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Calculating visual complexity in Peter Eisenman's architecture: A

computational fractal analysis of five houses (1968-1976)

Conference Paper · April 2009

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Ostwald, Michael J.; Josephine Vaughan (2009) Calculating Visual Complexity In Peter
Eisenman’s Architecture , Proceedings of the 14th International Conference on Computer
Aided Architectural Design Research in Asia / Yunlin (Taiwan) 22-25 April 2009, pp. 75-84

CALCULATING VISUAL COMPLEXITY IN PETER

EISENMAN’S ARCHITECTURE

A computational fractal analysis of five houses (1968-1976)

MICHAEL J. OSTWALD, JOSEPHINE VAUGHAN

The University of Newcastle, NSW, 2308, Australia.
{michael.ostwald / josephine.vaughan}@newcastle.edu.au

Abstract. This paper describes the results of the first computational

investigation of characteristic visual complexity in the architecture of
Peter Eisenman. The research uses a variation of the “box-counting”
approach to determining a quantitative value of the formal complexity
present in five of Eisenman’s early domestic works (Houses I, II, III, IV
and VI all of which were completed between 1968 and 1976). The box-
counting approach produces an approximate fractal dimension calculation
for the visual complexity of an architectural elevation. This method has
previously been used to analyse a range of historic and modern buildings
including the works of Frank Lloyd Wright, Eileen Gray, Le Corbusier
and Kazuyo Sejima. Peter Eisenman’s early house designs–important
precursors to his later Deconstructivist works–are widely regarded as
possessing a high degree of formal consistency and a reasonably high
level of visual complexity. Through this analysis it is possible to quantify
both the visual complexity and the degree of consistency present in this
work for the first time.

Keywords. Computational analysis; fractal dimension; box-counting;

Peter Eisenman.

1. Determining Visual Complexity

In this paper Peter Eisenman’s early domestic architecture is investigated using

a variation of the box-counting method for determining fractal dimension. This
research is part of a larger study using computational methods to reconsider
the formal characteristics of more than fifty iconic houses of the 20th century.
76 M. J. OSTWALD, J. VAUGHAN

Fractal geometry, emerging from Benoit Mandelbrot’s mathematical

proposals in the late 1970’s, has evolved from its initial domain in the sciences
of non-linearity and complexity to a much broader disciplinary field that now
includes architectural design and urban planning. Inspired by Mandelbrot’s
work, Bovill (1996) developed a “box-counting” method for calculating the
fractal dimensions of music, art and, significantly, architecture. Recently, this
method has been developed and refined for the computational analysis of the
fractal dimension of a small number of historical designs, ranging from isolated
ancient structures to complex twentieth century buildings. Despite this, the
method has not been comprehensively tested and without a much larger set of
published results, it is difficult to validate its usefulness. This validation process
is important because fractal analysis is one of only a small range of quantifiable
approaches to the analysis of the visual qualities of buildings. If valid, this
method could assist in the detailed analysis of historic structures or for the
design of buildings which might fit into a certain location or style. Previous
testing of this method indicates that Bovill’s proposal has potential merit but
that further analysis is required along with a much larger set of examples or
case studies.
This research describes the results of the first computational investigation
of the fractal dimensions of five of the house designs of Peter Eisenman.
Eisenman’s design approach became well known in the late 1960s with his
involvement in the “New York Five” (Eisenman, Meier, Graves, Hejduk and
Gwathmey) who were often known as the “Five Whites”; a reference to the
Modernist aesthetic of their early designs. As part of this group, Eisenman
presented his numbered series of house designs and became known for
producing abstract formal compositions that refused to acknowledge site
conditions or directly respond to the needs of human inhabitation. In the
following two decades Eisenman was in the vanguard of the Deconstructivist
movement, an approach intuitively regarded as having high visual complexity.
This paper provides an overview of the box-counting method and its
application in the analysis of house designs. It then briefly introduces the five
houses being considered in the present paper before discussing the results of
the research. Finally it undertakes a comparative analysis between Eisenman’s
results and those recorded in previous research for sets of house designs by
Frank Lloyd Wright, Le Corbusier, Eileen Gray and Kazuyo Sejima.

