Figure 12: A simple method for deriving geometric quarters and linear thirds using straight line geometries. Further divisions show the extension into eighths and sixteenths as well as sixths and twelfths. The resulting compound grid is often referred to as a tartan grid due to its resemblance to traditional Scottish fabric weaving.
Note that dividing the form down the horizontal and vertical centers yields linear halves and quarter areas (13·1). Similarly, linear thirds yield ninths as areas (13·2).
Diagram 13·1: Finding the cen-ter and quarcen-tering the square.
Diagram 13·3: Further subdivi-sions into fourths.
Diagram 13·5: Continued subdivisions into eighths.
Diagram 13·7: The resulting tartan grid.
Diagram 13·2: Finding the thirds within the square.
Diagram 13·4: Halving thirds to find sixths.
Diagram 13·6: Half again to determine twelfths.
Diagram 13·8: The grid and its geometry.
The familiar, by contrast, anchors us to reality. Distinguishing relationships between the familiar and the alien occupies much of the history of humanity and architecture. One of the first tasks in architecture is making a safe place, a familiar place, a defined place. Geometry arises from this need, and shapes its practice.
Dividing the square
As we observe a rectilinear figure, a square perhaps, we note the points of con-trast against a background – usually its corners. Thus, we establish the square as a figure differentiated from its ground. Once we connect the corners, we clarify its perimeter, but we also perceive its center, a particular form of mid-dle. Further on, implied lines crossing the center from the midpoints of its boundaries divide the original figure into four quadrants.
We may also perceive each one of the quadrants as smaller figures within the larger one. This makes of the original square a ground for the smaller fig-ures. Similarly, we may divide each of those smaller figures toward their cen-ters, creating a grid of yet smaller squares. The result of all of this subdivisions leaves us with a hierarchy related to the original figure – quarters and halves measuring linearly, or sixteenths, eighths, etc. measured by area.
As we begin with the drawing of the subdivisions – the halves – a diagonal that bisects the half also serves to subdivide the square into linear thirds. This likewise can generate sixths, twelfths, etc. (). We encounter this sort of sub-division folding paper to make origami, paper airplanes or cootie-catchers. A side-to-side or corner-to-corner fold will divide a page in equal halves. A top-to-side fold will measure a square from a rectangular page. Repeated folds will continue with subdivisions and reflect the underlying geometries.
Geometrical procedures enact how we might begin to measure and appor-tion the world. Visual mechanisms – figure-ground, percepappor-tion of center, etc.
– share this nascent geometry. Our front-back, left-right, top-bottom sense of our bodies – our cruciform selves – also play a part in this (). Our per-ceptions help construct our understanding and organization of place ().
Both our perceptions and our systems analyze and propose order. They relate to the architecture of the world for us, whether they cause it, or merely share in it. They are native to our selves.
Figure 13: Standing upright with our hands outstretched, we easily apportion the world to reflect our orientation. We perceive things as ahead or behind, left or right of and above or below us.
Figure 14: The diagram to the left shows the perceptual pat-tern of boundary and center in a square figure.
Figure 15: The variation to the right diagrams the proportion-ing of space from within the square figure.
*
The architecture of both the narrative and the fam-ily demonstrate that order, whether perceived and imposed, or perceived and distilled, participates in how we make sense of the world. It is inevitable per-haps that some forms of order come to be seen as either clearer or more desirable. Philosophy, aesthet-ics in particular, describes competing views about preferred, better constructions. From that discussion there emerges, almost inevitably, notions of an ideal.
The histories of both architecture and philosophy encompass much debate about whether there is an ultimate ‘best form’, an ideal. We encounter some of these as we describe architectural procedure. And although we are not without opinion, our goal is to present the terms of the argument without pretending to any ultimate conclusion.
As we measure things in relationship to one another, we eventually encoun-ter scale. The continuing subdivisions and relationships simply overwhelm our point of view. At some point, the web of lines outstrips our capacity to discern pattern, just as the size of the paper limits the number of folds. We might fur-ther ask what happens when one group of measurements overlaps anofur-ther.
Can we measure dissimilar things similarly?
One scenario is to impose the method first defined by Descartes, thereby ordering from without, using a neutral method or tool. The usefulness of the Cartesian grid resides in its abstraction and purity of method. The grid can provide a common expression for things with or without intrinsic form or shape. Algebraic formulae are just one example of how we gain form through the grid.
Aside from classes in geometry, algebra and physics, a contemporary expe-rience of the grid occurs at the computer monitor. The screen composes the world as dots of light, or pixels. The lower left-hand pixel of the viewing sur-face serves as the arbitrary zero point. Two numbers assign subsequent pixels to rows or columns. This is the practical Cartesian grid. It can count to infinity, although in practice it ends at the screen’s boundaries. An image can be larger or smaller, but it starts at zero and moves up in increments without real scale, albeit with practical size limitations.
The grid of the computer screen exists outside of meaning until we assign it a proper interpretation. As the digital environment evolves the character of its conventions becomes ever-more important. One reason to understand the evolution of earlier conventions – what we might call ‘local relational geom-etry’ – is that they provide a ground from which to critique the digital realm.
In this text, we discuss the universal Cartesian grid after we explore the fun-damentals of the relational grid. Half as many pixels is different from half of a form or shape. The difference is nuanced but vital to considering measure and order.
