^{EVERY THREEDIMENSIONAL FORM IS BORN FROM ITS PLAN AS A TREE IS BORN FROM ITS ROOTS.}
^{—DANIELE BARBARO, }^{LA PRATICA DELLA PERSPETTIVA}^{, 1569}
The corpus of plans we will be studying consists of those in Book II of Palladio’s Quattro Libri and some of his villa project drawings. Our first goal is simple: to generate plans that are horizontally symmetrical and modular, meaning that they are comprised of rectangles only.1 To begin, we must translate the rules about symmetry and modularity outlined in chapter 1—rules that, as we saw, can be written in different ways to achieve the same result—into language the computer can use. Planmaker, our computer program for generating plans, will construct its plans by splitting and resplitting the rectangle formed by the building’s perimeter into smaller rectangles. We define these splits as lines or sets of lines that divide a rectangular space into any number of smaller rectangular areas.
The direction of a split may be horizontal, vertical, or a combination of both. Any combination split could be reconstituted using a sequence of horizontal and vertical splits. However, we define the combination split as a unique, third variety because it always creates a grid; a sequence of horizontal and vertical splits does not always create a grid. (As we shall see, a grid is often the foundation of a Palladian plan.) Figures 2.1a, b, and c demonstrate the three split directions; figure 2. Id illustrates what will not be defined as splits. As we see in figures 2.1a and b, a split does not necessarily create rooms of equal size. Figure 2.1a also defines how we will use the terms “width” and “length.”
Three characteristics define a split: direction—horizontal, vertical, or both—number, and ratio. Number is the number of new rooms created by the split. Figure 2.1a illustrates a triple horizontal split that creates three rooms. Figure 2.1b is a quintuple vertical split, and figure 2.1c a double horizontal split combined with a triple vertical one. The split ratio defines the relative proportions of the new rooms. Using the same split direction and number (referred to in combination as the split type) but varying the ratio, one can divide a room many different ways. Figure 2.2 shows four of the many possible split ratios for a triple vertical split.
If we continue the splitting process, it creates what we will call split trees, namely sequential splits done in stages like the generations in a family tree. Split trees are analytic as well as generative devices. We can use them to describe any of Palladio’s existing rectangular plans. (As we shall see in chapter 4, we could also split so as to create nonrectangular rooms, but for now we must limit ourselves to the standard rectangular room.)
In figure 2.3 we illustrate the split tree of the plan of the Villa Valmarana at Lisiera (Quattro Libri, 2.59). The actual plan is shown in figure 2.3a; the remaining figures illustrate progressive stages of the split tree. The sequence begins with one large horizontal 4:3 area without interior walls (fig. 2.3b).
1. In the first stage we map onto 2.3b all continuous interior walls from the original design. (For the present we will ignore doors and windows; they come later.) The Villa Valmarana at Lisiera has two such walls running horizontally its entire width. We isolate them in figure 2.3c. The top and bottom rooms are each as wide (measuring across) as the middle room, but only ⅜ as long (measuring downward). So our new walls define a triple horizontal split with a ratio of 3:8:3.
2. We now consider independently each new room created by the first split. (We show old splits in gray lines, new ones in black.) There is no obligation to split each of these rooms in the same way, but the same criterion as in the first stage determines what new splits we will identify. For each new room we will include any continuous wall from Palladio’s plan that creates smaller rooms within the three spaces in figure 2.3c. Figure 2.3d is the result. It was produced by applying a quintuple vertical split in the top and bottom rooms (in which we simplified the original 15:7:50:7:15 to its close but simpler relative 2:1:7:1:2) and a triple vertical split (with ratio 1:1:1, simplified from 32:30:32) in the middle room.2
3. Now another split occurs, a simple double horizontal one, with a ratio of 1:1 (fig. 2.3e). In keeping with the rule for bilateral symmetry, it is identically applied to the two rooms flanking the central space.
4. In the last stage (fig. 2.3f), we split the upper section of each of these flanking rooms vertically, and symmetrically, using the ratio of 4:3; the larger room thus created is on the outside in each case. The original plan of the villa is now completely duplicated.
Let us return for a moment to figure 2.3e. This stage marks the first moment in our process in which certain rooms created by the previous stage were not split. These rooms, then, have no further branches in the split tree. Further, as suggested earlier, a single split may be both horizontal and vertical at the same time. In figure 2.4 we see the plan of the Villa Angarano (Quattro Libri, 2.46). It has a combination horizontal/vertical split in the first stage (fig. 2.4c) that creates its defining grid (both horizontal and vertical split lines run the full length of the building). By contrast, we used no combination splits in duplicating the Villa Valmarana, and it does not have the same rigid gridlike structure as the Villa Angarano. The rest of figure 2.4 maps out the same process we went through in figure 2.3b–f.
