In print: Environment and Planning B: Planning and Design
The dynamic façade pattern grammar
S D Kotsopoulos ¶
Department of Architecture, Design Laboratory, Mobile Experience Laboratory, Massachusetts Institute of
Technology, Cambridge MA 02139-4307, USA; email: skots@mit.edu
G Carra
Department of Building Environment Science and Technology, Polytechnic of Milan, Milan 20133, Italy;
email:guglielmo.carra@mail.polimi.it
W Graybill
Department of Computer Science, Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute
of Technology, Cambridge MA 02139-4307, USA; email: wgray496@mit.edu
F Casalegno (casalegno@mit.edu) ¶
Department of Architecture, Design Laboratory, Mobile Experience Laboratory, Massachusetts Institute of
Technology, Cambridge MA 02139-4307, USA; email: casalegno@mit.edu
Abstract: This paper presents a generative grammar producing a language of patterns for the south façade of a
prototype sustainable house. The patterns are produced through the activation of the electrochromic material that is
applied on the windowpanes of the façade. The class of the performatively effective configurations of the façade is
approached as a visual language and the productive (generation), combinatorial (enumeration) and performative
(verification) attributes of this language are examined. Random, performance driven, patterns could supply
sufficient interior daylight without acknowledging the visual potential of façade pattern generation. The uniqueness
of the chosen approach is that the shape grammar encodes the performative constraints pertaining the generation of
façade patterns in a visual manner by associating principles of 2D pattern generation to levels of illuminance.
1. Introduction
Light is an essential aesthetic and performative aspect of architecture. The lighting of interior
spaces typically involves natural and artificial light sources. Given that artificial lighting
contributes considerably to energy consumption, natural lighting is favored during daytime. This
paper presents an example of how shape grammars, Artificial Intelligence (AI) methods to
building control, and electrochromic technology can be combined to regulate natural lighting in
the interior of residential buildings. The association between a specific architectural element of a
¶ Current address: School of Humanities, Arts and Social Sciences, Comparative Media Studies, Mobile Experience
Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA
prototype house – a façade involving 100 electrochromic windows and the adjustment of the
natural lighting conditions in the house interior, is treated algorithmically. The façade operates as
a dynamic filter between interior and exterior, permitting the modification of the incoming solar
radiation and heat through the adjustment of the chromatism and the light transmittance of each
individual windowpane. Changing the number and the distribution of the active electrochromic
windows on the façade affects the intensity of the incoming sunlight, thus permitting more
effective management of the high thermal mass house envelope, but also transforms how the
house is perceived from the public street. A shape grammar drives the activation of the
electrochromic windows equally based on performative and visual criteria.
The increasing cost and scarcity of non-renewable energy sources promote the use of
sustainable principles in the design and operation of buildings. However, the principles of
orientation and plan organization for breeze, and sectional organization for cross ventilation and
cooling can turn into a design tyranny. And although new optimization and automation
techniques promise to secure economical management and compliancy of building performance,
they usually ignore building aesthetics. Motivation for this research was to provide an
environmentally conscious mode of building an original design vocabulary that is in alignment
with technological innovation so as to supply architecture with new tectonic means. An
architectural solution was proposed employing shape grammars, AI methods to building control
and electro-sensitive materials, which satisfies the environmental requirements of its function
and makes elegant architecture out of the provisions needed for their satisfaction.
Traditional buildings combine conservative, selective and regenerative modes of
environmental management to conserve heat, to admit selectively elements from the exterior
environment, and to restore favourable conditions by artificial means (Banham 1969). Thick
2
walls conserve heat and return it to the interior environment after the heat source is no longer
active in the winter, and delay the effects of solar heat in the summer. Glazed windows admit
light but exclude the direct sun, and louvered grilles admit air but exclude visual intrusions.
Heating Ventilating and Air Conditioning (HVAC) systems and artificial lighting restore desired
temperature and light settings at energy expense. Recent building paradigms use productive
systems to harvest the power of the sun, the wind, or biomass, and responsive systems to provide
adjustability of performance "in response to" real conditions (Klein and Kaefer 2008). In the
presented prototype the form and the fabric of the building envelope are used as a filter of the
external environment, in combination with dynamic control and energy production capabilities,
to provide an energy saving, environmentally sound, and architecturally rich living environment.
The prototype consists of an open plan interior measuring 7.75 m
20.00 m with a
programmable solar wall facing south. The house envelope secures high thermal resistance and
low conductivity, conserving heat during the winter and preventing excessive heat during the
summer. A single pitched roof stands 2.95 m high on the north and pitches up to 3.65 m on the
south thus exposing a wide façade area to the sun. The south façade is a solar wall of
individually addressable windows whose configurations enable the precise adjustment of light,
heat, air, and view. A solar cogeneration plant produces electricity, hot water and heated/cooled
air, while a central, autonomous control system offers parallel, integrated adjustment of all house
systems in response to a broad spectrum of natural conditions and user needs. Hence, on a hot
summer day the control sets the electrochromic material of a number of windows to its minimum
solar transmittance to protect the house interior from direct sun exposure, while on a cold winter
day it sets it to maximum solar transmittance in order to expose the interior to the winter sun.
The adjustment of the number and distribution of the active electrochromic windows on the
3
façade causes the formation of visual patterns. Since there is no standard class of façade patterns
satisfying all weather conditions the adjustment of the façade remains dynamic. Randomly
generated performance driven patterns could supply adequate interior daylight without
acknowledging the visual attributes of the patterns. The presented grammar is applied
dynamically by the control system of the house to generate façade patterns complying both on
performative constraints of daylight adjustment and visual principles of 2D pattern generation.
After an exposition of the research background, this presentation is organized into three
sections: generation, enumeration, and verification. Generation presents the productive scope of
the façade pattern language including the grammatical rules producing the patterns.
Enumeration presents the combinatorial scope of the pattern language including the count of
patterns and their classification based on symmetry and illuminance. Verification presents the
performative scope of the language by determining how variation in the number and distribution
of active windows affects the levels of interior illuminance. The sequencing generation –
enumeration – verification is retrospective. The generation of façade patterns and the verification
of their performance were advanced in parallel, whereas enumeration was conducted last.
2. Background
The connected sustainable home concept (Figure 1) aims to provide a living environment that
remains constantly well tuned to the comfort levels of the inhabitants. A key aspect to this is the
fine management of the house system dynamics. Natural conditions vary and so do the activities
of the inhabitants, but an intelligent control can always supply the desirable comfort levels at
minimal energy cost. A full-scale prototype with these capabilities is in the final stage of
construction in Trento, Italy.
4
Figure 1. Rendition of the prototype connected sustainable home in Trento, Italy.
The dynamic façade covering the patio and the kitchen on the south elevation of the house is
a matrix of 5
The lower 3
20 windows 700 mm
700 mm in size, arranged in columns and rows (Figure 2).
20 rows include operable windows, which can be opened and closed at precise
angles using electronic actuators, so that the permeability pattern to air flow is automatically and
precisely adjusted. The upper 2
18 rows include inoperable windows tilted by 75° to form the
patio-roof, while the upper right 2 corner windows are inoperable and not tilted. Each tripleglazed window involves an overlay of two electronically switched materials and has an overall
thickness of 43 mm. The glazings are separated by two Argon filled gaps, 12 mm and 6 mm. The
electrochromic coating applied on the external glazing enables the adjustability of solar radiation
and permits precise light and thermal management. The polymer dispersed liquid crystal film
(PDLC) applied on the internal glazing supplies adjustability of visibility and secures privacy.
Figure 2. The south elevation of the prototype, incorporating the dynamic façade.
5
The control system of the house applies grammatical rules to dynamically adjust the levels
of the admitted daylight by generating façade patterns that conform to performance and visual
principles. The control system compiles feedback from sensors, statistical climatic data and
ambient data to provide real time building performance evaluation. The residents specify ranges
of room temperature and their schedules and the control minimizes the energy consumption
while guaranteeing that the desired comfort levels are always maintained. To minimize the risk
of constraint violation the control allows operating an uncertain system within acceptable risk
bounds that are specified by the residents (Graybill 2012, Ono 2012).
2.1. Generative apparatus
The presented generative grammar determines the states of the dynamic façade equally based on
performance and aesthetics. In the past, Stiny and Mitchell (1978) described a shape grammar
encoding stylistic principles of generation, enumeration and verification of Palladian villa plans.
An insightful discussion of the dual character, generative / expressive of spatial rule systems
exists in Knight (2005). Two noteworthy articles on optimization and performance-driven
generative design are Luebkeman and Shea (2005), and Shea et al (2005). Luebkeman and Shea
(2005) show how navigating the performance space of a design solution promotes design
thinking and exhibit the association between variations and performance. Shea et al (2005) use
performance-driven generative methods in producing designs based on modeling of conditions
and performance. This paper extents the previous contributions in two ways. First, like in Stiny
and Mitchell (1978) it presents the generation, enumeration and verification of a design
language. However, instead of simply encoding stylistic conventions of form generation the
grammar captures conventions of daylight adjustment and encodes them visually. Second, the
6
rules of the grammar determine the states of a architectural element that is made out of a specific
variable transmittance material. Hence, the grammar exploits the visual potential of the
performance conventions applied to the specific façade throughout the four seasons.
