The Generative Dynamics of X, Y & Z Coordination


Manuel A. Báez, Architect, B. Arch., M. Arch.

Co-ordinator: Form Studies Unit, Foundation Year and Admissions

School of Architecture, Carleton University, Ottawa, Ontario, Canada







The architectural work-in-progress titled the Phenomenological Garden has been exploring the morphological and integrative potential of cellular units generated by fundamental processes within natural phenomena.  As part of the overall objectives of this project and the Forms Studies Unit at Carleton University’s School of Architecture, students in the Crossings Workshop have been carrying out these explorations through projects that incorporate hands-on procedures derived from the research.  The inherent properties of the cellular units, along with the nature of the materials and processes involved in these projects, allow for a generative and intuitive learning process to occur.  Previously, the generative dynamics of two-dimensional cellular units have been explored (see the Paper: Generative Dynamics: Process, Form and Structure, 2004 Generative Arts Conference Proceedings).  This paper will present the work that has emerged from the exploration of the X, Y and Z coordinate system as a fundamentally dynamic relationship within a generative cellular process.



1. Introduction


“In this brief account of coordinate transformations and of their morphological utility [The Theory of Transformations, or The Comparison of Related Forms] I have dealt with plane coordinates only, and have made no mention of the less elementary subject of coordinates in three-dimensional space . . . And that it would be advantageous to do so goes without saying, for it is the shape of the solid object, not that of the mere drawing of the object, that we want to understand; . . . But this extended theme I have not attempted to pursue, and it must be left to other times, and to other hands.”

D’Arcy Wentworth Thompson [1], also see Fig. 1



Throughout the natural environment we find fundamental processes that generate versatile systems and patterns.  These highly fertile, self-organizing and regulatory processes inherently exist within, and generate, the rich realm of natural phenomena.  Simultaneously, they are composed of, and self-generated by, elemental geometric relationships that gradually evolve into versatile integrative systems with startling form and structure generating capabilities.  Through the systematic analysis of the versatility and generative potential of these systems and their interrelated cellular patterns, new insights can be revealed into the emergence of complex morphological structure and form.  The intrinsic nature of these dynamic patterns reveals that they are cellular configurations of highly ordered relationships.  The extremely dynamic undulations of the flow of energy are constrained within the apparently static stability of the pattern.   This versatile constrained activity fluently encodes the emergent pattern with complex potentiality offering a multitude of possible or alternative ‘readings.’  The cellular units comprising these patterned networks innately contain the intrinsic attributes of the versatile processes that generate them.  Inextricably, we are participants in, and surrounded by, this rich and dynamic matrix of natural phenomena.  The inherent properties and characteristics of this generative matrix can be systematically explored allowing for the possibility of insightful understanding of its fluent potentiality.  This analytical process offers new insights into the nature of the reciprocal relationship between matter, developmental processes, growth and form.  Rich educational methodologies are offered through new procedures and techniques that can inherently allow for intuitive learning through self-discovery.  The Phenomenological Garden project is a work-in-progress and in-process that has been inspired by these insights and the working procedures that they can reveal regarding nature’s developmental processes.  It seeks to explore the form and structure generating potential of these dynamic processes along with their elemental components, emergent integrative properties and pattern generating capabilities.



2. Elemental Cellular Dynamics


Through systematic analysis, the dynamic potential of basic geometric relationships has been explored, leading to the development of a series of flexible cellular units and hands-on analytical

procedures.  Inherently, this allows for intuitive discovery to occur regarding the interrelationships between form, structure, and generative process.  The cellular units are constructed using bamboo dowels and joining them together with rubber bands thus creating a very malleable joint.  By combining these very flexible units together into three-dimensional configurations, the form generating potential of both the individual cells and the cellular assemblies can be easily explored.  The flexibility of the joints and their complex three-dimensional relationships, generate a wealth of forms and structures through the emergent, transformative and organizing properties of the integrated assembly.  The dynamic properties of initially two-dimensional cellular units have been explored (see the paper: Generative Dynamics: Process, Form and Structure, 2004 Generative Arts Conference Proceedings).  These have been hands-on dynamic explorations of what were primarily graphic, two-dimensional and static explorations by D’Arcy Thompson [1] through his “Theory of transformations or the Comparison of Related Forms” (see Fig. 1 below).