Although we do not know if Palladio actually designed plans using splits, he did use a method of subdivision and resubdivision—splitting—in other arenas. One example is his instructions for generating entablatures; figure 2.5 shows his directions for the Ionic entablature.3 Our splitting of plans is analogous to Palladio’s technique here of beginning with a starting area, subdividing it using a ratio, and then treating each resulting part as an entity that can in turn be subdivided or else left whole. The principal difference is that entablature sections, as here, have only one divisible dimension, height, while plans have both length and width.
Palladio uses this system for laying out a whole range of friezes and entablatures in the Quattro Libri. Consequently, he avoids the complex fractions that Vitruvius uses—or rather, he translates those fractions into simple geometrical terms. It is far more comprehensible to use this split system than to say, for example, that the second fascia is $\frac{4}{45}$ of the total height of the entablature, that the modillions are $\frac{10}{93}$ of the total height, and so forth. And for our purposes, it is important to note that just such a sequence, if alternated between horizontal and vertical splits, could be applied to a house plan.
Theoretically, then, we can duplicate Palladio’s plans by using the split system. But so far this is simply a way of giving Planmaker a language in which to describe Palladian plans. Let us now see if we can engage it in the more complex task of creating them. Planmaker will first choose, at random, the length and width of the starting perimeter. It will then pick an arbitrary split type and split ratio, draw out the resulting rooms, and, for each further room, pick another arbitrary split type and ratio, and draw out those resulting rooms. It will continue until the plan is complete.
But what split types and ratios should Planmaker choose? There must be some method to our madness. Since Planmaker will operate truly at random, left to its own devices it might happily divide a 20 × 40 room using a decuple horizontal split, thus creating ten exceedingly narrow rooms each 20 feet long and 4 feet wide (fig. 2.6a)! We must get rid of any tendency to do such things. Planmaker could also make the mistake, after selecting a reasonable split type like triple horizontal, of then choosing a ratio of 1:4:5 (fig. 2.6b)—and again create a worthless room 4 feet long.
To prevent these mistakes we supply Planmaker with fixed lists of possible split types from which to choose (triple horizontal, double vertical, etc.) and possible ratios for each (double splits of 1:1, 1:2, 2:3, etc.; triple splits of 1:1:1, 1:2:1, 2:3:2, etc.). What types and ratios should our lists contain? The answer, of course, lies in the Quattro Libri. We compile a split tree description of every relevant plan in Book II. From these trees we then tabulate all possible split types, all possible split ratios, and their statistical frequencies. For example, all basically sixpart plans, like figure 2.4, produce trees with a double horizontal/triple vertical split type. Tables 2.1 and 2.2 list our results. These tables, in turn, will serve as a steppingstone to other rules, rules that govern all of Palladio’s plans.
In determining split ratios, however, we encounter the wellknown fact that it is hard to get precise measurements and proportions from the illustrations in the Quattro Libri. Indeed the rooms in Palladio’s plans are rarely split precisely into the ideal ratios shown in table 2.2. It was necessary for us to round off many of the values. For example, if we split a room 44 Vicentine feet wide, as Palladio did in the Quattro Libri, into three rooms with widths of 10, 22, and 10 feet, we recorded that ratio as 1:2:1 instead of 5:11:5. Even simple ratios such as 8:3:3 were further rounded, in this case to 3:1:1. In many instances these odd ratios result from having to account for such things as wall thickness or the vagaries of the engraver. In any case, there is no harm in rounding off Palladio’s values. As one can see in figure 2.7, at the scale of Palladio’s villa plans the eye cannot register the difference between, say, 25:10:8, 8:3:3, and 3:1:1.
