This is the third in a series of posts exploring the ideas of the mathematician and design theorist Nikos Salingaros, and by extension those of his collaborator, the architect Christopher Alexander.
The first and second posts in this series covered Nikos Salingaros’ theories on fractals and fractal scaling in architecture. Here we will build on this foundation by looking at Salingaros’ exploration of how fractal structures are generated, via his analogical ‘physical’ model.
The basic model Salingaros proposes is the ‘push-pull’ model, which incorporates the two axial forces of tension (pulling) and compression (pushing), and their respective effects, perforation and folding, to account for how various architectural (and fractal) elements are generated in architecture. The role of bending forces and processes, in generating ‘boundaries for space’ such as curves and domes, is touched upon but not in great detail, so will not be covered here. Nor does Salingaros consider shear in relation to architecture, and it would be interesting to explore the possibilities of incorporating this force into his models.
Perforation, like the theoretical example seen in the Sierpinski gasket discussed in the previous post in this series, is the process of generating openings in a surface. In architecture, perforation results in windows, doors and other openings. Perforated elements have the property of semi-permeability; they let some elements through and prevent others from passing. Salingaros gives the example of the bollard, which permits pedestrians but blocks cars.
Perforations are generated by tension. Imagine a strip of rubber coated with sealing wax. If you pull on both ends of the rubber strip (i.e. apply tension), the wax will crack at regular intervals along the length of the strip - first into larger pieces, then each larger piece into smaller and smaller pieces. Note here that these cracks or ‘perforations’ are oriented perpendicularly to the axis of the tensile force - the importance of this point will become apparent in the next post in this series. Applying this analogy to architecture, we see that if we metaphorically ‘pull’ a wall horizontally, it will perforate into vertical openings - firstly windows and doors, and then, if we keep pulling, the wall will further separate out into arcades, then columns.
Diagram by Nikos Salingaros illustrating the process by which increasing horizontal tension creates first windows, then arcades, then columns.
Folding, by contrast, is the process of generating elements like folds, meanders, thickenings, hollows, and bulges by applying compression. Whereas perforation removes material from a plane, to fold is to fill space: ‘folding the line is the first step to filling the space slightly’. Architecturally, folding finds expression in articulating elements like pilasters on a wall, the capital, base, and fluting of a column, and thick door and window frames. Salingaros also gives the example of alcoves in a temple wall.
An alcove in a wall, flanked by pilasters.
Again, note for later that compression creates folds that are oriented perpendicularly to the axis of compression, so horizontal compression creates vertical fold lines, and vertical compression creates horizontal elements, like the ‘bulging’ of a classical column at its head and base.
Diagram by Nikos Salingaros illustrating the process by which increasing horizontal compression creates folds in a plane at successively smaller intervals and scales.
Classical columns embody the effect of vertical compression: bulges at the head and base, and entasis of the shaft.
From a structural perspective, folds create strength or stiffness in a material by moving parts of it away from its central axis; this is why steel sheeting is corrugated. Salingaros points out that a floor with beams exposed on its underside is visually expressive of the same idea: the beams can be regarded as the locations of strengthening ‘folds’ in the plane of the floor.
From a structural perspective, the beams in the floor plane are equivalent to the corrugations in a sheet of corrugated iron.
In the next post, I will consider the implications of Salingaros’ push-pull model on approaches to architectural design, in particular in explaining the traditional emphasis on verticality over horizontality in architecture, which, as Salingaros demonstrates, is not merely a superficial stylistic preference, but has an objective basis in physics and evolutionary biology.