A shape grammar consists of a calculating part and a syntactic-interpretive part. The
calculating part engages an algebraic framework in which elements of 0, 1, 2 and 3 dimensions
(points, lines, planes, solids, or combinations of these) are used in calculations that may happen
in a space of 0, 1, 2 or 3 dimensions. The syntactic-interpretive part consists of production rules
confining the syntactic (structure) and semantic (meaning) attributes of sets of products, which
are conventionally called languages. An algebra X i j formalizes the interaction of i-dimensional
elements on a j-dimensional space (Stiny 1991). For example, the algebra U12 formalizes the
graphic computations that designers execute with lines on the graphic plane: it captures the
interaction of one-dimensional elements, lines (i =1), on the two-dimensional plane (j=2). This
treatment can be expanded to include algebras with labeled points (Vi j), which serve the naming
of elements, and algebras with colors and properties like weights (Wi j) which serve their visual
distinction in desired ways (Stiny 1992). Product algebras can be formed as combinations of any
of the above algebras to allow the execution of calculations with labeled and colored forms.
Within this framework, the elements are composed with the aid of production rules. A product
algebra < U12 U22> including black lines and planes is used in the next example of a rule that is
similar to the rules used in the dynamic façade pattern grammar
.
Within the semantic context of façade pattern generation a rule ri checks a number of
neighboring cells depicted on the left hand side, and “activates” a number of cells by modifying
their state from off to on as depicted on the right hand side. The rule is of the general form:
7
x
prt(x) + (b-1 (prt(x))) .
If x is the shape appearing on the left hand side of the rule, then some part of x is "activated"
by applying a uniform tone to its area (b-1 (prt(x)), while another prt(x) remains intact (Stiny
2011). The rule of the example applies on a shape C capturing two neighbouring window cells
to produce shape C'
in two steps. First, a transformation t is used to recognize through matching, some part of C,
which is geometrically similar to x – the shape that appears on the left hand side of the rule – and
second, the same transformation t is used to modify C. It substitutes t(x) with t [ prt(x) + (b-1
(prt(x))) ] to produce C'. Concisely,
C'= [C – t(x)] + t [prt(x) + (b-1 (prt(x))) ] .
Rule ri can apply on C to produce C' under a transformation t in two ways, plain or under mirror
reflection (or 180° rotation). These correspond to the ways the shape on the left side of the rule
can be “matched” on C
.
2.2. Performative premises
The number and distribution of the active electrochromic windowpanes on the façade can be
determined to always supply levels of interior daylight illuminance above a preset value. Two
performative premises assure the generation of effective façade patterns. The premises were
extracted via simulation with Relux Professional and Relux Vision by Relux Informatik AG,
assuming that there were no neighbouring buildings casting shadows. Climatic data for the
8
location of the prototype and the lighting standards determined by Italian law were provided by
the database of the software. The specifications of the electrochromic material were provided by
Sage Electrochromics. In the simulations transmitance was set to 62% for glass at state off and
to 3.5% for glass at state on.
Several recent papers describe the state of the art research in electrochromic technology. Lee
et al (2006) present a study in which the effects of electrochromic technology are monitored in a
cube 3.0 m
3.0 m
3.0 m. Hausler et al (2003) present a technical comparison of
electrochromic glass. Selkowitz et al (2003) offer an overview of automated lighting and energy
control systems. Mardaljevic et al (2009) review the history of building compliance methods
determining energy efficiency and comfort and propose a new basis of more efficient metrics.
Two models established by the Commission Internationale de l'Eclairage (CIE) were used
in the simulations. The Standard Overcast Sky model provides an account of the natural light
emitted through cloudy sky. The Standard Clear Sky model computes natural light assuming that
the sun is the single lighting source without calculating the diffused and reflected light by the
sky. After compiling the meteorological records of the last decade for the province of Trento, the
percentage of rainy days was determined to be 36% and the percentage of sunny days 64%.
These numbers determined the days per year in which the Overcast Sky and Clear Sky models
were used. However, both models provide incomplete understanding of the phenomenon of
illuminance as they exclude the assessment of transitory conditions, such as the passage of
clouds across the sun (Wienold 2007). For this reason, simulation data for each day of the year
were evaluated and classified for the integration of an optimization algorithm in the control
system. Combined with real time input from sensors made the efficient management of the
façade possible even in conditions that cannot be captured by the static models.
9
Typical outputs of the simulations included the minimum, maximum and average values of
illuminance, the uniformity values and average daylight factor, the Isolux maps for assigned
planes, and the tridimensional illuminance diagrams. The interior daylight conditions are
determined by the average illumination Eave, the uniformity G1 and the daylight factor Dav. Italian
law adopts both a national law Circ. Min n° 3151 22/5/67 and the UNI EN 12464, which is a
norm defining the lighting of workplaces. Although the minimum daytime value of illuminance
Emin for residential buildings is 300 lux this threshold was raised in the prototype to 500 lux to
reach higher levels of visual comfort. Uniformity G1 captures the smoothness of daylight
distribution defined by the ratio Emin/Eave. A satisfactory value is G1 = 0,5. Lastly, the daylight
factor represents a physical feature of windows, which is constant and set to Dav= 3. The
simulation tests indicated that in order to achieve Emin 500 lux with smooth interior daylight
distribution Emin / Eave 0,5 the façade must satisfy two provisions, namely:
Provision I: In an average luminous day, out of 100 electrochromic cells, 50 – 75 cells
need to be active (on) to secure luminosity levels above the threshold value.
Provision II: No four consecutive cells can be concurrently active on the same row.
Dense accumulation of consecutive active cells horizontally casts linear shadows disrupting
smooth light distribution. Hence, if n is the number of consecutive active cells in any row n
3,
to secure fine distribution of active cells.
3. Generation
Each electrochromic glass cell of the façade can be set on (active) or remain off (inactive). Three
general use modes were determined for the façade based on illuminance performance: (a) 0 %
active, a single rule is required, do nothing, (b) 50 % - 75 % active, 12 rules organized into a
10
grammar generate façade patterns, (c) 100 % active, a single rule is required, activate all. Hence,
the façade can remain clear, be activated in a 2/4 – 3/4 ratio of its area, or be fully active. The
grammar generates patterns for (b) by activating a number of electrochromic cells within the
range from 50 - 75. Starting from an initial state a new state is produced after a rule is applied. A
rule may apply while taking into account the state of a single cell and the state of at most four
more cells arranged in the same row. A neighbourhood of cells involves at most m cells arranged
in the same row, with m
5.
3.1. Modes of rule application
Modes of rule application enforcing the generation of visually distinct patterns are explained
next. The 5 20 matrix of inactive electrochromic cells is depicted below in the algebra U12.
Auxiliary markings are introduced, namely boundaries and axes depicted in dashed grey
line, in the algebra W12. The axes organize the matrix into four partitions of 2
10 glass cells
each. A pair of labels (V, V) in the algebra V02 allows the distinction between vertical and
horizontal direction. Neighbouring partitions are bilaterally symmetrical along the vertical or
horizontal central axes. This augmented description is formed in the product < U12 W12 V02>.
Rule modes determine general ways after which the rules ri can be applied on the matrix.
Seven modes of rule application are outlined based on the above partitioning. Rule modes (i) 11
(vii) involve the application of a rule ri and oftentimes parallel application of copies of ri under
explicit transformations t*. This allows for multiple similar parts of the matrix to be “activated”
in parallel, and for various symmetries to be used in the generation of façade patterns. Based on
the schema of § 2.1. the rule modes have the general form
x
t* [prt(x) + (b-1 (prt(x)))] .
The rule modes are divided in two classes based on the position of the cells they affect. To
generate a façade pattern it is mandatory to use at least one mode from each of the two classes.
The first class includes modes affecting only cells positioned along the horizontal axis.
In the first class a rule ri can be applied along the horizontal axis in two ways
i.
ii.
.
The second class includes modes affecting cells in the remaining four partitions of the matrix.
In the second class a rule ri can be applied on the four 2
iii.
iv.
vi.
vii.
10 partitions of the matrix in five ways
v.
.
12
Modes (i) and (iii) involve application of a single rule ri , respectively. Modes (ii) and (iv)
involve parallel application of a rule ri and of a copy of ri under a transformation t that is a
mirror reflection along the vertical axis. Mode (v) involves parallel application of a rule ri and of
a copy of ri under a different transformation t that is a mirror reflection along the horizontal
axis. Mode (vi) involves parallel application of a rule ri and of a copy of ri under a transformation
t that is 180° rotation around the center point of the matrix. Mode (vii) involves parallel
application of ri and of three copies of ri under transformations t , t , t , i.e., mirror reflection
along the vertical and the horizontal axes and 180° rotation, respectively. Table 1 presents the
rule schemata expressing the modes (i)–(vii) based on the general rule schema of § 2.1.