2. Fractal Analysis

Fractal geometry may be used to describe irregular or complex lines, planes

and volumes that exist between whole number integer dimensions. This implies
that, instead of having a dimension, or D, of 1, 2, or 3, fractals might have a D
 CALCULATING VISUAL COMPLEXITY IN PETER EISENMAN’S... 77

of 1.51, 1.93 or 2.74 (Mandelbrot, 1982). Fractal geometry came to prominence

in mathematics during the late 1970s and early 1980s. In the years that followed
fractal geometry began to inform a number of approaches to measuring and
understanding non-linear and complex forms. At around the same time,
architectural designers adopted fractal geometry as an experimental alternative
to Euclidean geometry. However, it was not until the late 1980s and the early
1990s that fractal dimensions were applied to the analysis of the built
environment (Ostwald, 2001; 2003). For example, Batty and Longley (1994)
and Hillier (1996) have each developed methods for using fractal geometry to
understand the visual qualities of urban space. Oku (1990) and Cooper (2003;
2005) have used fractal geometry to provide a comparative basis for the analysis
of urban skylines. Yamagishi, Uchida and Kuga (1988) have sought to determine
geometric complexity in street vistas and others have applied fractal geometry
to the analysis of historic street plans (Hidekazu and Mizuno, 1990).
Significantly, the calculation of fractal dimensions allows for a quantitative
comparison between formal complexity in buildings, a factor which is important
in a discipline that usually relies on qualitative evaluations.
The box-counting method is one of the most common mathematical
approaches for determining the approximate fractal dimension of an object.
Importantly, it is the only method currently available to analyse the fractal
dimension of an architectural drawing. In its architectural variant, the method
commences with a drawing of, for example, an elevation of a house. A large
grid is then placed over the drawing and each square in the grid is checked to
determine if any lines from the façade are present in the square. Those grid
boxes that have some detail in them are recorded. Next, a grid of smaller scale
is placed over the same façade and the same determination is made of whether
detail is present in the boxes of the grid. A comparison is then constructed
between the number of boxes with detail in the first grid and the number of
boxes with detail in the second grid. This comparison is made by plotting a
log-log diagram for each grid size. By repeating this process over multiple
grids of different scales, an estimate of the fractal dimension of the façade is
produced (Bovill, 1996; Lorenz, 2003). The software programs Benoit and
Archimage, the latter designed and co-authored by the Authors especially for
this purpose, automate this operation.
There are several variations of the box-counting approach that respond to
known deficiencies in the method. The four common variations are associated
with balancing “white space” and “starting image” proportion, line width,
scaling coefficient and moderating statistically divergent results. The solutions
to these issues that have been previously proposed by Bovill (1996), Lorenz
(2003), Foroutan-Pour, Dutilleul and Smith (1999) and Ostwald, Vaughan and
Tucker (2008) are adopted in the present analysis.
78 M. J. OSTWALD, J. VAUGHAN

3. Analytical Method

Five of Eisenman’s house designs were selected for the present research. These
are; House I (1968), House II (1970), House III (1971), House IV (1971) and
House VI (1976). House V, the only one missing from this sequence, was not
completed in enough detail to be used for the current research. For the rest of
the designs, all of the elevations used for the analysis were redrawn to ensure
consistency and were sourced from the published sets of drawings produced
by the office of Peter Eisenman (Dobney, 1995).
The standard method for the fractal analysis of visual complexity in houses
is as follows.
a) The elevational views of each individual house are separately grouped
together and considered as a set.
b) Each view of the house is analysed using Archimage and Benoit programs
producing, respectively, a D(Archi) and a D(Benoit) outcome. The settings for
Archimage and Benoit, including scaling coefficient and scaling limit are
preset to be consistent between the programs. The starting image size
(IS(Pixels)), largest grid size (LB(Pixels)), and number of reductions of the
analytical grid (G(#)), are recorded so that the results can be tested or
verified. Archimage results are typically slightly higher than those produced
by Benoit although the variation is consistent.
c) The D(Archi) and D(Benoit) results for the elevation views are averaged together
to produce a separate D(Elev) result for each program for the house. These
results are a measure of the average fractal dimension of the exterior facades
of the house. Past research suggests that D(Elev) results tend to be relatively
tightly clustered leading to a high degree of consistency.
d) The D(Elev) results produced by Archimage and Benoit are averaged together
to produce a composite result, D(Comp), for the house. The composite result
is a single D value that best approximates the characteristic visual
complexity of the house.
This process is repeated for each house producing a set of five D(Comp) values.
These values are averaged together to create an aggregate result, D(Agg), which
is a reflection of the typical, characteristic visual complexity of the set of the
architect’s works. (See Table 1)

4. Peter Eisenman and the Five Houses

Greg Lynn (2004) describes Peter Eisenman’s early house designs as being
wholly concerned with “layered traces and imprints of orthogonal movement
and transformation within a turbulent but nonetheless closed system of
 CALCULATING VISUAL COMPLEXITY IN PETER EISENMAN’S... 79

TABLE 1. Abbreviations and definitions.