TABLE 2.1  

SPLIT TYPE  FREQUENCY (%) 
Double horizontal  20 
Triple horizontal  18 
Quadruple horizontal  3 
Quintuple horizontal  2 
Double vertical  5 
Triple vertical  27 
Quintuple vertical  7 
Double horizontal and triple vertical  8 
Double horizontal and quintuple vertical  2 
Triple horizontal and triple vertical  5 
Quadruple horizontal and triple vertical  3 
TABLE 2.2  

DOUBLE RATIOS  %  TRIPLE RATIOS  %  QUADRUPLE RATIOS  %  QUINTUPLE RATIOS  % 
1:1  29  1:1:1  25  1:2:2:1  50  1:1:2:1:1  34 
4:3  14  1:2:1  25  1:1:1:1  10  2:2:1:2:2  34 
3:2  14  2:3:2  13  2:1:1:1  10  2:1:1:1:2  8 
2:1  29  3:2:3  13  3:1:1:1  10  2:1:2:1:2  8 
3:1  14  1:4:1  6  4:1:1:1  10  2:1:3:1:2  8 

 4:1:4  6  5:1:1:1  10  2:1:4:1:2  8 

 1:5:1  6 





 5:1:5  6 




But we have not yet addressed one further basic question: When should Planmaker stop splitting? How does it decide between continuing to split and leaving a room whole? To rephrase these questions in more familiar terms, what is the “proper” size of a room? Palladio’s answer is vague. Except in one specific instance that we will note shortly, his only instructions regarding room size are that a building should contain “large, middlesized, and small rooms.”4 Although his specific rules are hazy, his underlying principle, easily observed from the plans themselves, is clear: Villas must contain rooms of logically varying size.5
Vitruvius also implies, without giving any very specific directions (except for atria), that in a house the rooms should be of clearly different sizes. As with Alberti and Palladio, Vitruvius’s differences are largely based on the rooms’ functions. Here we might note one sort of room that puzzles the modern eye: the interior room with no windows. It is found in Palladio’s own plans, like figure 2.3a, and will be found in ours as well. Such rooms were used in sixteenthcentury villas for storage, toilets, and the like, and also as waiting rooms for servants attending their masters who were using the larger, welllighted rooms.
And so we will have rooms of properly different sizes. We can base Planmaker’s decision to split or not on the area of the room in question. But our guidelines cannot be absolute. None of the quoted authorities specifies how many rooms there should be of each size. For example, if we were to tell Planmaker to split all rooms with areas larger than 800 square feet, then we would eliminate all such large main rooms from its plans and severely restrict variation. We would also be going against Palladio’s own example. The problem is solved if Planmaker makes its decisions to split randomly, based on a percentage. It could, for example, split rooms measuring 700 to 1000 square feet 40 percent of the time. Translating this idea into a computerusable concept, we would have Planmaker, confronted with the decision to split or not to split, pick a random number between 1 and 100 and then split the room in question only if the number were between 1 and 40.
TABLE 2.3  

ROOM SIZE (SQUARE FEET)  PERCENT SPLIT 
<300  3 
300–500  20 
500–700  22 
700–1000  40 
>1000  66 
We will adjust this percentage, of course, to accord with what we glean from Book II as to the frequency with which Palladio splits rooms with areas between 700 and 1000 square feet. We use the same principle for rooms of all other sizes: for example, how often does Palladio split rooms with areas of less than 300 feet? To collect this information we recycle the split trees that we have already compiled from the Book II plans. For each new room in any stage we now record the room’s area and whether or not it was split in the following stage. After dividing the rooms into ranges by area, we calculate the percentage that were split within each range. Table 2.3 lists the results. As one might expect, the smaller the room the less frequently it was split.
But how do we enforce the second principle we mentioned at the outset? Aside from limiting ourselves solely to rectangular rooms, we also opted, in accordance with Palladio’s plans, for bilateral reflective symmetry. To achieve it, let us now classify all rooms into one of the three groups shown in figure 2.8: rooms to the left of the vertical axis, rooms lying on that axis, and those to its right. Planmaker will split rooms along the center string as usual. However, it will insist that splits in the lefthand and righthand room groups be mirror images of each other.
A preliminary example is shown in figure 2.9. Suppose Planmaker is examining a room 40 × 17, or 680 square feet. After selecting the number 19 at random, it splits the room because table 2.3 instructs it to split rooms between 500 and 700 square feet 22 percent of the time. It then chooses a split type, also at random, say triple vertical. Next it selects a ratio for the split type, say 1:1:1. It is impossible to divide the room integrally, so Planmaker divides the 40foot width into three new rooms respectively 13, 14, and 13 feet wide. To preserve symmetry it adds the extra foot to the center room.
Now that we have instructed Planmaker as fully as possible, for the present anyway, we can begin generating our first set of “Palladian” plans. Planmaker has clear guidelines for all decisions and will operate without our aid. But, as we have warned, we foresee as yet unidentifiable mistakes and hope to learn from them. Sure enough, as soon as Planmaker is let loose using our incipient “Palladian” rules, it comes up with plenty of mistakes from which to learn.
