Proportional Spatial Relationships in Greece and Italy

Horizontal Relationships

Arguably the three most important design theorists in Italy that had direct effect on the evolution of squares in England were Vitruvius, Alberti and Palladio.

Vitruvius, writing in Italy 1400 years before Alberti, referred back to Greece when discussing the design of Roman forums. He believed there was a preference in Greece to create agoras  of a certain horizontal shape (in the proportion 1:1) – even though most of actual antique Greek town spaces reflect more of an organic response to simple terrain conditions. Raymond Unwin reflected: “Where the Greek would adapt his arrangement to the site, the Roman would adapt the site to his arrangement, carving away the rocks and leveling the ground to obtain a clear field for his work.” Raymond Unwin “Town Planning in Practice,” (Princeton Architectural Press, New York, N.Y., 1994) 45.)

Instead of his assumed Greek predilection for square agoras, Vitruvius recommended a horizontal proportion for Roman forums in the proportion of 2:3 and added that the actual scale of the space should not be so small that it was not useful nor too large for lack of population (Book V Chapter I.2).. In his choice of proportion, he was driven by both esthetic and functional issues regarding the various uses that a forum had to accommodate.

Concerning the shape of the residential atrium, another precursor of the community square, Vitruvius in his sixth book (chapter III.1) outlined three alternative spatial proportions:

“The width and length of atria are devised by three methods.

The first is laid out by dividing the length into five parts and giving three parts to the width [in other words, 3:5]; the second, by dividing it into three parts and assigning two parts to the width [2:3]; the third, by using the width to describe a square figure with equal sides, and giving the atrium the length of this diagonal line.” [1:1.414 (the square root of 2)]

Vitruvius thus left four horizontal spatial recommendations for outdoor spaces. in order of ascending relationships:

2:3 (for forums)

3:5 [1:1.6666] (for atria)

2:3 [1:1.5] (for atria)

1:1414 (for atria)

These can be reorganized according to ascending relationships and in relationship to the numerical value one:

1:1.414

1:1.5

1:1.666

It was not until the early Renaissance that designers returned to these design principles articulated centuries earlier by Vitruvius. Designers once again began to consider the spaces between buildings, the shapes of the spaces and their dimensions. Indeed, God himself was portrayed during this time as the Grand Geometer. The investigation of ideal proportions went beyond mere observation.

In 1450, Alberti, only a few years after the start of Brunelleschi’s building project adjoining Santissima Annunziata, published a critique of Vitruvius’ spatial concepts, repeating Vitruvius’ understanding of Greek squares:

“The Greeks made their forums or markets exactly square, and encompassed them with large double porticos…. Among …the Italians, the Forums used to be a third part longer than they were broad [3:4]…. For my part, I would have a Square twice as long as broad [1:2], and that the Porticos and other Buildings about it should answer in some Proportion to the open Area in the middle, that it may not seem too large, by means of the lowness of the Buildings, nor too small, from their being too high.” Leon Battista Alberti, The Ten Books of Architecture, Book VIII, Chpt. VI, from the 1755 Leoni Edition as republished by Dover Publications in 1986, Toronto, pp. 173.

120 years after Alberti published his treatise, Palladio weighed in on ideal proportions in his Four Books of Architecture.

“There are seven types of room that are the most beautiful and well proportioned and turn out better: they can be made circular, …square, or their length will equal the diagonal of the square of the breadth, or a square and a third; or a square and a half, or a square and two thirds, or two squares.” Palladio: the Four Books of Architecture, Book II, Chapter XVI,  translated by Robert Tavenor and Richard Schofield, MIT Press, Cambridge,  1997, p. 57.

“…square plazas are rare and do not look very good, and also that overly long plazas in which the ratio of length to width is more than three to one already begin to lose charm.”  George R. Collins and Cristiane Crasemann Collins “Camillo Sitte:  The Birth of Modern City Planning” (Rizzoli, New York) 182.

“[Sitte] recommends that places should not usually be square but rather oblong, the length and the width bearing some definite proportion to one another. Usually the width should not be greater than three times the length. Such rules, however, can at best only indicate one method likely to prove successful.” Raymond Unwin “Town Planning in Practice,” (Princeton Architectural Press, New York,1994) 208.