Modes
Rule Schemata
(i), (iii)
x
(ii), (iv)
x + t (x)
[prt(x) + (b-1 (prt(x)))] + t [prt(x) + (b-1 (prt(x)))]
(v)
x + t (x)
[prt(x) + (b-1 (prt(x)))] + t [prt(x) + (b-1 (prt(x)))]
(vi)
x+t
[prt(x) + (b-1 (prt(x)))] + t
(vii)
x + t (x) + t (x) + t
prt(x) + (b-1 (prt(x)))
(x)
(x)
[prt(x) + (b-1 (prt(x)))]
[prt(x) + (b-1 (prt(x)))] + t [prt(x) + (b-1 (prt(x)))] +
t [prt(x) + (b-1 (prt(x)))] + t
[prt(x) + (b-1 (prt(x)))]
Table 1. Rule schemata capturing the general form of the rule modes (i) – (vii).
All seven modes of the rule application may be used to generate patterns with no particular
symmetry. Combination of the mode (ii) with the mode (vi) or (vii) generates patterns with
rotational symmetry. To generate a pattern with reflectional symmetry along the vertical central
axis, the modes (ii) and (iv) can be used. To generate a pattern with reflectional symmetry along
the horizontal central axis the modes (i) or (ii) can be combined with (v) or (vii). To generate a
pattern with reflectional and rotational symmetry only the modes (ii) and (vii) can be combined.
13
3.2. Algebras and descriptions
The necessary graphic elements for deriving a pattern are finalized and presented based on their
corresponding algebras Ui j, Wi j and Vi j. Initial shape is the augmented description of the 5
20
matrix, including grey line axes and labels. The pair of labels (A, A) along the horizontal axis
indicates the ongoing stage of the derivation.
Three possible graphic illustrations of window cells are determined, on (active), off
(inactive), and the auxiliary state, excluded. Cells are marked excluded to become inaccessible.
In this way they are distinguished from remaining inactive cells, which can still become active.
At the terminating stage of a derivation all excluded cells turn into inactive cells. The graphic
illustrations of the three possible window states are depicted below, on (active-left), off (inactivecenter), and excluded (right).
.
In the calculations to follow all graphic elements are manipulated on the plane (j =2). Active
cells are depicted as black squares in the algebra U22. Inactive cells are depicted with black lines
in the algebra U12. Excluded cells are depicted as grey squares in the algebra W22. Auxiliary axes
are depicted with grey lines in the algebra W12. Letters A, B, and C are used as labels in the
algebra V02. The overall algebraic component is a product: < U12 W12 U22
A typical step C
< U12
W12
U22
W22
V02 >.
C' in a derivation is formed in the product
W22
V02 >
< U12
W12
U22
W22
V02 >.
14
Components W12 , W22 , or V02 in this product algebra can be substituted with the empty
shape. For example, when the excluded window cells turn into inactive their grey tone is erased
(W22
) and their linear boundaries are redrawn in the U12 component of the product. Hence,
the terminating step, where all the auxiliary graphic elements are erased, is expressed
< U12 W12
3.3.
U22
W22
V02 >
< U12
U22
>.
Derivation stages
Façade pattern generation involves 12 rules satisfying provisions I and II. Derivation is
organized in three stages A, B, and C the productive objectives of which are explained next. All
three stages include a number of productive rules and a terminating rule, applying at the end to
introduce the consecutive stage or to terminate the derivation.
Stage A – the initiating stage – includes 4 rules in total (not shown here). These apply on
the outmost 4 columns of the matrix. Stage A is terminated when all the outmost cells are active
or excluded. The cells that can be affected in stage A are depicted below.
Stage B – the main productive stage – includes 5 rules in total. Generation at this stage
proceeds until all available cells are either active or excluded. The cells that can be affected in
stage B are depicted below.
15
Stage C – the terminating stage – includes 3 rules in total. Generation at this stage proceeds
until all available excluded cells are turned into inactive and all axes are erased. The shapes that
can be affected in stage C include the entire matrix.
3.4.
Generative rules
Rules are named with the letter ri (i = 1, 2,…12). Each productive rule scans a neighbourhood of
cells, as depicted on the left hand side and modifies the state of a number of cells as depicted on
the right hand side. Productive rules always apply until they exhaust the available window cells.
The number of the activated cells can be 0, 1, or 2. Rules apply in three stages. Stage A initiates
the derivation. Stage B is the main productive stage. Stage C terminates the derivation. The
positioning of grey axes and labels in the rule expressions is parametric.
3.4.1. Stage A
Productive rules 1-3 initiate the process. They are parametric and apply on the outmost left or
right window cells lying next to the vertical boundary axes. Rule 1 scans one cell and excludes it
by applying a grey tone to its area. Rules 2 and 3 scan and activate one or two cells respectively.
The performance of each rule is 100%. Rules 1-3 apply until all the outmost left and right cells
are activated or excluded. Rule 4 terminates stage A and introduces stage B as shown below.
r1
r2
r3
r4
16
3.4.2. Stage B
Productive rules 5-8 are parametric. They affect the entire matrix except from the outmost left
and right cell columns. Rules 5 and 6 scan two cells and activate or exclude one. The percentage
of contribution of each of these rules is 0
x
50. Rule 7 scans three cells, activates one cell and
excludes one cell. Rule 8 scans five cells, activates one cell and excludes one cell. The
percentage of contribution of these two rules is 0
x
33.3. Rules 5-8 apply until all available
cells are either active or excluded. Rule 9 terminates stage B and introduces stage C.
r5
r6
r7
r8
r9
3.4.3. Stage C
Productive rules 10, 11 are also parametric. Rule 10 erases the grey markers from the excluded
cells anywhere on the matrix and turns them into inactive cells. Rule 11 eliminates the auxiliary
grey axes. Rules 10, 11 apply until all excluded cells are turned into inactive, and all axes are
erased. Rule 12 applies last to erase the labels and terminate the process.
r10
r11
r12
17
3.5. Derivation
The generation and illuminance performance of the façade pattern depicted bellow are calculated
in the Appendix I and V respectively. The pattern includes 74 active windows, it involves rotation
around the center of the matrix and it is produced after the modes (ii), (vi) of the grammar.
4.
Enumeration
The enumeration computes the number of the patterns and it places them into subclasses based
on illuminance and symmetry. The enumeration is computed assuming the provisions I, II. First,
a calculation is performed of the total number of patterns with density between 50 and 75 active
windows. Then, the consideration of their symmetry further refines the enumeration.
If n denotes the number of active windows, the interest is for 50
n
75, due to provision I.
The façade is decomposed into rows numbered = 1, 2, ...5, with r denoting the number of
active windows in row . The n active windows are distributed across the rows with the
requirement
r = n and 0
r
20. Given a fixed number of active windows r it is
calculated the number of ways in which these can be arranged across any row . The function
P(r, k) is defined to compute the number of arrangements of r active windows across a row of
length k, subject to provision II. The goal is to calculate P(r , 20). Two cases are distinguished.
Case 1: The leftmost window is inactive. Then, there are P(r, k – 1) ways to distribute
the r active windows across the remaining k – 1 windows, noted as
k
.
1
k-1
18
Case 2: The leftmost window is active. Then, three sub-cases are distinguished.
First, if the consecutive window is inactive the leftmost portion of the row will
not violate provision I. The state of the 3rd window may be chosen from the left
without restriction. This leaves r – 1 windows to activate across a total of k – 2
remaining windows in the row. Hence, there are P(r – 1, k – 2) possibilities,
noted as
k
.
1
2
k-2
Second, if the consecutive window is active the two leftmost windows are active
and the 3rd window from the left is inactive. This leaves r – 2 windows to activate
across a total of k – 3 remaining windows in the row. Hence, there are P(r – 2, k
– 3) possibilities, noted as
k
.
1 2 3
k-3
Third, if the consecutive window is active, the three leftmost windows are active.
Then, the 4th window from the left is necessarily inactive. This leaves r – 3
windows to activate across a total of k – 4 remaining windows in the row. Hence,
there are P(r – 3, k – 4) possibilities, noted as
k
.
1 2 3 4
k-4
19
In total, a recursive equation can be formed
P(r, k) = P(r, k – 1) + P(r – 1, k – 2) + P(r – 2, k – 3) + P(r – 3, k – 4).
The base cases of the recursion are P(1, 1) = 1 and P(1, 2) = 2. The argument establishes an
expression for the productive possibilities on a single row. For a given assignment to r1,… r5 this
number is denoted by the product of these expressions
5
P(r , 20).
