author’s image

D’Arcy Wenthworth Thompson, On Growth And Form, 1917

“The finest work of literature in all the annals of science that have been recorded in the English tongue” as described by the novelist Peter Medawar, is really just all preface from beginning to the end- a preface to allometry. The “On Growth and Form” shook the scientific explanation of  morphogenesis despite the fact that it does not really include a single unifying thesis. Nor does the visual thinker- Sir D’arcy Thompson- attempt to establish a resulting relationship between the forms seen in biology and forms based on physics.

Thompson tries to break the processes by which patterns are formed in plants and animals into its simplest components using descriptive and analytical methods. The study of form starts with describing it by words and ends in a definition in the precise language of mathematics. The common misconception is that mathematical definitions, being very concise, are too strict for common use. We apt to think their rigor is limitative, while in fact it offers endless freedom. The functions of mathematics within its laws are bound to underlie all parts of physical science and all composite physical phenomena. The curves of higher order, conic section and other akin curves are interpretations of forces in the static and dynamic relations- being  the merging point of mathematics and physics (simple example- catenary curve- a hanging chain with two points of arrest – 0-growth points). A raindrop should in theory be a regular perfect sphere because water always forms in the minimal surface, but because physical forces are applied the shape deforms obliquely. Although it is not regular anymore, it can still be mathematically defined. Amorphological forms are nothing but identical functions of different coordinate systems and simple graphic expressions with the occurrence of localized variations, extensions not uniform at all distances etc. Thompson strives to look for mathematical explanations in defining plants’ and animals’ patterns using the method of coordinates, interchange and substitution of x and y coordinates etc. In the schemes only mathematics and physics are considered, therefore he only depicts the ideal sample state and is not symptomatic of what the shapes really undergo in the nature.

Thompson points out – in example after example – correlations between biological forms and mechanical phenomena, like the observation of  numerical relationship between spiral structures in plants and the Fibonacci sequence. In the following threads he explores the extent to which differences in the forms of related animals could be described by means of relatively simple mathematical transformations.

In his instructive cases of grid deformation and transformation he draws an analysis of a common por-cupine fish, Diodon. After converting the vertical coordinates into a system of concentric circles and making the horizontal lines provisionally resemble a system of hyperbolas, the old outline, transferred in its integrity to the new network, appears as a manifest representation of closely allied but different looking sunfish. And as such it accounts, the system being now non-isogonal can only be visually valid. Using the same principal, Thompson uses non-uniform deformations, elongated shifts and rectilinear coordinates transformed to coaxial circles in the mathematical analysis of mammalian skulls. By curving the lines of intersection on a human skull grid, Thompson arrives to a simian type skull. This exertion however can only be performed within listings of topological similitude. In other words, a beetle and a cuttlefish do not fit in the same framework and although the shape could perhaps be distorted, there are no invariant basis for this generic transformation. For all contractual his work is, his permutations of organic forms find a common base with the work of Albert Durer featuring face profile analysis.

The geometrical analogies weigh heavily against Darwin’s conception of endless small continuous variations, because the assumed mutations creating new types are excluding many physico-mathematical possibilities. The book has never conformed to the mainstream of biological thought. Thompson claimed Darwinism to be an inadequate explanation for the origin of new species. He did not reject natural selectionbut regarded it as a secondary to the origin of biological form.The central conclusion could be that biologists of the author’s day overemphasized evolution as the fundamental determinant of the form and structure of living organisms, and underemphasized the roles of physical laws and mechanics.

A research that could be done would evaluate mathematics’ influence in terms of architectural design nowadays. Mathematics is applied in all 3D programs and digital machines like 3D printers that pioneer the emerging architectural styles. The contemporary design and architecture are driven by the mathematical definitions and formulas translated into the computer language which makes it a lot easier for all kinds of users- even those lacking knowledge in the field of mathematics. If nowadays, everyone is able to ‘produce’ architecture, what should be the next step in our pursuit of form? An important issue to analyze would be the genesis of form now- based on mathematical definitions- and the evolution of the tools and methodological strategies. Mathematics is indeed endless in its possibilities, however new tools could include more incorporation of physics and chemistry in creating the form, or even, conducting simulations inside the programs. If Thompson pioneered in his independent insight into the form genesis in 1915 and somehow predicted the computational methods, how much does it take for us to rise up to our century’s level? It is not enough to use the digital tools in a repetitive manner because a prospective mean for revolutionary thinking lays far beyond applied techniques we have already learned about. My question is: are we there yet?