Figure 2.10 illustrates six plans that look more like wall treatments by the art nouveau artist C. R. Mackintosh than like anything by Palladio. (Perhaps that is appropriate, given the type of computer we work with.) And yet the plans do unquestionably conform to our two most fundamental Palladian principles, rectangular rooms and bilateral reflective symmetry.
But the mistakes leap off the page. Our first complaints concern room shapes. These plans contain a plethora of overly stretched, slivershaped rooms. Palladio’s text offers advice regarding such problems, for as we have noted he lists seven ideal room shapes. These start with the square and move by simple added fractions to a maximum of two squares (i.e., the long dimension equal to twice the short dimension).6 Although Palladio does occasionally employ spaces of more exaggerated dimensions, usually porches or loggias, rooms of more than two squares are in fact rare. In contrast, the plans that Planmaker is now spewing forth (and these are only a tiny sample) contain many of these overlong room shapes. Splits only consider one dimension at a time and ignore the consequences of the other dimension. Our rules must be more specific.
We can correct this problem, as usual, by rewriting one of Palladio’s own rules for Planmaker. We will demand that it split any room, regardless of size, that is more than two squares in proportion. Yet we know already that the new rule will not work in isolation. For one thing, Palladio does sometimes call for threesquare (or threesquareplus) rooms, as in the Villa Valmarana (fig. 2.3a). But suppose, furthermore, that the rule causes Planmaker to split the overly long room in figure 2.11a. Planmaker might well decide as in figure 2.11b to split it vertically into two even narrower rooms. In short, Planmaker’s method of selecting the split type randomly from a table, without considering the room’s proportions, is inadequate. What we need to do is coerce it to “split toward the square,” that is, to pick split types such that the resulting rooms are less than two squares. Figure 2.12a illustrates what we consider to be splitting toward the square; figure 2.12b, on the other hand, illustrates what is from now on discouraged.
To put it another way, we are asking Planmaker to pick split types based on the rooms’ ratios of length to width. Exactly what should the final proportions of the rooms be? Palladio’s six canonical options (we are not considering circles for the moment), we recall, are ${\sqrt {2}:1}$, 1:1, 4:3, 3:2, 5:3, and 2:1. Since neither we nor Planmaker know which room is to be a loggia as opposed, say, to a courtyard, we will blur these proportions into a continuum. We accordingly divide all rooms into the four categories shown in figure 2.13: length greater than twice the width (a), length less than twice the width but greater than the width (b), width greater than the length but less than twice the length (c), and width greater than twice the length (d).
Hereafter, in contrast to its earlier practice, Planmaker will classify a given room in terms of these categories. Only then will it choose a split type. Each category of proportions, in turn, will require a different set of split types. The split types for categories (a) and (b) should be predominantly horizontal, and those for (c) and (d) predominantly vertical.
To further decide just what split types to choose for each kind of ratio, and what their frequencies should be, we will refine our analysis from table 2.1. We again compile a split tree of every relevant plan in Book II, but we now sort each split type into one of the four new categories of lengthwidth ratio shown in figure 2.13. The final tabulations in table 2.4 show that Palladio’s own plans, when interpreted in a computerfriendly way, do in fact have a marked tendency to split toward the square, just as we suggested above. The tendency is not absolute, however: note the frequency of triple vertical split types in category (b), and of double and triple horizontals in category (c). But these anomalies occur, we also note, only in the two middle categories, (b) and (c), where the room shapes are less extreme. Our original rule, then, which forced Planmaker to split all rooms greater than two squares, will now have its intended effect.
TABLE 2.4  

SPLIT TYPE  FREQUENCY (%) 
(A) LENGTH GREATER THAN TWICE WIDTH  
Double horizontal  10 
Triple horizontal  20 
Quadruple horizontal  50 
Quintuple horizontal  20 
(B) LENGTH GREATER THAN WIDTH BUT LESS THAN TWICE WIDTH  
Double horizontal  27 
Triple horizontal  27 
Triple vertical  20 
Quadruple horizontal and triple vertical  13 
Triple horizontal and triple vertical  13 
(C) WIDTH GREATER THAN LENGTH BUT LESS THAN TWICE LENGTH  
Double horizontal  25 
Triple horizontal  10 
Double vertical  10 
Triple vertical  15 
Quintuple vertical  5 
Double horizontal and triple vertical  25 
Triple horizontal and triple vertical  5 
Double horizontal and quintuple vertical  5 
(D) WIDTH GREATER THAN TWICE LENGTH  
Double vertical  7 
Triple vertical  73 
Quintuple vertical  20 
TABLE 2.5  

ROOM SIZE (SQUARE FEET)  PERCENT SPLIT 
CENTER ROOMS  
<300  0 
300–500  24 
500–700  21 
700–1000  25 
>1000  33 
OFFCENTER ROOMS  
<300  4 
300–500  17 
500–700  22 
700–1000  100 
>1000  100 
But there are more mistakes in the plans in figure 2.10. We now notice their conspicuous lack of a largescale central focus. Palladio’s own plans all center on main spaces—courtyards, rotundas, or oversized rooms—that he refers to as sale. They always lie on the center axis. We should once again employ our ability to nudge Planmaker, this time toward larger rooms on the center axis. We nudge rather than force because, following Palladio, not every room lying on the axis is oversized, only most of them. Such central spaces will result if Planmaker splits the rooms along the center axis less frequently than it does other rooms. Again, the answer to the question “how much less often?” lies in a refinement of a previous rule.