=1
To calculate the total number of arrangements for a given n sum over all values r1,… r5 to yield
an equation for the number of arrangements E as a function of the number of active windows n
5
E(n) =
P(r , 20).
r1,…r5
r =n
0 r
20
=1
Restricting the production to any n in the range 50 -75 yields 1.285
1024 configurations.
The results of the computation for E are provided in Table 2.
n
50
1.751
10 27
n
60
1.285
10 24
n
70
1.285
10 16
51
1.191
10 27
61
3.891
10 23
71
1.285
10 15
52
7.566
10 26
62
1.061
10 23
72
1.285
10 13
53
4.484
10
26
63
2.590
10
22
73
1.285
10 12
54
2.470
10 26
64
5.611
10 21
74
1.285
10 10
1.265
10
26
1.069
10
21
75
550,731,776
10
25
66
1.774
10
20
10
25
67
2.529
10 19
25
68
1.285
10 18
69
1.285
10 17
55
56
57
E(n)
5.986
2.610
58
1.048
10
59
3.844
10 24
65
E(n)
E(n)
Table 2. Enumeration of distinct façade patterns, within the range of 50%-75% activation.
20
Five subclasses account for the symmetry of the patterns in the language, namely: (1) no
symmetry, (2) rotational symmetry, (3) reflectional symmetry about the vertical axis, (4)
reflectional symmetry about the horizontal axis, and (5) rotational and reflectional symmetry.
Class (1) includes patterns that can be generated through translation on the plane. Class (2)
includes patterns that remain identical upon 180° rotation. Classes (3) and (4) include patterns
that remain identical when reflected about the central axis of the façade, either vertical or
horizontal. Class (5) includes patterns for which the distinction between horizontal and vertical
axis is unnecessary. The five classes are denoted S0, Srot, Sref v, Sref h, and Sref, rot.
Let n denote the number of tinted windows and let ki denote the number of windows tinted
in row of the 5 x 20 matrix. The function P(k, m) is defined to be the number of possible ways
to tint k cells in a row of length m cells, according to the rules of the grammar.
Let S(k) denote the number of arrangements with k tinted cells in a row of length 20 that are
bilaterally symmetric about the center. Two cases can be distinguished in calculating S(k).
First, the two-centermost cells are tinted. Then, the cells to either side of the two centermost
cells must remain clear. Thus, there are P(k/2
2, 8) possible arrangements
k
.
k/2 - 2
k/2 - 2
Second, the two-centermost cells are clear. Then, there are P(k/2, 9) possible arrangements
k
.
k/2
k/2
21
Therefore:
S(k) =
P(k/2, 9) + P(k/2 – 2, 8) if k even.
0
if k odd.
It is first enumerated the subclass S0 of patterns that do not involve any symmetry. All
modes of rule application may be used to generate a pattern in S0 and all possible configurations
are included in the class. The total number of configurations in S0 is the product of P(k , 20) over
each row summed over all possible partitions of n tinted cells across the 5 rows
Es0 (n) =
P(k , 20).
k1,…k5
k =n
0 k
20
For the subclass Srot of patterns involving rotational symmetry the analysis proceeds
separately for the window cells of the horizontal axis and those of the remaining two 2 x 20
partitions (see §3.1). Only mode (ii) can be used on the horizontal axis, since the center row must
be bilaterally symmetric. The number of arrangements is S(k3). The modes (vi) and (vii) can be
used on the remaining two 2 x 20 partitions. This implies that the configuration of the bottom
two rows is determined by that of the top two rows (or vice versa). Hence, it is sufficient to count
the arrangements of one section. Let E2(n) denote the number of arrangements of n tinted cells
across two rows. This value can be calculated similarly to ES 0
E2 (n) =
P(k , 20).
k1, k2
k1 + k2 = n
0 k
20
The enumeration of configurations in the subclass Srot is calculated as follows
Esrot (n) =
S(k3) · E2(k12).
k12, k3
2k12 + k3 = n
0 k3 20
0 k12 40
22
where k12 denotes the total number of tinted windows between rows 1 and 2.
For the Sref v subclass of patterns involving reflectional symmetry about the vertical axis, the
calculation is similar to S0 except that only the modes (ii) and (iv) can be used, since each row is
bilaterally symmetric. For even n, the enumeration in the Sref v subclass is calculated as
Esref v (n) =
S(k ).
k1,... k5
k =n
0 k
20
To generate a pattern in the subclass Sref h of patterns involving reflectional symmetry about
the horizontal axis, the modes (i), (ii), (v), and (vii) can be used (see §3.1). The modes (i) and (ii)
can be used on the center row and the number of arrangements is P(k3, 20). For the remaining
four partitions it is sufficient to count the number of arrangements for one section of top or
bottom rows. As in the Srot, subclass this number is E2(n). Summing over all partitions of n tinted
cells across the rows calculates the enumeration
Esref h (n) =
P(k3, 20) · E2(k12).
k12, k3
2k12 + k3 = n
0 k3 20
0 k12 40
For the Sref, rot subclass of patterns involving rotational and reflectional symmetry, only the
modes (ii) and (vii) can be used (see §3.1). The mode (ii) implies that there are S(k3, 20)
arrangements for the center row. To calculate the arrangements for the remaining four 2 x 10
sections, define E2S(n) to be the number of arrangements of n tinted cells across 2 rows such that
each row is symmetric. E2S is calculated similarly to ES 0
E2S (n) =
S(k ).
k1,... k2
k1 + k2 = n
0 k
20
23
Summing over the partitions of n across the rows, the enumeration is calculated as
Esref rot (n) =
S(k3) · E2S(k12).
k12, k3
2k12 + k3 = n
0 k3 20
0 k12 40
The complete numeric results of the enumeration of façade patterns in each symmetry
subclass, is provided in Appendix II.
Verification
This section demonstrates the validity of the adopted performance constraints that are encoded in
the grammar. The constraints are pertinent to the adjustment of daylight illuminance at the
examined location in Trento, Italy, throughout the four seasons. The demonstration exposes the
simulation steps and the analysis that had lead to provisions I and II of the grammar.
Simple façade patterns were tested first, including the patterns with all the windows clear,
or tinted, or with consecutive active windows arranged in rows, or in columns, or in a
"chessboard" formation. Composite façade patterns were tested after, including combinations of
the simple façade patterns. A link was determined between pattern configuration and interior
illuminance and general principles applicable to any façade pattern were extracted. The notion of
an equivalence class was introduced to designate classes of façade patterns having the same
number of active windows and reaching proximate values of interior illuminance independently
of pattern configuration. Equivalence classes allow switching from one pattern to another while
maintaining a desired level of illuminance in order to satisfy other factors of performance such as
thermal comfort or aesthetic preference. Lastly a predictive model was specified associating
coverage ratio and illuminance, and having the capacity to project the performance of any façade
pattern during any day of the year.
24
5.2. Daylight performance with the façade in clear state
The distribution of interior daylight was simulated at the simplest possible condition, with all the
windows inactive and the façade in a clear state. The detailed results of this simulation, including
the minimum, maximum and average illuminance values calculated at hourly intervals during
four days of the year, winter/summer solstice and spring/fall equinox, in Overcast Sky
Conditions, are presented in Appendix III. Figure 4 summarizes in a comparative diagram the
average illuminance values from this simulation.
Figure 4: Average illuminance for the interval 8:00 a.m. to 8:00 p.m. in Overcast Sky.
Based on the results of the simulation, the highest values of illuminance were recorded
during the summer and the values decreased during the winter. On December 21st the
illuminance value was zero after 5:00 p.m. as the sun drops below the horizon. Zero illuminance
values also occurred on March 21st after 6:00 p.m. and on September 21st after 7:00 p.m. The
obtained uniformity G1 remained constantly three times lower than the threshold value G1 = 0,5.
The uniformity G1 defined by the ratio Emin/Eave captures the smoothness of daylight distribution.
The obtained G1 values capture the high density of windows towards the south. Since the east
and west elevations are blind and the north elevation has only a limited number of windows, the
north windows cannot balance the light intensity of the south windows, which are always
exposed to direct sunlight. In the tridimensional diagram of Figure 5, the illuminance values in
25
lux that correspond to the south façade are higher and they are decreasing towards the north
façade. Exceptions apply locally at values corresponding to the north windows, where the
illuminance distribution is slightly higher.
Figure 5. Tridimensional diagram of illuminance distribution for the interior of the
prototype. In the diagram the south façade corresponds to the light blue tone.
5.3. Daylight performance with the façade in various active configurations
Proper management of the façade permits the adjustment of the incoming daylight reaching an
interior work plane. The activation of a horizontal zone of windows causes a corresponding
“shaded zone” in the house interior. But what would be the effect of a more complex façade
pattern composed out of dispersed windows on and off, and how could we always ascertain the
appropriate number and distribution of active windows to obtain illuminance values that are
above the threshold? To examine these questions a limited number of façade patterns were tested
during specific days. These tests helped to establish a link between coverage ratio
illuminance Eave, where coverage ratio
and
corresponds to the ratio of the number x of active
windows versus the total number of windows (
= x / 100). The patterns included linear
arrangements (rows or columns) of active windows. The most critical lighting condition occurred
in June 21st, at 1 p.m. when illuminance reached its highest, during the summer equinox.
