical forests, global warming) are shifting the focus to this second phase.
  3. Focusing on the problematique as a whole: Having cultivated a more intuitive insight, enabling it to integrate its complementary modes of perception, humanity focuses directly on the problematique, recognizing its many manifestations as consequences of different forms of inappropriate human intervention. There are episodic exercises in focusing on the problematique as a whole (Brandt Report, Brundtland Report), although what they fail to take into account quickly condemns them as sub-systemic and inappropriate and encourages further initiatives of a similar nature.
  4. Encompassing the problematique: Humanity grapples with the problematique directly for the first time. The momentum of the problematique, developed over the long periods during which it was uncontained, and the pressures and habitual opportunities of an undisciplined social environment, make it extremely difficult to control. Severe disciplinary measures are necessary. The various development strategies, especially the current attempt at "sustainable development", correspond to this fourth phase, but only to the extent that efforts are made to implement them. On the national level, the structural adjustment required by the IMF is indicative of the political will required -- although typically such adjustment fails to take into account many facets of the problematique.
  5. Orienting the problematique: Every insight concerning the problematique leads humanity to further insights in an endless pattern. With discernment these will all be of value. But when humanity deceives itself, confusion will prevail and the problematique will reassert itself in an inappropriate manner. Constant vigilance is required to discipline the problematique and orient its manifestations within appropriate bounds. The seeds of this fifth phase may be seen in the increasing recognition of the need for a disciplined and radical change of life style, especially on the part of the industrialized countries.
  6. Using the problematique as a vehicle for sustainable development: The struggle of humanity with the problematique is over. Humanity is no longer traumatized by gain or loss, which are assimilated as phases in a larger process that is now the focus of attention. Rhythms of action in harmony with nature are cultivated. The problematique is used as a vehicle moving in sympathy with those rhythms towards the re-enchantment of the Earth. The old modes of action are not considered viable and their advocates are no longer heeded. Some indications of the nature of this phase are to be found in the writings of the "deep ecology" movement and in the preoccupations of some forms of sustainable agriculture -- although their obvious limitations lie in their inability to deal realistically with the conditions of industrialized, urban societies and the impoverishment of an overpopulated planet. The missing insight would seem to be how to achieve the transition to this stage by benefiting from the problematique itself.
  7. Transcending the realm of the problematique: Having used the problematique as a vehicle to reach a sustainable condition, it is no longer required. However, the necessary disciplines for humanity to handle it remain available. Humanity can now act with serenity guided by insight that is no longer obscured by the dynamics of the problematique. There are writings on paradigm shifts into a new consciousness (in which the problematique no longer figures) and these do offer clues as to the nature of this phase. However, their neglect of the problematique would seem to be more a question of avoidance rather than transcendence, indicating that such perspectives lack vital insights.
  8. Disappearance of both humanity and the problematique: The dualistic mindset through which humanity is perceived, in opposition to the problematique and to other species, is itself transcended, as are the disciplines through which that relationship is articulated. Confusion disappears. But there is no question of being either entranced by more integrative insights or entrapped by lesser ones. The nature of this condition does not lend itself to definition. Typically, any desire for it renders it unattainable or unsustainable.
  9. Expression of essential humanity: Grounded in its essential nature, humanity stands untouched by inappropriateness. Processes of integration and disintegration are witnessed from a perspective that enfolds them. Neither formulation nor reformulation are necessary to ensure sustainability. Change, as perceived, is necessarily appropriate however paradoxical it may appear.
  10. Human intervention in the world: Human action is no longer associated with any particular mindset, nor does it follow any recognizable path. It cannot be assessed by any form of conventional wisdom, nor does it depend on any particular tools. No special effort is made to preserve forms of any kind -- including those of humanity itself. Insight into the emptiness underlying form enfolds any form of action in a more meaningful context, thus enabling greater appropriateness to emerge as required.


Arabic origin

find a story / fable

mnemonic education


ecosystem network of design transforms (vect equil) with punctuation


memetic transfortms


when does 3D start?

Punctuated evolution:


David Bohm. Wholeness and the Implicate Order. London 1980

Spencer Brown

Grail dialogue

Q and Answer

David Pratt. David Bohm and the Implicate Order. Sunrise Magazine, February/March 1993. [text]






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