Abbreviation Meaning
D Approximate Fractal Dimension.
D(Archi) D calculated using Archimage software
D(Benoit) D calculated using Benoit software.
D(Elev) Average D for a set of elevation views of a house using a specified
program.
D(Comp) Composite D result (averaged from both Archimage and Benoit outcomes
for all elevations) is a measure of the average characteristic visual
complexity of the house (or average fractal dimension for the 2D visual
qualities of the design).
D(Agg) Aggregated result of five composite values used for producing an overall
D for a set of architects’ works.
IS(Pix) The size of the starting image measured in pixels.
LB(Pix) The size of the largest box or grid that the analysis commences with,
measured in pixels.
G(#) The number of scaled grids that the software overlays on the image to
produce its comparative analysis.

nonfigurative cubic grids” (162). Sanford Kwinter (1995) argues that in

Eisenman’s design work “structure always emanates from an initial pattern
that is knocked away from equilibrium. The disturbance then travels, reaches a
limit, then turns back toward itself to form a self-interfering wave” (13). This
tendency can be traced in Eisenman’s earliest works, his numbered series of
houses. The first of these, House I, is a modernist villa, with a focus on geometric
form and the introduction of voids within orthogonal plans. House I, also known
as the Barenholtz pavilion, was completed in Princeton, New Jersey in 1968.
Designed for the Barenholtz family, Eisenman (2006) describes House I as “an
attempt to conceive of and understand the physical environment in a logically
consistent manner, potentially independent of its function or its meaning.”(32)
House I, actually a pavilion, is sited next to the original Barenholtz house. It is
a small, timber-framed and timber panelled structure with some interior brick
walls.
House II, in Hardwick Vermont, was designed for the Falk family and
completed in 1970. The house is sited on the crest of a hill with views in three
directions. It is timber-framed, clad in painted plywood panels and it “lacks
[the] traditional details associated with conventional houses” (Davidson, 2006:
37). Cassara (2006) describes House II as being focused on the architectural
80 M. J. OSTWALD, J. VAUGHAN

expression of two types of volumetric and structural relationships. “To articulate

these ways of conceiving and producing […] information in House II, certain
formal means were chosen, each involving an overloading of the object with
formal references” (82). In the House II, the layering of voids within the structure
produces a series of perforated planes that intersect with each other leaving the
relationship between exterior and interior spaces ambiguous (see figure 1).

Figure 1. House II, Elevation 2 (South), Peter Eisenman [D = 1.501]

(Elev Archi)

House III was designed for the Miller Family in Lakeville Connecticut and
completed in 1971. Like Houses I and II, it is timber framed and clad, with a
painted finish. The house has been described as an attempt to “produce a physical
environment which could be generated by a limited set of formational and
transformational rules” (Dobney 1995: 34). House III’s position in Eisenman’s
formal vocabulary is associated with the introduction of the 45° angle in plan
into an otherwise orthogonal 90° system.
House IV, while designed around the same time as House III, marks a return
to the planning strategies of Houses I and II. Designed for a site in Falls Village
Connecticut, House IV is an elaborate investigation of the process of design
transformation wherein various structural systems are allowed to trace solids
and voids in the overlapping multi-level plan of the house. House IV is
significant because the formal transformations occur in three dimensions; prior
to this, the operations were essentially planar in nature.
House VI was constructed in Cornwall, Connecticut, in 1976 for the Frank
Family. Designed as a weekend house on a small rural site, it features the first
clear instance in Eisenman’s architecture wherein the trace of a form (its absence
represented in a void) takes precedence over its presence (the form itself). In
House VI Eisenman famously divided the master bedroom, and the bed itself,
in two with the trace of a missing beam; effectively cutting a void through the
floor and separating the married couple.
 CALCULATING VISUAL COMPLEXITY IN PETER EISENMAN’S... 81

5. Results and Discussion

If the original 20 views (five houses each with four elevations) are subjected to
two variations of the computational method, each using 520 data points, the
aggregate result for the visual complexity of Peter Eisenman’s early house
results is D(Agg) = 1.425. (See Table 2)

TABLE 2. D(Archi) and D (Benoit) results for all elevations, D(Comp) results for each House and D(Agg)
result for the complete set of works.