Vertical/Horizontal Relationships

Alberti had definitive thoughts about the relationship between the height of buildings and squares: “…and that the Porticos and other Buildings about it [the square]should answer in some Proportion to the open Area in the middle, that it may not seem too large, by means of the lowness of the Buildings, nor too small, from their being too high. A proper Height for the Buildings about a Square is one third of the breadth of the open area, or one sixth at the least.” Leon Battista Alberti, The Ten Books of Architecture, Book VIII, Chpt. VI, from the 1755 Leoni Edition as republished by Dover Publications in 1986, Toronto, pp. 173.

Camille Sitte, an influential Austrian town designer of the late 19th century, tempers reliance on such rules of thumb:

“We find…that the height of its principal building, taken once, can be declared to be roughly the minimum dimension for a plaza, the absolute maximum that still gives a good effect being the double of that height – provided that the general shape of the building, its purpose, and its detailing do not permit exceptional dimensions.” Camille Sitte “The Birth of Modern City Planning” (Rizzoli, New York, 1986) 182. More recent design professionals such as Andreas Duany continue to recommend enclosure ratios, consistant with Alberti and Palladio’s view that building height to width ratios of less than 1 to 6 lose their effective sense of enclosure. However, as is the case in several of the London squares, the ratio can be effectively exceeded when mature trees and/or specialty street lamps transform the sense of enclosure.

Paul Zucker in his late 1950’s book Town and Square argued for a classification of squares divided into five types: closed, dominated, nuclear, grouped and amorphous. (Town and Square, Paul Zucker (MIT, Cambridge, 1970) 7, 8.) Zucker further suggested that a square be perceived as made up of three space confining elements: the rows of surrounding structure, the floor and the sky above. The “closed“ square basically functions as an outdoor room. The height of a closed square (the “ceiling” in Zucker’s terms) is generally perceived by Zucker as three to four times the height of the tallest  surrounding buildings – with the exception of overly broad squares. Perception tests reveal that the human eye takes in a view corridor that spreads to about a 60 degree angle.

Vertically, vision cuts off at about 27 degrees as Zucker observes. If one desires to see the entirety of a building flanking a square, one must be away from a building at least far enough to duplicate this cone of vision.

Geometrically, this means that one must be at a point away that matches twice the height of the building to be grasped plus five feet (average eye level). This means that an observer standing in the middle of a square would perceive a natural sense of enclosure from a vertical point of view when positioned in a square whose horizontal dimensions were no more than four times the height.

This means a sense of full enclosure from a vision point of view occurs in squares which are proportioned at a width and breadth about four times the height IF viewed from the center. But, to see the entire breadth of one of the sides of a square, one must be back approximately 3.5 times the height of the opposing wall of buildings.

But the perception of a square is rarely, if ever, taken from its mid point. Rather, one often perceives a square as one walks along its edge. Here, the vertical restriction means that a square will be ideally no wider than 2 times the height. This dimension allows the viewer to see all of the opposing side of the square.

As critical as proportions are, actual distance is also an important issue in the design of a successful square. Probably the most important issue is the safety one feels as an actual resident of a district when he or she can actually recognize a friend (or a stranger) on the opposite side of a square. Beyond 250 feet, most people lose the ability to identify friend or foe, and this results in an uncertainty of who exactly is with you in the square (or coming toward you on the sidewalk). Further, fine grained architectural detail looses richness as distance increases. Mass and shadow and void become the only elements easily identified at great distances. So, on a general basis, smaller is better – up to a point. When the green itself is more smaller than about 3500 sq. ft.or more narrow than 55 feet, the scale of the ensemble to some degree will overpower the value of the void, and the square will suffer (Egerton Place, Beaufort Gardens).