26
5.3.1. Rows
The tested façade patterns include ten patterns obtained by switching on or off various
configurations of window rows and examining how they affect the distribution of interior
daylight. During the tests the north widows remained at state off and their transmittance was set
to 62%. The transmittance values are included in Table 3, and diagrams of the simulations
appear in Figure 6. Windows at state off appear as white squares and windows at state on as grey
squares. The simulations model the conditions of June 21, at 1 p.m. in Trento, Italy.
Skylight
transmittance (%)
Windows
transmittance (%)
Coverage ratio ( )
Line
1
2
1
2
3
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
10th
62
3.5
3.5
3.5
62
0
1
0,6
3.5
3.5
3.5
62
0,8
3.5
3.5
62
3.5
0,8
3.5
62
3.5
3.5
0,8
3.5
62
62
3.5
0,6
3.5
62
3.5
62
0,6
62
62
3.5
3.5
62
62
0,6
3.5
0,4
Table 3. Transmittance values for the various window rows of the south façade.
Figure 6. Comparative presentation of the Isolux diagrams for ten distinct façade patterns
involving active windows arranged in rows.
27
The illuminance values (minimum, maximum, average), the uniformity value (G1) and the
average daylight factor (Dav) are simulated next for the same ten façade patterns. The values are
calculated for a plane placed 0.75 m above the floor and in parallel to it. Table 4 summarizes the
obtained results. Figure 7 offers a comparative presentation of the obtained illuminance levels.
Emax
Emin
Eave
1st
4880
204
1400
2nd
1440
12
159
3rd
4340
180
1080
4th
2740
102
572
5th
1470
44
228
6th
2520
65
496
7th
2120
86
679
8th
4160
153
1020
9th
2400
128
768
10th
1510
41
593
G1
Dav
0.15
7.2
0.075
0.82
0.17
5.6
0.18
3
0.19
1.18
0.13
2.6
0.13
3.5
0.15
5.3
0.17
4
0.07
3.06
Table 4. Illuminance values (minimum, maximum, average) uniformity value (G1) and
average daylight factor (Dav) for a plane placed 0.75 m above the floor, for the ten patterns.
Figure 7. Comparative diagram of illuminance for the specified interior work plane.
Table 4 and Figure 7 demonstrate that the highest illuminance value is achieved in the 1 st
configuration, where all the windows are at state off. The lowest value is achieved in the 2nd
configuration, where all the windows are at state on. Variation in the number and distribution of
active windows causes variation in daylight distribution. This is captured by the fluctuation in
the uniformity G1 and average illuminance Eave. values. Alteration in the combinations of active
window rows affects interior daylight drastically. Long rows of tinted windows cast long linear
shadows disrupting smooth daylight distribution. This result was encoded by provision II.
28
5.3.2. Columns
This class of simulations shows how various vertical configurations of active windows (columns)
affect the distribution of interior daylight. Figure 8 presents simulation diagrams for eight façade
patterns. A “chessboard” pattern was simulated last and its values were compared to the
previous. Figure 9 offers a comparative presentation of the levels of illuminance and the
uniformity G1. The simulations model the conditions of June 21 at 1 p.m. in Trento, Italy.
Figure 8. Comparative presentation of the Isolux diagrams and tridimensional diagrams for
eight façade patterns, involving active windows arranged in columns.
Figure 9. Minimum, maximum and average illuminance (left) & uniformity G1 (right) for
the same eight patterns.
29
The diagrams of Figure 9 reveal that the variance in illuminance, average daylight factor and
uniformity G1 depends on the number of the active windows. Configurations with the same
number of active windows yield proximate values despite pattern dissimilarities. For example,
the 1st and the 8th configuration yield proximate values of illuminance, average daylight factor
and uniformity G1, despite pattern dissimilarity. The same is true for the 6th and 7th configuration.
5.3.3. Composite patterns
Additional simulations modeling different hours and days of the year demonstrate whether the
illuminance, the average daylight factor and the uniformity G1 values remain relevant under a
broader spectrum of conditions. The comparison of results for the 1st and the 8th configuration
offered a basis for generalizing the conclusions on illuminance and on whether two distinct
patterns can yield the same level of illuminance performance. The interior daylight conditions
were monitored in two daytime moments, at 10:00 a.m. and 4:00 p.m. on December 21 (Table 5),
March 21 (Table 6), June 21 (Table 7) and September 21 (Table 8). The specific daytimes were
selected because of their dissimilarity in solar radiation.
Emax (lx)
Emin (lx)
Eave (lx)
G1
Dav
1st
650
27
199
0,13
4,4
8th
679
28
197
0,14
4,3
21st
Dec
10:00
Emax (lx)
Emin (lx)
Eave (lx)
G1
Dav
1st
273
84
10
0,12
4,3
8th
285
83
11
0,13
4,3
st
21
Dec
16:00
Table 5. Interior daylight conditions for December 21, 10:00 a.m., 4:00 p.m. (patterns 1, 8).
Emax (lx)
Emin (lx)
Eave (lx)
G1
Dav
1st
1590
67
486
0,14
4,3
8th
1670
68
482
0,14
4,3
21 st
March
10:00
Emax (lx)
Emin (lx)
Eave (lx)
G1
Dav
1st
1300
54
398
0,14
4,3
8th
1360
54
395
0,14
4,3
st
21
March
16:00
Table 6. Interior daylight conditions for March 21, 10:00 a.m., 4:00 p.m. (patterns 1, 8).
30
Emax (lx)
Emin (lx)
Eave (lx)
G1
Dav
1st
2060
83
623
0,13
4,3
8th
2120
86
617
0,14
4,3
21st
June
10:00
Emax (lx)
Emin (lx)
Eave (lx)
G1
Dav
1st
2400
99
730
0,14
4,3
8th
2500
101
724
0,14
4,3
21st
June
16:00
Table 7. Interior daylight conditions for June 21, 10:00 a.m., 4:00 p.m. (patterns 1, 8).
Emax (lx)
Emin (lx)
Eave (lx)
G1
Dav
1st
1340
56
408
0,14
4,3
8th
1400
57
404
0,14
4,3
21st
Sept.
10:00
Emax (lx)
Emin (lx)
Eave (lx)
G1
Dav
1st
1630
68
495
0,14
4,3
8th
1690
68
490
0,14
4,3
st
21
Sept.
16:00
Table 8. Interior daylight conditions, September 21, 10:00 a.m., 4:00 p.m. (patterns 1, 8).
The results of the Tables 5-8 establish that the performance of two distinct façade patterns
can be invariable throughout the year. Hence, a façade pattern can be modified from one
configuration to another while maintaining constant levels of interior illuminance. Next it was
examined whether any two distinct patterns with the same coverage ratio
could yield invariable
illuminance values. If performance and coverage ratio were analogous then the visual
configurations could vary while retaining a constant coverage ratio . This result would establish
that façade pattern generation could follow a range of generative rules while satisfying any preset
value of illuminance, average daylight factor, and uniformity G1.
Figure 10, presents three combinations of the 1st and the 8th pattern. These composite
patterns are formed after substituting a number of active windows arranged in columns with an
equal number of active windows arranged in chessboard formation. Testing a limited number of
composite patterns was sufficient to generalize the results. The tables and diagrams of Figure 10
present the results for July 21 at 1:00 p.m. Variation in daytime or season does not affect
illuminance, average daylight factor, or uniformity G1 under constant coverage ratio conditions.
31
Figure 10. Patterns 1 and 8 appear at the top row. Bellow, 3 composite patterns appear on the left
column. A comparison of results, including tridimensional diagrams, appears on the right.
5.3.4. Equivalence classes
In Overcast Sky Conditions the 1st and the 8th patterns and their three composites with constant
coverage ratio , reach proximate values of illuminance (minimum, maximum, average), average
daylight factor, and uniformity G1. These results are presented in Figure 11.
Emax (lx)
Emin (lx)
Eave (lx)
G1
Dav
1st
2780
113
840
0.14
4.3
8th
2850
115
833
0.14
4.3
M1st
2850
110
811
0,14
4,2
M2nd
2870
113
802
0,14
4,1
M3rd
3090
112
785
0,14
4,1
Figure 11. Diagram and table of minimum, maximum and average illuminance, uniformity
G1 and daylight factor values for patterns 1, 8 and their three composites, in Overcast Sky.
32
Patterns with the same total number of windows at state on yielding a specific value of
illuminance belong to the same equivalence class. This provides an option of switching from one
configuration to another while maintaining a desired coverage ratio , in order to satisfy other
factors related to aesthetics or performance.
5.4. Determining the desired coverage ratio
The option of modifying a façade pattern within an equivalence class based on coverage ratio
requires determining what the desired coverage ratio is, for every daylight condition of the year.