Table 2.3 lists the percentages of rooms split, sorted by size. We now break this information down into two categories, one for rooms on the center axis, the other for rooms off it. As previously, we then divide the rooms into ranges by area and calculate the percentage that are split within each range.
Table 2.5 shows the result. Note that the groups of smaller rooms (those less than 700 square feet) are split with roughly the same frequency for center and offcenter rooms. Once a center room is split to a small size it no longer serves as a central focus, and so is not treated differently from an offcenter room of equivalent size. Conversely, since a larger center room can be a focus, we split it less often than we would its large counterpart that does not not lie on the central axis. Using this table, Planmaker will decide how often to split a room, a decision now based on the room’s position relative to the center axis as well as on its size.
Having adjusted the rules so as to insist on rooms less than two squares and to favor, but not insist on, large central spaces, we can now generate our second set of plans. Six samples are shown in figure 2.14. Each has a central focus and there are no slivershaped rooms, so we have succeeded in correcting two of Planmaker’s more wayward tendencies. But now, just as we foretold, the solution of old problems makes new ones obvious.
To begin with, the central rooms in figures 2.14a and b still do not look really Palladian. A quick survey of the Book II villa plans confirms our intuition: none of Palladio’s plans contains a rectangular room running the plan’s entire length, and only one has a room that stretches the plan’s entire width.7 Most often, rooms that would stretch across the entire length or width of a plan would have to be oblongs greater than two squares, and our previously minted rule therefore tells Planmaker to split them. Occasionally, however, as in figures 2.14a and b, such rooms are not greater than two squares; left to its random devices, Planmaker may legitimately choose not to split them. Therefore, to insure the eradication of these planspanning rooms, we will institute a new rule that forces Planmaker to split any room that is as long or wide as the entire plan.
Another taboo is broken in figures 2.14c and d. In each plan there is one interior wall that lies directly on the center axis. In fact, both of these walls touch the front facade edge of the plan. As a result, neither of the houses could have a central front door, which in turn would destroy the bilateral symmetry of their main facades. This Palladio would never do. Looking at the plans in Book II once again, we discover no walls lying on center axes, whether touching the facade edge or not. (We do occasionally spot centeraxis walls that do not touch the facade edge in, for example, Francesco di Giorgio’s symmetrical plans.) Apparently Palladio wanted to preserve the powerful effect of entering a villa and having a sight line straight through the entire building. These offensive, sight linedestroying walls result when Planmaker splits a center room into exact halves using a double vertical split. As a cure, we will prohibit Planmaker from choosing this split type when dividing center rooms.
Lastly, figures 2.14e and f adhere to every rule devised so far, and yet they are still unPalladian: they have far too many tiny rooms. Planmaker has gone too far in its subdividing. In fact, no room in the Book II plans is less than 40 square feet; some of the rooms in figures 2.14e and f are as small as 15 square feet. The percentages in table 2.5 do strongly discourage Planmaker from splitting small rooms into tiny ones, but recall our other rule, which forces it to split rooms greater than two squares. This rule overrides Planmaker’s usual process of choosing whether or not to split, a decision made on the basis of room size. Thus we are inadvertently forcing Planmaker to split small slivers into tiny squares.
We will legislate against these unwanted tiny rooms as follows. Assume Planmaker has decided to split a room and has chosen a split type and ratio. It will then look ahead and calculate the dimensions of the resulting rooms. If any dimension is less than 7 feet it will choose a new split type and ratio. If, after several attempts, it still has not selected a satisfactory type/ratio combination, it will then abandon its attempt to split the room and let it be.