Shape

The shape of a “square” directly effects how it is perceived as well as how easy it will be to build buildings to enclose the square. The circular form offers the greatest sense of envelopment (the Circus at Bath or example), but every building is left to deal with the cost implications of the circle. Obviously, concave forms yield a sense of envelopment compared to convex, which should be avoided where possible. The clever combination of the concave crescent with a linear side provides some degree of right angle building opportunities (Wilton Crescent for example). Triangular spaces keep a sense of envelopment while providing right angle building opportunities (Chesham Place). The square shape itself, though still creating a sense of enclosure by the shape alone, is less so.

Organization

Leon Krier reminds us that “Urban space is a void, a structured and structuring void; it has a hierarchy, it has dimensions and character, it cannot be just a left-over between haphazard building operations. Too much of it is a waste, a false luxury; too little of it, a false economy. All buildings have a public facade, acting positively or negatively on public space, enriching or impoverishing it. Streets, squares, and their numerous declinations are the optimum forms of collective space. Neither public nor private enterprise produce public space naturally as a mere by-product of their activities. Public space, the public realm in general, its beauty and harmony, its aesthetic quality and socializing power, never result from accident, but from a civilizing vision…” LÉON KRIER Published in: “Building Cities”, Edited by Norman Crowe, Richard Economakis, and Michael Lykoudis, Artmedia Press, London, 1999, pages 40-41.

“Just as it is good that there should be many squares distributed around the city, so it is absolutely essential that there is one that  is large and prestigious which would be the principal square and which really can be called public.” Palladio: the Four Books of Architecture, Book II, Chapter XVI,  translated by Robert Tavenor and Richard Schofield, MIT Press, Cambridge,  1997, p 193.

A Walking Scale and Pattern

Leon Krier had it exactly right when he proposed in The City Within the City that each planned neighborhood should be no broader than a willing inhabitant could easily walk, and that each neighborhood should be organized around at least one square or green. This yields an idealized pattern where squares or greens are dispersed no further from any neighborhood residence than a five minute walk – approximately 1100 feet (335 meters).

Plus, we find that the walk distance between the twenty squares (including crescents) surveyed is, on average,  much closer than Krier’s theoretical 1100 feet — being only 590 feet (180 meters) apart, a measurement that means several squares fall within the limits of a pedestrian defined neighborhood.  It is no wonder that W. Weir, writing around 1850, recalling Milton’s “Paradise Lost,” exclaimed  that “in the …westward from Belgrave  Square … squares are to be found ‘thick as the leaves in Vallombrosa strewed.’” W. Weir, Charles Knight, ed, “London,” Vol. VI, Henry G, Bohn, London, 1851. But is this frequency a good thing?

Walk distances in feet.

The Frequency Issue

The rich frequency of the placement of squares in Belgravia  and Kensington introduces another key issue: economics. Considerable work has been done over the last few years in “proximate theory,” the theory that housing real estate values differ in accord with the proximity of a specific house to an amenity – particularly a square or green. Andrew Miller’s thesis at MIT analyzed realized premiums for lot values adjacent or near squares or greens in and around Dallas, Texas. Most of his research concerned monocentric park neighborhoods where he found that lot and house values tend to increase on or very close to an amenity, though Miller cautions that one has to be very careful about extrapolating to other geographic areas.

Ideal Spacing

Assuming Miller’s findings, are reasonably valid for most emerging suburban areas around the country, do the economics of square infrastructure cost coupled with lot revenue loss (lots not put into a neighborhood because the land on which they would have been placed instead has been devoted to the square itself) exceed the surrounding increase in value caused by the square. My theoretical calculations suggest that a developer operating in markets similar to those survey by Miller will be better off including squares (assuming they are sized similarly to an efficiently developed residential block) so long as their  frequency is no closer than a walk distance of approximately 1200 feet (365 meters) or, as the crow flies,  950 feet (290 meters).

It is possible that the “proximity to two parks would produce a price premium greater than seen in the monocentric park models Miller researched.  If that’s the case, then the math would tighten the optimal spacing and/or configuration.

Commenting on the 950 feet conclusion, Miller observed: “I see a couple of scenarios under which additive park premiums would seem reasonable (though I didn’t have data to study this except by inference):

1) social status benefits of living in a more park-rich environment (with attendant price premiums)

2) heterogeneity of the park amenities producing separate and additive proximity gradients.