A predictive model of this kind is obtainable. The model associates coverage ratio
and daylight
illuminance Eave during various time intervals for each day of the year in Overcast Sky
Conditions. A model for Clear Sky Conditions is specified in Appendix IV. Eleven base
configurations were tested, including the two marginal conditions with the façade clear and
tinted. The table of Figure 12 shows the number x of active windows for each configuration and
the corresponding value in lux. The graph depicts the association between Eave and coverage ratio
. The simulations model the conditions of winter solstice on December 21 at 1 p.m.
0
1
2
3
4
5
6
7
8
9
10
Base
configuration
x
Extreme
Mixed
7th vertical
3rd vertical
8th vertical
1st vertical
6th vertical
5th vertical
2nd vertical
4th vertical
Extreme
0
40
45
50
50
50
55
60
70
75
100
Eave
0,00
0,4
0,45
0,50
0,50
0,50
0,55
0,60
0,70
0,75
1,00
497
335
308
305
303
293
286
241
221
188
82
Figure 12. Table and graph associating Eave in lux and coverage ratio
for December 21, at 1
p.m. in Overcast Sky. Extreme configurations are, clear (first) and fully tinted (last).
33
The interpolation of values of the table in the graph determines an equation that calculates
the illuminance value in lux corresponding to a coverage ratio
associated to it. This expression
determines the number of active windows yielding a specific illuminance level. It is a linear
equation of the form y =
x + b, where y represents the lux value corresponding to the number
x of active windows, with x =
100 (100 is the total number of façade windows). Four
expressions calculate the values at 1:00 p.m. for solstice and equinox.
December 21: y = -411.86x + 500.69
March 21: y = -839.84x + 1020.4
June 21: y = -1113.1x + 1365.4
September 21: y = -848.63x + 1033.1
Expressions yielding the y value for every hour of the year can be formed and the required
number of active windows can be specified for any condition. Hence, it is possible to determine
the required number of active windows for reaching the illuminance threshold and confine the
patterns to this threshold. The tables and graphs of Figures 13, 14, and 15 capture the association
between Eave and coverage ratio
0
1
2
3
4
5
6
7
8
9
10
Base
configuration
Extreme
Mixed
7th vertical
3rd vertical
8th vertical
1st vertical
6th vertical
5th vertical
2nd vertical
4th vertical
Extreme
x
0
40
45
50
50
50
55
60
70
75
100
for the remaining days of solstice and equinox at 1:00 p.m.
Lux
0,00
0,4
0,45
0,50
0,50
0,50
0,55
0,60
0,70
0,75
1,00
1010
682
625
621
617
597
583
491
452
382
167
Figure 13. Table and graph associating Eave in lux and coverage ratio
for March 21, at 1:00
p.m. in Overcast Sky. Extreme configurations are, clear (first) and fully tinted (last).
34
0
1
2
3
4
5
6
7
8
9
10
Base
configuratio
n
Extreme
Mixed
7th vertical
3rd vertical
8th vertical
1st vertical
6th vertical
5th vertical
2nd vertical
4th vertical
Extreme
x
0
40
45
50
50
50
55
60
70
75
100
Lux
0,00
0,4
0,45
0,50
0,50
0,50
0,55
0,60
0,70
0,75
1,00
1340
923
846
840
833
808
788
665
611
516
226
Figure 14. Table and graph associating Eave in lux and coverage ratio
for June 21, at 1:00
p.m. in Overcast Sky. Extreme configurations are, clear (first) and fully tinted (last).
0
1
2
3
4
5
6
7
8
9
10
Base
configuration
Extreme
Mixed
7th vertical
3rd vertical
8th vertical
1st vertical
6th vertical
5th vertical
2nd vertical
4th vertical
Extreme
x
0
40
45
50
50
50
55
60
70
75
100
Lux
0,00
0,4
0,45
0,50
0,50
0,50
0,55
0,60
0,70
0,75
1,00
1020
693
635
631
624
607
591
499
459
387
169
Figure 15. Table and graph, associating Eave in lux and coverage ratio
for September 21 at
1:00 p.m. in Overcast Sky. Extreme configurations are, clear (first) and fully tinted (last).
The presented predictive model calculates the required number of active windows to reach the
value of 500 lux. The error between prediction and simulation is 3-4 %, but the error rises up to
5-6 % for patterns with coverage ratio beyond 50-75 %. It is still possible to activate a lower
number of windows, thus achieving Eave higher than 500 lux. Using the inverse process, if the
desired illuminance Eave is 500 lux and Eave= y, then reaching 500 lux in Overcast Sky in
December requires all windows to be off. This is necessary due to the low levels of sky radiation
35
during December in Trento. In all, the values x ensuring illuminance value of 500 lux during
December, March, June, and September, are:
December 21:
y = -411,86x + 500,69
x = y / 411,86 – 500,69 / 411,86
March 21:
y = -839.84x + 1020,4
x = y / 839.84 – 1020.4 / 839.84
x
62
June 21:
y = -1113.1x + 1365,4
x = y / 1113.1 – 1365.4 / 1113.1
x
78
September 21:
y = -848.63x + 1033.1
x = y / 848.63 – 1033.1 / 848.63
x
63
x
0
If the number of active windows ensuring the threshold of 500 lux is known, then it is possible to
vary the façade configurations without affecting the levels of illuminance. The calculation can be
extended – for hourly intervals – over the entire year, and a general database can be obtained
determining the required number of active windows to reach the threshold value. The generation
of façade patterns halts when the number of active windows reaches the value 75. This condition
was encoded by provision I of the grammar. Figure 16 includes a table and a graph summarizing
the numbers of active windows x ensuring 500 lux on the 21st day of each month, at 1:00 p.m. in
Trento, Italy. Based on the diagram an increase of the illuminance levels occurs only for few
hours yearly without affecting the performance.
1
2
3
4
5
6
7
8
9
10
11
12
13
1p.m. -21st
December
January
February
March
April
May
June
July
August
September
October
November
December
x
0
18
45
62
72
75
75
75
72
63
45
16
0
0,00
0,18
0,45
0,62
0,72
0,75
0,75
0,75
0,72
0,63
0,45
0,16
0,00
Figure 16. Table and graph showing the maximum number of active windows on the 21st day of
each month, at 1:00 p.m. in Standard Overcast Sky. In dotted line the 75% limit.
36
6. Discussion
This paper has presented a generative approach in the production of patterns for the primary
façade of a prototype house in Trento, N. Italy. The façade is a dynamically controlled solar wall
including 100 individually addressable, electrochromic windows 700 mm
700 mm in size,
enabling the precise adjustment of daylight, heat, view and ventilating air at the house interior,
and affecting the way the house is perceived from the public street. After compiling feedback
from sensors, statistical climatic data, and ambient data, the control system of the house provides
real time performance evaluation and generates electrochromic patterns on the façade in response
to the conditions and the needs of the inhabitants for various combinations of privacy, visibility
and view. The uniqueness of the presented approach lies on the deployment of a shape grammar
to configure the states of the dynamic façade equally based on performance and aesthetic criteria.
The grammar treats the class of the effective façade patterns as a design language, where the full
spectrum of visual and performance attributes of the configurations is taken into account.
The increasing cost and scarcity of non-renewable energy sources promote the use of
sustainable principles in the design and operation of buildings. The accumulation of precise
knowledge on the association between man and environment, and the availability of new
optimization and automation techniques force the reassessment of the energy management
methods. But to be adopted, the new techniques need to be integrated in engaging ways into
building aesthetics and not just to be efficient. Unfortunately, the acquisition of precise
knowledge and control capabilities is rarely accompanied by the required sensitivity to apply
them in effective and aesthetically pleasant ways into the built environment. Motivation for this
research was to provide an environmentally conscious mode of building an original tectonic
vocabulary that is in alignment with technological innovation. An architectural solution was
37
proposed employing generative grammars, AI methods to building control and electro-active
materials, which satisfies the environmental requirements of its function and makes elegant
architecture out of the provisions needed for their satisfaction.
Buildings often become the embodiment of specialized technological innovation in response
to given problems. However, a sophisticated approach to sustainable architecture should not
simply rely on new machinery. It should be able to express unique features related to the
environment, the culture, and the context it is situated. The prototype house in Trento is an
example of environmentally responsible architecture that points to certain visionary
technological possibilities without disregarding their impact in the habits of the local people. In
the context of Trento, the principle façade of a building has predominantly expressive purpose as
the meeting surface between interior and exterior. The management by the means of a generative
grammar of the electrochromic technology that is deployed on the dynamic façade of the
prototype acknowledges this purpose by equally addressing performance and aesthetic factors.
Façade patterns produced randomly based only on performance would supply adequate daylight
while disregarding the aesthetic potential of electrochromic technology. The chosen approach
both takes full advantage of this potential and is computationally elegant. The grammar is
applied dynamically by the intelligent control system of the house to produce façade patterns by
linking principles of 2D pattern generation to constraints of daylight adjustment.