Figure 1: D’Arcy Thompson, grid or coordinate transformations of graphic depictions of biological forms [1].

As he recommends in the introductory quote above, Thompson’s “extended theme” of three-dimensional coordinate transformations have been explored through investigations of the generative dynamics of such complex assemblies.  The following is a presentation of some of the forms and structures generated from the emergent properties of several intrinsic combinations of a cellular unit that is a dynamic three-dimensional assembly of the X, Y and Z system of co-ordination.



2.   X, Y and Z Co-ordination


In Figure 2 we see four views of a cellular unit constructed with 12" and 5" bamboo dowels and joined together with rubber bands.  The unit is composed of three surfaces (or planes) at right angles to each other with each surface being defined by four 12" dowels assembled into a grid of two pairs at right angle to each other and four 5" dowels, one at each end of the 12" pairs (see Figure 2 D).  The three surfaces have a high degree of transformability due to the flexibility of the joints and each surface defines one of the X, Y and Z coordinate directions in three-dimensional space.  Each surface can fully collapse along the two orthogonal diagonals of the assembled grid.  They can also be warped into a transformable, collapsible and highly flexible hyperbolic paraboloid.  Three-dimensionally, this cubic cellular unit (or module) is composed of several interacting degrees of freedom through the combination of flexible joints (a total of 42).  From another perspective, this complex intermingling is also the interactions of the three flexible hyperbolic paraboloids within the three-dimensional assembly.  In Figures 3 and 4 we see several configurations that can be generated from this dynamic interplay.


                        A                                  B                                 C                                  D


Figure 2: Views of the X, Y & Z Cellular Unit: 12" and 5" bamboo dowels and rubber bands.  Three planes at right angles to each other: D clearly shows one of the planes with the central diagonal edges of the other two; B & C show views through the four diagonals of the cubic assembly.



In Figures 4 and 5 we see several of the transformations that can be generated from the cellular unit through a systematic hands-on investigation of its dynamic properties.  In the Crossings Workshop, students have been exploring this cellular unit along with the forms, structures and dynamic properties that emerge when several of these units are combined.  The numerous possible combinations lead to unexpected overall patterns and dynamic arrangements that generate new and diverse developmental directions for the assembling process.


                A                               B                    C             D

Figure 3: The Cellular Unit and several of its basic transformations. A: The Cellular Unit.  B: Flattened assembly along one of the four diagonals of the cubic assembly.  C:  Collapsed assembly centered around one of the four diagonals.  D: Collapsed X, Y and Z axes with 5" dowels removed (see Fig. 5).


                       A                                 B                                 C                                 D

Figure 4: Different transformations of the Cellular Unit.  In A the 5" dowels have been removed.  Each one of these configurations becomes the “modified” cellular unit that is then assembled together.


                                        A                                 B                                   B

                                        C                                  D                                 E

Figure 5: Different stages of a cellular unit that can completely collapse into the X, Y and Z axes (A & B) and gradually expand into a tetrahedron (C, D, E and F).





Figure 6:  Two views of the same construction, by M. Báez, using the cellular unit shown in Fig. 3.  The construction is a dodecahedron that emerged from the assembling process.  Throughout the structure and the generated patterns one can discern the squares, pentagons, triangles, hexagons, cubes and tetrahedrons that are intrinsically embedded within the dodecahedron.





Figure 7: Cellular Constructions.  Left: By M. Báez, constructed with the same unit as in Figure 6 and exhibits the same properties.  Right: By Sarah Amirault, constructed using the unit shown in Fig. 4 B.  Different patterns are revealed throughout these constructions.  The X, Y and Z axes can be clearly seen in the overall pattern of the construction on the right.