But now we return to an earlier problem. Suppose the old rule forces Planmaker to try to split a room greater than two squares. Then, in accordance with the new rule, after several unsuccessful attempts to choose a proper split type and ratio, it decides not to split the room. Does the old rule override the new, or vice versa? In other words, which is worse, a misshapen room or an undersized one? The answer is that while we find no tiny rooms in Palladio’s plans, we do, on rare occasions as noted, find rooms that are “too long.” When forced to choose between the two evils, then, Planmaker will opt for the sliverlike but larger room.
Having unmasked three more Palladian taboos, and having instructed Planmaker not to violate them, we now commission it to produce a third set of plans. Four samples are illustrated in figure 2.15. Although they do conform to all the rules identified so far, most observers would probably again classify these plans as unPalladian. Their Palladian source is clear, but they lack a certain rigor that we will now have to define.
Let us compare a specific Book II plan with the entire stepbystep process by which we can now generate one of our “school of Palladio” arrangements. On the left in figure 2.16a is the plan of the Villa Zeno (Quattro Libri, 2.49). In the righthand column is an approximation of that plan that was randomly generated by Planmaker using all the laws we have formulated so far, but still lacks that undefined Palladian rigor. Although not precisely the same as those in the Villa Zeno, the rooms in the computergenerated plan are all of acceptable dimension, size, and position. So the rigor we are searching for probably does not concern the characteristics of individual rooms. Rather, it seems to concern how rooms relate to each other.
Let us suppose for a moment that the Villa Zeno had been computergenerated. By simultaneously retracing the splits that Planmaker would have used to create that plan (the lefthand plan of each pair) and the splits it actually did use to generate the plan at right in each pair, we can detect the point at which the two plans diverge. Figure 2.16b illustrates the starting areas. The first splits are both double horizontal (figure 2.16c). The proportions in the right plan, although different, are acceptable, so we proceed to figure 2.16d. The roomtoroom relationships are still the same, as they also are after the next split in figure 2.16e. So we advance to the fourth and final set of splits (fig. 2.16f).
Now we notice a distinct difference between the two plans. In Palladio’s plan, the newly added walls align with existing vertical walls. In Planmaker’s plan, on the right, they do not. Palladio’s plan is more than a mere symmetrical bundling of smaller rectangles within a larger one. Established walls have a tendency—a tendency, not an absolute obligation—to continue when new rooms are added, even if there is an interruption between old and new wall. (Note that the question of alignment affects vertical subdivisions more often than horizontal, because mirror symmetry already assures most crosswise wall alignment.)
As Wittkower and others have pointed out, Palladio’s plans conform to underlying grids.8 But in designing plans room by room, as we have been doing, there has been no consideration of such a grid. Let us call this focus on the design of a single room “microstyle.” For example, when Vitruvius describes the Roman house he speaks exclusively in terms of microstyle; he is concerned only with the size and proportion of individual rooms. What we will call “macrostyle,” on the other hand, embodies the relationship of room to room and of room to whole. When we think of macrostyle in a villa, the key elements are conformity to an underlying grid, enclosure of the plan in a rectangular envelope, and, of course, symmetry.
Until the fifteenth century, macrostyle in this sense applied only to public buildings. Unlike the temple, basilica, or bath, the Roman house, as we have noted, was nearly always irregularly bounded, ungridded, and asymmetrical. We might say that chapter 1 of this book showed how Renaissance architects attempted to infuse domestic architecture with the macrostyle of public architecture. Indeed Palladio defines what we call macrostyle when he writes on the first page of the Quattro Libri: “Beauty will result from the form and correspondence of the whole, with respect to the several parts, of the parts with regard to each other, and these again to the whole; that the structure may appear an entire and complete body, wherein each member agrees with the other.”9 “Parts with regard to each other, and these again to the whole” is the definition of macrostyle. And Palladio applies this definition to every “fabrick,” not just to public structures. Yet as we saw earlier, microstyle cannot be ignored. After all, it is explicitly Vitruvian, and its concentration on variety of room shapes is what keeps villas from becoming monotonous grids. Macrostyle in domestic planning, which is not Vitruvian (but became Vitruvian via the reinterpretations of Cesariano, Barbaro, and Palladio), restricts variety because it encourages uniformity of interior spaces. In injecting macrostyle into domestic architecture, therefore, Renaissance architects created a conflict between variety and uniformity that had previously not existed. Palladio’s work elegantly resolves this conflict, though he may be observed struggling. (Anyone who doubts the intensity of the struggle should try teaching a computer to mimic Palladian plans and facades. It is a constant battle between anarchy and monotony.)