If we’re talking about neighorhood parks, their limited size is going to preclude more than a subset of typical uses.  Homeowners should, one would imagine, be willing to pay more for a house with proximity to two parks – one of which might have tennis courts and the other a ballfield – than they would for proximity to only the park with tennis courts.  At that point, heterogeneity (both of use and of configuration) among the parks… [would] be key. This is not so different from any model for municipal or regional park systems – or, for that matter, the clustering of business centers based on different levels of specialization as seen in Central Place Theory. In either case, the value of an individual parcel will reflect the impact of multiple gradients to different activity clusters with different amenities.” (Andrew Miller correspondence,  Jan. 19, 2007)

None the less, the rich initial spacing of squares and parks in the early 19th century in Belgravia and Kensington probably was a little too close for economic comfort. Certainly, several developers encountered severe absorption slowdowns in the 1800s. An additional factor also must be taken to mind: the exact demographic served. When we run the numbers with greater increases in value than Miller experienced in suburban Dallas, the separation between squares may tighten.

If the rate of value increase for lots directly fronting the square is significantly greater than Miller’s average of 22%, then the frequency between squares could tighten.

Miller found an average increase of 22% for frontage on a Square or park in properties tested in Texas. My own development experience in the Memphis area suggests that high net worth individuals will pay significantly more,  but likewise, as one goes down the economic scale, the increase in value can be below Miller’s average. Depending on the quality of design and how a square is used, the impact can even be negative.

But using Miller’s average, we can create a diagram that allows us to analyze the theoretical loss of standard lots in a standard urban situation caused by inclusion of a green against the increase of value that accrues to the lots that are near the green. Here is the base diagram for a test of 25 blocks where the center block has been given over to a green.

Each urban block in the analysis is 260 feet long and wide including a center block alleyway with 20 foot wide townhome lots (13 per side and not shown) run perpendicular to the alley way system.

The importance in the analysis is not the ideal block size, but choosing a dimension that allows the calculation of the number of lots lost when a green is inserted. and whether that loss of potential developer revenue is offset by the increase of lot values immediately bordering the square, near to the square (within 300 feet) or between 300 feet and 600 feet. At a distance beyond 600 feet, Miller found that there was almost no (less than 5%) increase in value. Lots of frontage type A directly face the square; lots of frontage type B are within a 300 foot walk distance, and type C lots are between 300 and 600 feet away. D lots would be those beyond 600 feet of a square or green.

There is a point (from a developer’s viewpoint) where you have included too much green and your overall economics begin to crumble.

Bottom line, if you assume the value averages outlined below, there is a 32% increase in net revenues as calculated against the loss of revenues for the loss of lots in this 25 block analysis. I asked myself several other questions, one of which is also dealt with below in Alternative Two. The actual projected market increase that one would accrue in a specific location can vary a lot. So what percentage increase in value for a lot sitting on the amenity is the LOWEST percentage increase to at least cover the loss of the lots removed from the grid. The answer I came up with is about 17%. In other words, if you expect only to get a 10% increase in value for having a lot sitting on the amenity, you will have lost money for including the amenity. You need to get at least a 17% increase to warrant the introduction of the square. At least in this simulation. The introduction of the green hurt your economics rather than helped you.

I think the answer (assuming the 22% increase for A lots Andrew came up with) is not much closer diagonally than 900 feet. I ran two scenarios. One Square was 500 feet away diagonally and one was 950 feet away. The 950 feet diagonal spacing eliminated all “D” lots. Though the 500 spacing eliminated all D lots and upgraded others, the overall effect caused a net loss in value.

Andrew has a wonderful comment on page 99 and 100 of his thesis regarding the danger of overestimating how much value a park will yield to lot values from which I quote: “…parks that are not economically justifiable will appear to be so, encouraging developers to make bad investments.” Further, my simple spread sheet assumes that all lots are for the same market, but you would never do that over a 25 block area. You would mix things up as much as possible, and I doubt seriously have a continuous grid of the type I have diagramed. But, the reality is that varying widths will not significantly change the differential we are calculating. It would change the absorption (improve it) which we have also not taken into account. The effort is just to understand that the frequency in Belgravia and Kensington would be possibly TOO frequent from a purely economic point of view were densities more similar to our USA examples. Plus, there are more complex shapes to consider that do not eliminate so many otherwise developable lots. One form I find particularly interesting from that point of view is the crescent. You will remember that there is an incredibly long and slender low density version of a crescent at the back of Seaside. But London has many of them and they provide great frontage as well as wonderful “display,” a developer’s dream while taking up very little land.