Elevations analysed with Archimage Elevations analysed with Benoit

House D(Elev)1 D(Elev)2 D(Elev)3 D(Elev)4 D(Elev)1 D(Elev)2 D(Elev)3 D(Elev)4 D(comp)

I 1.337 1.348 1.450 1.290 1.330 1.362 1.399 1.302 1.352

II 1.540 1.501 1.521 1.267 1.514 1.481 1.475 1.184 1.436
III 1.525 1.534 1.579 1.611 1.481 1.492 1.546 1.577 1.528
IV 1.456 1.474 1.460 1.459 1.334 1.339 1.327 1.335 1.398
VI 1.451 1.455 1.476 1.483 1.359 1.380 1.342 1.371 1.415
D(Agg) = 1.425

For all images: IS(Pix) = 1200 x 871, LB(Pix) = 300 and G(#) = 13.
Of these five early houses of Peter Eisenman, House III had the highest
average value for visual complexity with a result of D(Comp) = 1.528. The lowest
result is for Eisenman’s first design, House I; D(Comp) = 1.352. The most complex
facades are typically in House III and are lead by Elevation 4 (D(Elev, Archi) =
1.611 and D(Elev, Benoit) = 1.577) (see figure 2). Indeed, it is relatively rare in the
fractal analysis of modern architecture to produce a result which is close to or
above D = 1.6. Elevations with this level of complexity have previously been
found in the highly decorative designs of the Arts and Crafts movement of the
late 19th and early 20th centuries and are less common in the late 20th century.

Figure 2. House III, Elevation 4 (West), Peter Eisenman [D = 1.611]

(Elev Archi)
82 M. J. OSTWALD, J. VAUGHAN

In general, the results recorded in the present paper confirm the intuitive or
qualitative reading of the five houses with House 1 being relatively simple in
form, Houses II, IV and VI having a similar level of complexity and House III,
with its 45° angled cross structure, as the most complex. How then do these
results compare with other results developed using the same method?
In five previous studies using this method, sets of houses by Wright, Le
Corbusier, Gray and Sejima have all been examined. Curiously, despite having
a reputation for producing architecture that is over-complex in its form, in the
case of Eisenman’s early houses at least, his works are less complex than those
of Wright previously tested. However, there are similarities between the results
for Eisenman and those for Le Corbusier’s early Modernist houses and Eileen
Gray’s house designs of a similar period. In essence, the two sets of Modernist
works and Eisenman’s early houses, which were experiments with Modernism,
have relatively similar results. Frank Lloyd Wright’s Prairie-style architecture
is more visually complex and Le Corbusier’s Arts and Crafts style works are
also slightly more complex. As anticipated, Kazuyo Sejima’s Minimalist works
produced a lower result than Eisenman’s (see table 3).

TABLE 3. Comparison of results for Eisenman with those of past research.

Architect Focus D(agg) D(Comp) D(Comp) D(Comp - clustering)

lowest highest closest 3 results
Frank Lloyd 5 houses 1.543 1.505 1.580 0.015
Wright (1901-1910) (<1% variation)
Le Corbusier 5 houses 1.495 1.458 1.584 0.031
(1905-1912) (~2% variation)
Le Corbusier 5 houses 1.481 1.420 1.515 0.015
(1922-1928) (<1% variation)
Eileen Gray 5 houses 1.378 1.289 1.464 0.087
(1926-1934) (~5% variation)
Peter Eisenman 5 houses 1.425 1.352 1.528 0.017
(1968-1976) (~1% variation)
Kazuyo Sejima 5 houses 1.310 1.192 1.450 0.116
(1994-2003) (~8% variation)

An additional dimension that arises from the present research concerns the
clustering of results for visually similar works. When an architect has the
opportunity to focus on a series of projects over a sustained period of time and
then develop them to a similar level of resolution, it might be anticipated that
these projects would exhibit similar levels of visual complexity. This is certainly
the case for Wright’s Prairie houses and for Le Corbusier’s early modern houses,
 CALCULATING VISUAL COMPLEXITY IN PETER EISENMAN’S... 83

both of which have a D range, for the best cluster of results, of less than 1%
visual difference (around D = 0.015). This means that, despite different sites,
variations on program and levels of complexity, each of these architects’ sets
of works consistently featured almost identical results for their respective
projects. Peter Eisenman’s early works have a similar magnitude of variation.
Houses II, IV and VI are all tightly clustered together with a D(Comp) range of
between 1.398 and 1.415 resulting in a difference in visual complexity, between
these three designs, of less than D = 0.017 or around 1% in comparative terms.
This result is significant not only because it confirms the standard qualitative
reading of each of these three sets of projects as being consistent within each
architect’s oeuvre, but it also affirms the usefulness of the method for such
consistent works (a result with a less than 1% mathematical difference is
significant). Conversely, the results for Gray and Sejima were less well clustered
(5-8%), a result supported by an intuitive reading of those works as less
consistent in their visual complexity (Ostwald, Vaughan, Chalup, 2008; Vaughan
and Ostwald 2008).
While there are few quantitative methods available for the analysis of visual
complexity in architecture, the box-counting method, and its computational
variation, remains a significant approach. The method is not only repeatable,
but it can produce accurate results that typically support more conventional
qualitative, semantic and historical readings of the same buildings.

Acknowledgements

This research was supported by an Australian Research Council, Discovery

Project grant (ARCDP): DP0770106.

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