The limitations on the architect in this struggle between microstyle and macrostyle, then, first involve setting a plan into a rectangular boundary, which is much harder than creating a freeform bundle of rooms. Bilateral symmetry then cuts the architect’s field of choice roughly in half. But macrostyle’s constantly implied underlying grid is the most constricting demand of all. One can partially mediate between variety and conformity by using a flexible grid—that is, a grid containing modules of various sizes, as in some of the plans illustrated in chapter 1. Palladio uses this approach in the Villa Emo (figure 4.31) and the Palazzo Antonini (figure 4.32). To resolve the conflict using a flexible grid only, however, is Procrustean. Palladio only does this in the two examples cited. If the flexible grid is the architect’s only tool, then his role is reduced to that of deciding on the scale of the grid.
Palladio therefore employs more than just a flexible grid. In their article “The Palladian Grammar,” Stiny and Mitchell construct grids of squares ranging from 3 × 3 to 7 × 5. The grid in figure 2.17a is a 5 × 3 example. Having set up this matrix, they then simply remove walls, as in figure 2.17b, or shift walls, as in figure 2.17c, until a given plan from the Quattro Libri has been replicated.10
This method nicely highlights the struggle between variety and conformity, micro and macrostyle. Wall removal creates variety in room sizes by catenating two spaces into one, but only mildly violates the grid because it creates merely a temporary break. Wall shifting provides much greater flexibility in proportioning and sizing rooms, but it violates the grid more egregiously. In figure 2.17c the wall shift flouts the grid’s very existence. Of the two methods, Palladio uses wall removal more often than wall shifting, but he does shift with some frequency. His most pronounced use of wall shifting is in the nowdestroyed Villa Sarego at Miega di Cologna (fig. 2.18; Quattro Libri, 2.68), where the horizontal walls in particular are situated without heed to an underlying grid.
Our splitting method, on the other hand, is inherently microstylistic. Since it only considers one room at a time, it often utterly fails to make the “parts,” as Palladio says—i.e., the rooms—correspond to each other and to the whole. Until now, therefore, Planmaker has been completely unconcerned with aligning walls to an underlying grid as an end in itself. Only rarely, and by chance, do two walls align in the plans we have generated so far. Our next task, then, defines the architect’s conflict: how do we align walls in such a way as to preserve variety of room shapes and not turn our plans into monotonous grids?
To begin with, many existing rooms do not actually have walls with which new walls may be aligned. In these cases, Planmaker must still be free to choose its split type and ratio at random. Figure 2.19 shows a number of partially completed plans. In any room containing an X, Planmaker would calculate the split type and ratio so as to align the new walls with old walls. In all other rooms it would select the split data at random, and its decision, as before, would be based on the proportions of the room in question.
However, Planmaker should not split rooms simply because it has the opportunity to create aligned walls. Nor should the number of surrounding walls influence Planmaker’s decision as to whether or not to split a room. Only after making a decision to split will Planmaker check for surrounding walls with which the new walls might or might not align. If there are such walls, Planmaker, in most cases overriding its usual random process for choosing a split type and ratio, will align the new walls with some existing wall. To retain the variation we find in Palladio, it will choose at random how many aligning walls it will create—and hence how many new rooms. In figure 2.20a the room marked X is an obvious candidate for splitting. Figures 2.20b–d illustrate Planmaker’s three options. Finally, however, Palladio does sporadically “misalign” walls. And so, on random occasions, Planmaker will deliberately do the same. It will ignore surrounding walls and its own tendency toward alignment, choosing a split type and ratio at random.
Having established the practice of wall alignment, we commission a fourth set of plans from Planmaker. Some of these invite comparison with Palladio. Now that we have developed a satisfactory formula, we have the potential to create millions of these “Palladian” plans. Since Palladio himself only produced a tiny proportion of all possible Palladian plans, most of the new designs will not have direct analogues in his oeuvre. In the four examples illustrated in figure 2.21, Planmaker’s plan is on the left; in three cases (2.21a, b, c), the closest Palladian parallel is on the right.
In figure 2.21a Planmaker has transformed the Villa Comaro by filling in its Lshaped side wings, eliminating the upper layer of three spaces—a pair of oval stairs flanking a porch—and horizontally subdividing the square tetrastyle hall in the center. A Palladio drawing is on the right in figure 2.21b; Planmaker’s scheme on the left continues the line of the stairs partially into the reapportioned body of the villa. The lefthand plan in figure 2.21c abstracts Palladio’s interpretation on the right of a Roman house with a “Tuscan atrium” (Quattro Libri, 2.18). It further idealizes his already idealized interpretation by making it reflectively symmetric across both the horizontal and vertical axes, and by aligning it more rigorously with the underlying grid. Figure 2.21d has no close analogue, but its inspiration is clear.