In Figures 6, 7, 8, 9 and 10 we see several forms and structures that can emerge as the assembling process gradually evolves into more complex configurations.  Figures 6 shows two axial views of the same construction.  This particular assembling process generated a dodecahedron that was not preconceived nor initially anticipated.  Cellular units (as shown in Figure 3 A) were assembled together using their inherent properties as the guiding principles.  Within the resulting three-dimensionally dynamic pattern of the form one can discern the complex interweaving of the rich geometric properties of the dodecahedron: cubes, tetrahedrons, hexagons, pentagons and golden rectangles (to name a few) in a reciprocally complex relationship.  Several of these shapes can be discerned in the two views provided.  Figure 7 is another construction generated through the same process as in Figure 6 and also reveals the same level of complex multilayering of forms.  On the right side we see a construction that incorporates the cellular unit shown in 4B and the X, Y and Z axes of the initial cellular unit are equally prevalent at this level of evolving complexity. The different modifications to the original unit in Figure 2 lead to the emergence of totally different complex patterns and dynamic properties. 





Figure 8: Cellular Constructions, Study models by M. Baez, constructed with the unit in Fig. 3.  On the left are two views of the same model and on the right are three views of another.



Figure 9: Cellular Constructions. Work by Ana Lukas constructed with the unit shown in Figure1.  Top view on the left and under construction on the right.






Figure 10: Cellular Constructions.  Work by Michael Putman, Patrick Bisson and Rheal Labelle, constructed with the unit in Figure 3.  Top view on the right and a side view on the left.



Figure 8 shows study models constructed with the cell shown in 3A.  On both the right and left sides we see different views of the same construction.  Figures 9 and 10 show views of two other constructions that have been developed to a more complex level than the ones previously shown.  The transformability of the cellular units generates very different overall complexity throughout the larger assemblies.  Figure 9 shows a toroidal construction that was assembled with the same unit and procedures used in Figures 6 and 7 (left side).  The form shown on the left side of Figure 6 fits directly into the central opening of the form shown in Figure 9 (left side).  By comparing the two views shown in Figure 9, one can see the dynamic diversity within the elaborate pattern.  Figure 10 shows the most complex construction that has been made with the cellular unit used in Figure 6.  On the right side we see the top view through the main vertical axis of the elaborate assembly and on the left, a partial side view.  The elaborate pattern is ever changing throughout the structure.  Overall, the emergent patterns are at times reminiscent of the patterns generated by vibrations in liquids and in thin layers of fine powder.  Throughout all of these constructions, dynamic patterns emerge with an ever-evolving intricate level of complexity.    Paradoxically, within the integrative interactions of this complexity lurks the simplicity of the original cellular units.



2.    Conclusion


“We have been trained to think of patterns, with the exemption of those in music, as fixed affairs.  It is easier and lazier that way but, of course, all nonsense.  In truth, the right way to begin to think about the pattern which connects is to think of it as primarily (whatever that means) a dance of interacting parts and only pegged down by various sorts of physical limits and by those limits which organisms characteristically impose.”

Gregory Bateson [2]


     The rich diversity found throughout nature’s processes challenges our creative imagination and common sense because of its reciprocally related combination of dynamic complexity and simple organizing principles.  The work-in-progress presented here, along with the broader goals of the Phenomenological Garden, inherently address this fundamental paradox through multidisciplinary research and an integrative working process.  Such an approach offers new possibilities and directions to the fields of morphology, architecture and other creative disciplines at a time when there is an increasing interest in the broad implications of our deeper understanding of Bateson’s “dance of interacting parts” throughout the physical world.




 [1] D’Arcy Wentworth Thompson, On Growth and Form.  Complete Revised Edition: Dover, p. 1087.  1992.  Chapter XVII covers The Theory of Transformations, or The Comparison of Related Forms (P.1026-1095).


[2] Gregory Bateson, Mind and Nature, Bantam Books, p. 13-14.  1980