Three general modes were determined for the façade, namely, façade fully inactive, façade
active in a ratio equal to 2/4 – 3/4 of its area, and façade fully active. The grammar determines
the configuration of the façade when 2/4 – 3/4 of its area is active. This is possible through the
application of 12 rules that apply under 7 different modes, while satisfying 2 performative
premises. The first premise specifies that in an average luminous day between 50 and 75 of the
38
100 windows need to be active, in order to achieve interior luminosity levels above the threshold
required by Italian law. The second premise specifies that to ensure smooth daylight distribution
no 4 consecutive windows can be concurrently active on the same row. In clouded sky conditions
the generative grammar exhibits great flexibility in producing patterns ensuring interior daylight
comfort. The façade pattern language extends to 1.285
1024 patterns, which are more than the
stars in the observable universe. In clear sky conditions full activation of select façade sub-areas,
leaving others inactive, could allow for optimal daylight conditions to occur locally while heat
flow through the inactive façade sections will be still possible.
The absence of adequate real building data from experiments characterizing the performance
of environmental management systems similar to the one that it was presented in this paper,
dictates that the assessment of the proposed apparatus will be subject of future research, after
quantifiable data from the prototype become available. And yet, it is likely that in the future the
use of programmable materials and AI methods to building control will attain a higher degree of
influence in the architecture of buildings. Similar to how the supply of electricity had profoundly
affected architecture at the close of the 19th century, real time supply of computational power
into the built environments and material components will gradually transform their aesthetic and
physical attributes, giving rise to new conceptions of space. The absorption of this capacity
provides an opportunity for producing buildings that are energy saving, environmentally sound
and architecturally rich. The design of the prototype house in Trento is an eloquent application of
algorithmic design methods, AI methods to building control, and material engineering research,
pointing to original tectonic possibilities without disregarding the consequences of the ongoing
transformation in environmental management.
39
Acknowledgments
This paper is dedicated to William J Mitchell (1944 – 2010), who pioneered multidisciplinary
research in design. Thanks are due to Profs. Terry W Knight and George N Stiny for technical
remarks. The research was conducted within the Green Home Alliance between the Mobile
Experience Lab at the Massachusetts Institute of Technology and the Fondazione Bruno Kessler
in Trento, Italy.
References
Banham, R, 1969, The Architecture of the Well-tempered Environment, Architectural Press, 23.
Graybill W, 2012, Robust, “Goal-Directed Planning and Plan Recognition for the Sustainable
Control of Homes”, Master's Thesis, Massachusetts Institute of Technology.
Hausler T, Fischer U, Rottmann M and Heckner K H, 2003, “Solar optical properties and
daylight potential of electrochromic windows”, International Lighting and Colour
Conference, Capetown.
Klein C and Kaefer G, 2008, "From Smart Homes to Smart Cities: Opportunities and Challenges
from an Industrial Perspective", in Balandin et al. (eds), Lecture Notes in Computer Science
vol. 5174, Springer-Verlag, Berlin, Heidelberg.
Knight T, 2005, “Creativity. Rules”, Proceedings of HI’ 05 Sixth International Roundtable
Conference on Computational and Cognitive Models of Creative Design, Heron Island,
Lee E S, Di Bartolomeo D L, Klems J H, Yazdanian M and Selkowitz S E, 2006, “Monitored
energy performance or electrochromic windows for daylighting and visual comfort”,
ASHRAE Summer Meeting, Quebec City, Canada.
40
Luebkeman C, Shea K, 2005, “CDO: Computational design + optimization in building practice”,
The Arup Journal, 3.
Mardaljevic J, Heschong L and Lee E S, 2009, “Daylight metrics and Energy savings”, Lighting
Research and Technology, 41:261.
Ono, M, 2012, “Robust, Goal-directed Plan Execution with Bounded Risk,” Ph.D. Dissertation,
Massachusetts Institute of Technology, 2012.
Selkowitz S, Lee E and Aschehoug O, 2003, “Perspectives on advanced facades with dynamic
glazings and integrated lightings controls”, presented at CISBAT, Lausanne, Switzerland.
Shea K, Aish R, Gourtovaia M, 2005, “Towards integrated performance-driven generative
design tools”, Automation in Construction, 14, 253-264
Stiny G, 2011, “What Rule(s) Should I Use?”, Nexus Network Journal Vol.13, No. 1, 15, 15-47
Stiny G, 1992, “Weights” Environment and planning B: Planning and design 19 413-430.
Stiny G, 1991, “The algebras of design”, Research in Engineering Design 2 171-181.
Stiny G, Mitchell W J, 1978, “The Palladian grammar”, Environment and planning B: Planning
and design 5 209-226
Stiny G, Mitchell W J, 1978, “Counting Palladian plans”, Environment and planning B:
Planning and design 5 189-198
Stiny G, Mitchell W J, 1978, “An evaluation of Palladian plans”, Environment and planning B:
Planning and design 5 199-206
Wienold J, 2007, “Dynamic simulation of blind control strategies for visual comfort an energy
balance analysis”, International Building Performance Simulation Association, Beijing,
China, 1197-1204.
41
Appendix I
The derivation of a façade pattern is presented. The reading order is from top to bottom and left
to right. Derivation arrows are not shown. The number of the rule ri that is applied at each step
appears on the upper center of each façade diagram. Rules ri are noted only once, although they
apply twice, in parallel, at each derivation step after the modes (ii) and (vi). The transition of rule
modes is indicated on the upper left side of the façade diagrams each time a rule mode is put into
use. The derivation starts from the initial shape, the inactive window matrix, at stage A.
Stage A
(after mode ii)...
r3
(after mode vi)...
r1
r3
r3
r3
Stage B
(after mode ii)...
r4
r8
r5
r5
r8
r5
r5
r5
r6
r8
r5
r5
(after mode vi)...
r8
r5
r5
r8
r5
r5
r5
r7
r6
r8
r5
r5
r5
r7
r8
r5
r5
r8
r5
r5
r9
r11
r11
Stage C
(after modes ii & vi)...
r10
r12
2
Appendix II
The complete enumeration of the patterns in each symmetry subclass is provided in Table 9.
n
50
51
52
53
54
S0
Srot
1.751 x 10 27
1.466 x 10 13
1.191 x 10 27
0
7.566 x 10 26
9.541 x 10 12
4.484 x 10 26
0
2.472 x 10 26
5.397 x 10 12
Sref h
6.971 x 10 15
5.664 10 15
4.433 x 10 15
3.338 x 10 15
2.415 x 10 15
Sref v
5.069 x 10 12
0
2.961 x 10 12
0
1.488 x 10 12
0
7.948 x 10 6
Sref, rot
1.521 x 10
7
0
1.155 x 10
7
n
55
56
57
58
59
S0
Srot
1.265 x 10 26
0
5.986 x 10 25
2.628 x 1012
2.612 x 10 25
0
1.048 x 10 25
1.087 x 10 12
3.844 x 10 24
0
Sref h
1.675 x 10 15
1.113 x 1015
7.067 x 10 14
4.278 x 10 14
2.463 x 10 14
Sref v
0
6.351 x 10 11
0
2.266 x 10 11
0
6
6
Sref, rot
0
5.111 x 10
0
2.893 x 10
n
60
61
63
64
65
S0
Srot
1.285 x 10 24
3.759 x 10 11
3.891 x 10 23
0
2.590 x 10 22
0
5.611 x 10 21
2.406 x 10 10
1.069 x 10 21
0
Sref h
1.345 x 10 14
6.943 x 10 13
1.541 x 10 13
6.562 x 10 12
2.597 x 10 12
Sref v
6.616 x 10 10
0
0
2.717 x 10 9
0
5
0
Sref, rot
6
0
0
0
2.708 x 10
n
66
67
68
69
70
S0
Srot
1.774 x 10 20
4.180 x 10 9
2.529 x 10 19
0
3.053 x 10 18
5.300 x 10 8
3.063 x 10 17
0
2.494 x 10 16
4.534 x 10 7
Sref h
9.468 x 10 11
3.167 x 10 11
9.578 x 10 10
2.606 x 10 10
6.248 x 10 9
Sref v
3.434 x 10 8
0
2.753 x 10 7
0
1.049 x 10 6
Sref, rot
8.550 x 10 4
0
2.150 x 10 4
0
4.096 x 10 3
71
72
73
74
n
S0
1.523 x 10
15
13
7.700 x 10
2.285 x 106
2.615 x 10
0
12
75
10
5.547 x 10
5.018 x 10 4
5.507 x 10 8
0
Srot
1.597 x 10
0
Sref h
1.295 x 10 9
2.299 x 10 8
3.131 x 10 7
3.537 x 10 6
1.756 x 10 5
Sref v
0
0
0
0
0
Sref, rot
0
0
0
0
0
Table 9. Enumeration of façade pattern subclasses based on symmetry.