Rules of engagement
The history of architecture intertwines with the histories of mathematics and philosophy. Criteria for how to do things well compete in the marketplace of ideas.* Relational calculus and the grid both make use of scale and order to
Figure 16: Stones strewn ran-domly, encountered by chance in a field.
Figure 17: Viewed as a set of four, they may describe an irregular quadrilateral figure.
Diagram 19·1: Viewed in plan, the stones define this irregular figure.
Diagram 19·2: The simple assumption of corner diagonals allows us to bound the form using a larger square with the same center point.
Diagram 19·3: This second variation shows the diagonal center point locating a smaller, enclosed square.
Diagram 19·4: This alternative sharing of the diagonal center point defines a square of equal scale to the original.
Figure 18: The drawing shows the boundaries and axes center point of the figure identified by the four stones.
represent the world. Stories commonly have beginnings, middles and ends.
Modern narratives may rearrange the parts, but their structures remain a matter of scale, moving from the episode to larger compositional entities:
fables, epic poems and novels. Similarly, one might begin with an individual and move on to siblings, parents, etc., continuing to extend the family tree by branching forward, backward or laterally. Such an enlarged set of relation-ships might construct a clan, community or even a nation. These sorts of cal-culus emerge from both a starting point and perceived boundaries or rules of enclosure.
Imagine coming across several stone markers in an open field (–). If we identify the stones as a group, we might start to perceive figures and bound-aries in the open space (). In doing so, we begin to structure the hitherto unstructured terrain. We begin to make architecture within space.
In our example, the group may be seen as an irregular quadrilateral – four sides neither parallel nor at right angles. Such a figure might result from toss-ing four stones from a stoss-ingle place while attempttoss-ing to construct a square.
If this were our intention, it would also be possible to adjust the stones to achieve the more regular figure of a square. This is not simply a rhetorical ges-ture; regular figures follow sets of rules by definition. In the case of the square, the rules are exceedingly simple. All four sides are of equal measure, as are the two diagonals. Middle divisions follow accordingly. We make sense of an irreg-ular quadrilateral by overlaying it with the simpler architecture of a square. In doing so, we construe both figures as diagrams.
With diagrams in mind, we can easily perceive three possible relationships.
The square can surround the first figure, or vice versa; the two figures can share alignments at their diagonal center or share corner adjacencies. They might also be merely near one another. Diagrammatic geometry helps us to organize and analyze the forms as well as to visualize and act on the rela-tionship. The diagrams to the left (·–) illustrate the relationship of three squares that share an axial center with the irregular figure.
In practice, irregular figures commonly appear within a site, the tangible context for building design. Site boundaries may result from irregular land-forms, prior orientations of roadways and rivers or from more abstract and circumstantial conditions arising from a site’s particular history. Strategies for
Figure 19: The drawing to the left shows an amalgam of multiple orthogonal relation-ships between the figures.
These include edge and center relationships, groupings and shifted dynamics.
The diagrams to the right (20·1–8) identify each separately.
Diagram 20·1
Diagram 20·3
Diagram 20·2
Diagram 20·4
Diagram 20·5 Diagram 20·6
Diagram 20·7
Diagram 20·9 Diagram 20·10
Diagram 20·8 working through the nuances of irregularity derive from overlays drawn from
the simpler architecture of regular figures.
Returning to the four stones, we can demonstrate how elemental spatial relationships clarify such strategies (). Here, the relationships hinge upon three simple notions: center, edge and boundary. We can identify centers and edges of each individual stone and the entire composition. Boundaries derive from relationships between two or more stones. As we characterize those relationships, we discover four fundamental visual concepts: ,
-, and . In our diagrams, we note that the four stones are approximately the same size, thereby exhibiting little contrast in dimension (·). The image of four similarly sized objects arrayed in a roughly orderly manner prompts an acknowledgement of repetition (·).
Visualizing the relationships between centers and edges depends on ideas of alignment (·–). Comparing the intervals between objects in their arrange-ment defines their proximity (·–).
Contrast includes a larger number of differences, as we shall discover in
. Here, as we measure the irregular forms of the stones, we see that they are approximately the same scale and likeness. The square boxes drawn around the shapes in · help us approximate and represent that similar-ity. Recognizing that the arrangement easily fits within a square wherein the stones roughly occupy the four corners, suggests an interpolation of basic repetition on both horizontal and vertical axes. Diagram · compares the ordered repetition to the actual placement of the stones, while · illustrates the ideal with all elements aligned and evenly spaced.
Measuring the of the forms, we discover that alignment rela-tionships outside of that ideal nonetheless use the same boundaries of each stone to construct sight lines from edges and centers. Comparing · with
·, we note the former gathers two edge relationships at the fulcrum of the farthest outlying stone while the latter organizes a grouping in which the cen-ter of one stone relates to the inner edges of two others. As we continue our observations by completing the boundary lines of the upper-left and lower-right stones, we see that the edge relationships (·) govern a broad horizon-tal and vertical area, whereas the centered relationships (·) form a more tightly constrained visual area.
Figure 20: Four squares placed at the corners of a square – a positive shape – also create a cruciform negative space.
Our last observations measure horizontal and vertical distances between pairs of stones (·). These comprise the proximity of elements within the whole arrangement.