We now examine two last elements of Palladio’s villaplanning practice: doors and windows. As one might expect, Palladio does not locate them arbitrarily. Their placement is subject to precise lateral symmetry and alignment11 and derives directly from the placement of walls. Thus doors and windows add another layer of geometric structure to the plans.
Palladio’s openings, whether doors or windows, almost invariably lie on major or minor axes. Figure 2.22 illustrates this in the Villa Valmarana at Lisiera (Quattro Libri, 2.59). Note that each axis coordinates with a wall: major axes run parallel and adjacent to major walls, while minor axes do the same with shorter walls. The only exceptions, the top and bottom horizontal axes, do not stretch the entire width of the plan because they are blocked by stairs, and Palladio’s stairs only have one entrance per floor. However, note that beyond the sets of stairs there are outer windows lying along the same axes. Finally, most of these axes symmetrically bisect the smallest room in their path.
Palladio’s other plans show the same hierarchy of major and minor walls, along with associated door axes, that we see in the plan of the Villa Valmarana. We have seen that the language of splits is similarly hierarchical. A split tree, after all, can hardly be otherwise. The succession of walls and axes, then, follows naturally from the split analysis of a given plan. Major walls are those comprising earlystage splits; minor walls appear in later stages. Each new room created by a split should therefore contain a door axis that bisects the room and parallels the new wall.
This being so, Planmaker can simultaneously generate door axes as it splits rooms. Later, after finishing its plan, it can use the door axes to cut openings through the walls. Figure 2.23 illustrates how Planmaker does this. The starting area, shown in figure 2.23a, contains one axis cutting vertically through the top wall, down the center, and through the bottom wall. Superimposition of this axis on the completed plan will create front and back entrances as well as a line of doors running down the center of the villa.
Except in one specific case discussed below, we will create a door axis for each new room. Bisecting the new room, the axis parallels the direction of the split that created the room and is exactly as long as the wall that has just been created. All door axes, except the initial vertical one, run to, but not through, the end walls of their corresponding rooms. Figure 2.23b illustrates the first split and the axes created along with it.
Figure 2.23c illustrates a triple vertical split of the top room. We create new door axes for each new room except the center room. (Here is the exception mentioned above; an existing axis already runs through the center room, so a new, shorter axis is redundant.) Note that the top horizontal axis created in the previous split now does puncture two walls. In figure 2.23d we split the top left room using a 1:2 ratio (and reflect this split on the right). We create a door axis along with this upper new room. But, since an existing axis runs through the lower new room, once again we do not create a shorter duplicate. We do, however, shift this existing axis down into the center of the lower room. This is because, as noted, Palladio situates most axes, regardless of their length, so as to bisect the smallest room in their paths.
Returning to figure 2.23c, suppose that we executed the split of the top left and top right rooms with a 1:1 ratio instead of the 1:2 ratio used to create figure 2.23d. Now the new walls land directly on an existing door axis, as shown in figure 2.23e. Subsequent use of this axis would have the unintended result of erasing the entire wall. This conflict is resolved by shifting the axes up to the centers of the new top left and top right rooms, as shown in figure 2.23f. In splitting the top center room, however, Planmaker could again choose a split type and ratio that would place a wall in conflict with the recently moved axis. Given this scenario, it would reject that split type and ratio and select another. Planmaker will shift door axes only once. Otherwise, it could shift them back into conflict with existing walls.
Planmaker deals only in apertures in walls. We are the ones who read the central apertures in the back and front walls, and all apertures inside the house, as doors. We have yet to carve windows. Although most of these will lie on axis with doors, window placement on the main facade (always the bottom edge of the plan) is determined only partially by the plan itself. The facade elevation, as we shall see, is also a determinant. For example, it determines the width of all windows. In the next chapter we will define lines of communication through which facade and plan may amicably negotiate the question of window placement.
We have completed our study of Palladian villa plans. Our search has revealed the following rules:
Nearly exclusive use of rectangular rooms
Bilateral symmetry on the central vertical axis
Doors and windows on axes parallel and adjacent to walls
Wall alignment where possible
Larger rooms on the central vertical axis
No rooms greater than two squares
No rooms as wide or long as the entire plan
No room dimension less than 7 Vicentine feet
No walls along the central vertical axis
How comprehensive is this list? Are there elements we have failed to discover? In chapter 4 we will examine a set of computergenerated plans and facades ranging from the abjectly failed to the startlingly successful. We will also explore the wisdom, value, and limits of computerbased geometric analysis of Palladio’s villas.