3
Appendix III
A simulation of daylight distribution with the façade inactive is presented. Table 10 summarizes
the minimum, maximum and average illuminance values calculated at hourly intervals in four
days of the year, winter/summer solstice and spring/fall equinox, in Overcast Sky Conditions.
20.00
19.00
18.00
17.00
16.00
15.00
14.00
13.00
12.00
11.00
10.00
9.00
8.00
Emin
Emax
Eav
Emin
Emax
Eav
Emin
Emax
Eav
Emin
Emax
Eav
Emin
Emax
Eav
Emin
Emax
Eav
Emin
Emax
Eav
Emin
Emax
Eav
Emin
Emax
Eav
Emin
Emax
Eav
Emin
Emax
Eav
Emin
Emax
Eav
Emin
Emax
Eav
0
0
0
0
0
0
0
0
0
0
0
0
26
619
189
74
1770
544
112
2650
811
135
3140
967
138
3310
1010
125
3040
925
101
2400
730
58
1410
429
7
158
48
Dec
21st
0
0
0
0
0,14
0,14
0,14
0,14
0,14
0,14
0,14
0,14
0,14
G1
0
0
0
0
0
0
28
656
200
97
2270
695
162
3770
1140
210
4960
1520
249
5860
1780
264
6330
1940
275
6460
1960
257
6060
1850
221
5270
1610
179
4160
1270
117
2750
840
Mar
21st
0
0
0,14
0,14
0,14
0,14
0,14
0,14
0,14
0,14
0,14
0,14
0,14
G1
72
1640
500
134
3140
965
199
4680
1420
262
6040
1840
296
7190
2180
337
8060
2450
366
8540
2600
370
8570
2630
358
8330
2540
322
7630
2340
272
6660
2030
222
5400
1640
163
3920
1200
Jun
21st
0,14
0,14
0,14
0,14
0,14
0,14
0,14
0,14
0,14
0,14
0,14
0,14
0,14
G1
0
0
0
15
357
109
84
1990
610
152
3500
1070
201
4810
1470
247
5770
1760
268
6360
1940
275
6520
1990
259
6300
1910
241
5570
1710
188
4560
1390
138
3200
978
71
1650
504
Set
21st
0,14
0,14
0,14
0,14
0,14
0,14
0,14
0,14
0,14
0,14
0,14
0,14
G1
Table 10: Hourly illuminance values for the interval 8:00 am- 8:00 pm in Overcast Sky.
4
Appendix IV
In Clear Sky direct sunlight radiation raises the illuminance values higher than in Overcast Sky.
The simulation results show proximity to 500 lux when the façade is fully active. The tables and
graphs of Figures 17, 18, 19, and 20 capture the association between Eave in lux, and coverage
ratio
0
1
2
3
4
5
6
7
8
9
10
for solstice and equinox at 1:00 p.m. in Trento, Italy and Standard Clear Sky Conditions.
Base
configuration
Extreme
x
0
0,00
8340
Mixed
40
0,4
5070
7th vertical
45
0,45
4790
th
Lux
8 vertical
50
0,50
4600
1st vertical
50
0,50
4520
3 vertical
50
0,50
4390
6th vertical
55
0,55
4220
5th vertical
60
0,60
3500
rd
nd
2 vertical
70
0,70
3010
4th vertical
Extreme
75
0,75
2480
100
1,00
490
Figure 17. Table and graph associating Eave in lux and coverage ratio
for December 21 at 1
p.m. in Standard Clear Sky. Extreme configurations are, clear (first) and fully tinted (last).
0
1
2
3
4
5
6
7
8
9
10
Base
configuration
Extreme
Mixed
7th vertical
1st vertical
8th vertical
3rd vertical
6th vertical
5th vertical
2nd vertical
4th vertical
Extreme
x
0
40
45
50
50
50
55
60
70
75
100
Lux
0,00
0,4
0,45
0,50
0,50
0,50
0,55
0,60
0,70
0,75
1,00
11400
7180
6830
6610
6570
6460
6110
5130
4500
3760
675
Figure 18. Table and graph associating Eave in lux and coverage ratio
for March 21 at 1 p.m.
in Standard Clear Sky. Extreme configurations are, clear (first) and fully tinted (last).
5
0
1
2
3
4
5
6
7
8
9
10
Base
configuration
Extreme
Mixed
th
7 vertical
1st vertical
8th vertical
3rd vertical
6th vertical
5th vertical
2nd vertical
4th vertical
Extreme
x
0
40
45
50
50
50
55
60
70
75
100
Lux
0,00
0,4
0,45
0,50
0,50
0,50
0,55
0,60
0,70
0,75
1,00
7020
4520
4020
4340
4100
3920
4120
3170
3130
2400
565
Figure 19. Table and graph associating Eave in lux and coverage ratio
for June 21 at 1 p.m.
in Standard Clear Sky. The extreme configurations are, clear (first) and fully tinted (last).
0
1
2
3
4
5
6
7
8
9
10
Base
configuration
Extreme
Mixed
th
7 vertical
1st vertical
8th vertical
3rd vertical
6th vertical
5th vertical
2nd vertical
4th vertical
Extreme
x
0
40
45
50
50
50
55
60
70
75
100
Lux
0,00
0,4
0,45
0,50
0,50
0,50
0,55
0,60
0,70
0,75
1,00
11300
7260
6640
6730
6460
6200
6310
5090
4420
3800
672
Figure 20. Table and graph associating the Eave in lux and coverage ratio
for September 21
at 1 p.m. in Standard Clear Sky. The extreme configurations are, clear (first) and tinted (last).
The process of §5.4 can be extended for Clear Sky Conditions with a reduction in its
accuracy (the error raises to 5% -7%). Along these lines, a set of explicit equations is determined
next for solstice and equinox at 1:00 p.m. in Clear Sky Conditions.
6
December 21:
y = -7792,2x + 8343,1
x = y / 7792,2 – 8343,1 / 7792,2
March 21:
y = -10560x + 11641
x = y / 10560 – 11641/10560
June 21:
y = -6273,7x + 7148,5
x = y / 6273,7 – 7148,5 / 6273,7
September 21:
y = -10455x + 11553
x = y / 10455 – 11553 / 10455
x =100
x = 100
x = 100
x =100
Figure 21 summarizes the maximum numbers of active windows x ensuring 500 lux on the 21st
day of each month of the year, at 1:00 p.m.
1
2
3
4
5
6
7
8
9
10
11
12
13
1p.m. -21st
December
January
February
March
April
May
June
July
August
September
October
November
December
x
100
100
100
100
100
100
100
100
100
100
100
100
100
1
1
1
1
1
1
1
1
1
1
1
1
1
Figure 21. Table and corresponding graph presenting the maximum number of active windows
throughout the year on the 21st day of each month at 1:00 p.m. in conditions of Clear Sky.
Clear Sky demands the integration of additional parameters for the efficient management of
the façade. The façade aims at optimizing visual comfort, but also contributes to improving
energy efficiency by managing the high thermal mass properties of the house envelope. In the
winter the increasing need for sunlight flow at the interior for heat conservation purposes may
lead to a different set of conventions. Hence, full activation of select façade partitions, leaving
others inactive, could allow for optimal daylight conditions to occur locally while heat flow
through the inactive façade partitions will still be possible.
7
Appendix V
The verification of the derived pattern in Appendix I, is presented next. The façade pattern was
tested and a comparison between the values of Eave obtained through the predictive equations and
the software simulation was performed to ascertain the accuracy of the predictive model. The
equations listed in § 5.4. and in the Appendix IV were used for Overcast Sky and Clear Sky,
respectively. The simulations model the conditions of solstice and equinox at 1 p.m. in Trento,
Italy. A comparative exposition of the results appears in Tables 11 and 12.
December 21 st
March 21st
June 21 st
September 21 st
Prediction with
the model Eave
(lx)
195
398
541
404
Verification with
Relux Pro
Eave (lx)
186
379
513
385
Error
5%
5%
5%
5%
Table 11. Comparison of Eave between simulation and prediction in Overcast Sky.
December 21 st
March 21 st
June 21 st
September 21 st
Prediction with
the model Eave
(lx)
2586
3826
2505
3816
Verification with
Relux Pro
Eave (lx)
2630
4030
2490
3840
Error
2%
5%
0%
0%
Table 12. Comparison of Eave between simulation and prediction in Clear Sky.
The listed values verify the agreement between simulation and prediction. The coverage
ratio
of the façade pattern was 74 % and the error margin was 5 %. In Overcast Sky the 74 %
coverage ratio yielded acceptable interior illuminance only in June 21 at 1 p.m. In the other three
days of the simulation, the value of Eave remained below the threshold. However, in practice the
interior daylight levels can be adjusted by lowering the coverage ratio . In Clear Sky the 74 %
coverage ratio yielded illuminance levels above the threshold, but still acceptable.
8