Here are two other comparisons: squares placed every 950 feet (diagonally) versus a linear placement of green. You will remember that Andrew correctly noted that a linear configuration of green would yield more green fronting premium lots. I am surprised by what I found when I actually tested this. The linear approach is actually negative to value. I arranged the test so that almost all lots are within 600 feet of a green. Both diagrams have approximately 440 lots. The linear arrangement has blocks fronting on a linear green which is 1/2 a block in depth (130 feet) allowing us to calculate the loss of developable lots for the creation of the green.

Obviously, the linear scheme as Andrew predicted has a lot more green fronting lots than our diagram of small urban squares spaced at 950 feet apart. The linear arrangement has 110 lots (25% of all lots) fronting on a green, but the arrangement takes up three times as much green as the 950 diagram – therefore losing about 55 lots that could have been developed as opposed to 26 lots in the 950 foot scheme). In the spaced square approach, there are one only 52 lots fronting a green, but the loss of developable lots is only 26.



The problem is that we lose $836,000 in the linear scheme while gaining $388,000 in the spaced square scheme – over a million dollar difference with diagrams that have almost all lots within 600 feet of a green. I was really surprised with how negative the outcome was for a linear arrangement.



One final point regarding frequency. Even if the premiums are reduced because of frequency, they will be reduced in both instances and so the relative advantage of the spaced square approach will still remain.

When Andrew looked over my conclusions, he shared the following: “My argument, as best I recall, had been that a park that maximized perimeter would – for any given park area – tend to maximize the most valuable frontage and thus the cumulative premium.  Obviously, this won’t hold at some theoretical extreme of elongation. Though, to the extent that key benefits of a park include a) visual open space, and b) status signification for adjacent homes, that theoretical extreme might be further out than my data set would have permitted extrapolation. [In the linear versus square comparison], because the area of the park differs, they do something a little different than what my argument had intended.  In this case, the total space devoted to parkland  (and thus, the total opportunity cost of the parkland) under the greenway plan, is about twice as high as under the diagonal park scheme.  If you equalize the total park areas in each scheme in your model, the positive differential for the greenway plan would [be positive].

“Whether the proximity premiums for multiple parks will be additive or substitutive – that is, whether the premium for meaningful proximity to two parks would produce a price premium greater than seen in the monocentric park models we’ve been talking about.  If that’s the case, then the math might change on the optimal spacing and/or configuration.

I’d see a couple of scenarios under which additive park premiums would seem reasonable (though I didn’t have data to study this except by inference):
1) social status benefits of living in a more park-rich environment (with attendant price premiums)
2) heterogeneity of the park amenities producing separate and additive proximity gradients.  If we’re talking about neighorhood parks, their limited size is going to preclude more than a subset of typical uses.  Homeowners should, one would imagine, be willing to pay more for a house with proximity to two parks – one of which might have tennis courts and the other a ballfield – than they would for proximity to only the park with tennis courts.  At that point, heterogeneity (both of use and of configuration) among the parks in your grid will be key.

This is not so different from any model for municipal or regional park systems – or, for that matter, the clustering of business centers based on different levels of specialization as seen in Central Place Theory. In either case, the value of an individual parcel will reflect the impact of multiple gradients to different activity clusters with different amenities.”

My conclusion? The introduction of formal open space needs to be thoughtfully analyzed in economic terms. In addition to purely esthetic considerations, economics have to be taken into account. ALL of the London Square developments were speculative; some failed, and too great a frequency could have contributed to the problem. On the other hand, if well designed and executed formal open space has been introduced in an adjoining development, all property in close proximity will likely benefit.

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