Paths formed by livestock over time are an example of the natural formation of optimised detour path networks.

Schumachers’ Parametricism – A new global style for Architecture & Urban Design (2008) is an interesting text covering many of the broader topics of Parametricism. It provides the reader with a brief insight into the world of parametricism, it’s history, and it’s real world feasibility & applications.

 The initial paragraph deals with the lead up to Parametricism, claiming that it “closes the transitional period of uncertainty that was engendered by the crisis of modernism”.

This introduction provides us with an insight as to the role Schumacher envisions for parametricism in the 21st century. He talks of parametricism as the new style, to follow from post-modernism.

Parametricism is the great new style after modernism

From here, Schumacher talks about the benefits afforded by Parametricism in the post-fordism age. The ability to customise every object at the same price of mass production and repetition has lead to a “heterogenous society of the multitude, marked by proliferating life-style and career differentiation.”

The following definition of parametric sensibility leaves the reader with an insight as to the fundamentals of parametric design being employed to agglomerate complex features into a uniformed solution.

Aesthetically it is the elegance of ordered complexity and the sense of seamless fluidity.

The various taboos (rigid geometry; simple repition; juxtapositions of unrelated elements or systems) are explored, and compared to the  dogmas of parametricism (malleable forms; differentiation; inflections and correlations). From this comparison we begin to see that Schumacher stresses the point of parametricism being not about spaces themselves, but about the laws surrounding spaces. This is emplified by the tests undertaken by Frei Otto – tests that can be applied to any space and project, with an infinite number of outcomes and solutions.

The next topic covered is the relationship between modernist urbanism and parametricism. The comparison made is between Le Corbusier’s  limitation of only seeing order in strict geometry, as opposed to that of Frei Otto, whom sees order in natural complexity through the laws that govern the spacial qualities and forms.

Le Corbusier’s limitation is not his insistence upon order but his limited concept of order in terms of classical geometry. Complexity theory in general, and the research of Frei Otto in particular, have since taught us to recognize, measure and simulate the complex patterns that emerge from processes of self organization.

Schumacher provides numerous examples of Otto’s work, including the Magnet and Polystyrene chip test, and the wool thread model test. The latter test provides us with an brief explanation of optimised detour path networks, one of three fundamental types of  network configuration in Otto’s experiments -

Direct Path Networks; Minimal Path Networks; Minimizing Detour Networks.

From here, we are given an introduction into the laws controlling the relationships in design. Schumacher talks of Relational Fields, and the subsidiary layers creating these fields. The conclusive notion here is that patterns of occupation directly correlate to patterns of connection, and it is the relationship between these two patterns which can define a design.

To conclude his text, Schumacher provides us with the case study of a project being developed by his firm, ZHA. The Kartal-Pendick Masterplan, in Turky (2006), is an excellent example of the optimisation of path networks, and the habitation of the space provided by these networks. The seemingly complex masterplan is grounded by deep relationality.

”Ordered complexity here replaces the monotony of older planned developments and the disorienting visual chaos that marks virtually all unregulated contemporary city expansions.”

ANALYSIS

Schumacher deals well with the idea of Parametricism as a style which governs the laws of object, spacial and network design. He makes a strong comparison between Le Corbusier and Frei Otto, and their very different opinions of order and structure. Firstly Le Corbusier is used an example to the forces against parametricism. However, Schumacher provides us with a limit to this implied ignorance, by quoting Le Corbusier:

Nature presents itself to us a chaos… the spirit which animates nature is a spirit of order.

A major theme explored  is the adaptability of parametric design laws to any project. This theme however is limited in that once a project is constructed, the form cannot be changed, and hence a design must be layed out to accommodate various uses over a projects lifecycle.

The use of variations of path networks, and their relationships to the final form, are incredibly important in this text and evidently in Schumacher’s work.

Although the text provides us with a strong insight into parametricism, Schumacher attempts to explore and define too many topics in too little space. The result is that the text is very inconsistent in its exploration of the themes, and finally appears to be very broad and lacking of depth.