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33rd
eCAADe
Conference
TU Wien (AT)
09/2015
Real
Time
V2
Gradient-Analysis
Method and Software to Compare Different Degrees of Complexity in the
Design of Architecture and Designobjects
Matthias Kulcke1, Wolfgang Lorenz2
1Hamburg University of Technology / HafenCity University Hamburg2Vienna Uni-
versity of Technology
1matthias.kulcke@tu-harburg.de 2lorenz@iemar.tuwien.ac.at
The aim of the research presented in this paper is to provide an additional method
and tool for architects and designers as well as students and scholars to analyze
the degree of complexity of a design. Fractal analysis (box counting) e.g. is one
of these methods already used in architecture to measure the degree of complexity
of an architectural design, for example of the elevation of a building. The method
of semi-automated gradient-analysis described here focuses on the repetition of
gradients and thus of proportion-repetition in a given design as one of several
aspects of complexity reduction by redundancy.
Keywords: Gradient-Analysis, Design-Complexity, Redundancy, Spatial
Analysis, Form and Geometry, Proportion
INTRODUCTION
In order to analyze aesthetic quality of architectural
design, the complexity of a given architectural ob-
ject or object arrangement can be measured taking
different designaspects into account. The degree
of complexity detected applying the same methods
and rules to the analysis of two or more objects can
be compared to make verifiable statements about
this complexity in chosen areas of analysis as a part
of the overall aesthetic quality (and/or hypothetical
design-strategy that led to the materialization of the
object).
Based on the assumption that the complexity
of an objects appearance is reduced by redundancy
(Cube, 1965), which can be measured focussing on
different levels of comparison, the gradient-analysis
as described here, attempts to find repetition of pro-
portions in general, regardless of what kind of pro-
portion is repeated. In other words; the number of
similar relations between measurements on different
scales are in focus, not specific relations (e.g. Wag-
ner, 1981). The reduction of complexity is hereby in
itself not a goal, but a means to an end, which may
be a balance between complexity and readability of
form (on this see also Birkhoff, 1933 and Bense, 1971).
To achieve such a balance is the domain of the artist
or design-professional and, through feedback pro-
cesses, of the participating user or customer. There-
fore the presented tool may be used as one from a
greater palette to aid designaspect-analysis. The in-
tegration of such a tool into a responsive CAD sys-
tem combining different methods of design-analysis,
aimed at practicing designers and those learning the
trade alike, is a future possibility.
Shape, Form and Geometry - Concepts - Volume 1 - eCAADe 33 |415
From another angle, looking at the layers of com-
plexity of a design might also be interesting out of
a cognitive perspective, where redundancy plays an
important role e.g. in the cognitive segmentation of
figure and ground, enabling the viewer of an object
to distinguish it from its surroundings (Guski, 1996).
The area of comparison chosen in this research is
the frequency of repetition of gradients, comparing
every significant point with all the other points in a
2D representation of the object one by one, succes-
sively listing pairs of points and their gradients (re-
garding the problem of defining significant lines and
therefore also significant points see: Ostwald and
Vaughan, 2013). The authors do not distinguish be-
tween the relation of points that are connected by
material edges and those which are not. Points are
solely chosen as clearly identifiable references (such
as corners or intersections of lines). The authors as-
sume that Gestalt perception does not necessarily re-
quire edges, merely possible visual connection suf-
fices. Consequently in a first step the relation be-
tween every single point without differentiation is
analyzed. A next step in this research will include the
issue of further perceptual relations. A repetition of a
gradient is interpreted as an indication of complexity-
reduction by redundancy of the objects design as a
whole.
DEVELOPING THE SOFTWARE
The gradient-analyses tool is a makro programmed in
AutoCAD resp. BricsCAD using Visual Basic for Appli-
cations (VBA) and allows for applying the method. It
has been developed in two separate versions (resp.
algorithms), one by each author. This approach al-
lowed questions concerning data format, collection
of significant points, data refinement, mathematical
limitations and other factors of significance for the
software and underlying method to arise quickly and
thus fueled a fruitful discussion.
The gradient-analysis tool attempts to enable
verifiable statements about complexity; by compar-
ing the relation of vertical to horizontal distances
measured inbetween all the significant points of a
given objects representation in 2D one could also say
Figure 1
First analysis of
simple elevations
with successively
increasing
proportion-
complexity (a=red
... most repetitions,
b ... second most
repetitions, c ...
third most
repetitions)
416 |eCAADe 33 - Shape, Form and Geometry - Concepts - Volume 1
the degree of proportion-complexity is measured, i.e.
the number of proportions that are repeated and
how often these proportions are repeated, in this
case, proportions of rectangles that are enclosed by
edges as well as those that are solely defined by two
opposing corner points.
Testcase Selection
For the first experiments a simple and artificially to
the purpose designed façade-like 2D object with ar-
ranged proportions overall and then successively up-
graded entropie has been used (figure 1).
Angles are measured as diagonals of the virtual
reference rectangle from lower left to upper right
corner. They are then represented as a bundle of
lines, in figure 1 underneath the scrutinized objects.
Lower angles than 45° are mirrored over the 45° an-
gle to compare all angles as diagonals of standing
up rectangles. The different colors of the angle-
representations signify the number of repetitions.
Starting with red for the angles repeated most, then
yellow, green, blue for successively lesser numbers
and finally grey, to signify the lowest number of rep-
etitions.
These first gradient-analyses made repetitions
of proportions visible and showed that the chosen
abstraction might be a useful representation of de-
creasing proportion-redundancy in correlation with
increasing complexity of form distances and mea-
surements. In a next step the output was refined
(figure 2); the representation of angles and distances
was separated, an angle-redundancy and distance-
redundancy quotient (also referred to as length quo-
tient), allowing different margins of error (see also
Wiemer and Wetzel, 1994), automatically calculated
and an additional graphic output of detected con-
nections between points within the element under
scrutiny generated.
Figure 2
Succeeding analysis
of simple elevation
using different
margins of error
(a=red ... most
repetitions, b ...
second most
repetitions, c ...
third most
repetitions)
Shape, Form and Geometry - Concepts - Volume 1 - eCAADe 33 |417
In figure 2 the image on the left shows the ob-
ject itself, followed by the representation of angles as
bundles of lines. In contrast to figure 1 the lengths of
the lines are now normalized, since only their orien-
tation is looked at. The third image shows the differ-
ent distances between points, regardless of the ori-
entation of a connecting line, with the same coloring
of repetition as used for angles: Starting with red (in
figures marked as a) for the most repetitions and end-
ing with grey for the smallest number. The last image
finally represents the repetition of angles in the origi-
nal object preserving the length and using the color-
ing of the second image of normalized angles.
The angle-redundancy quotient (Rα) is calcu-
lated by the number of different angles (Cr, r: all rep-
etitions) divided by the total number of angles (C;
see formula 1 and 2), i.e. the number of every pos-
sible connection of points.
Rα=Cr
C(1)
C=(n·
n−1
2)⇔(2)
C=n!·1
k!·(n−k)! (3)
(with k = 2 and n = all single connections)
This is also true for the distance-redundancy
quotient. A tolerance coefficient takes into account
that angles that differ only very little may be per-
ceived as similar and/or that the drawing of the ob-
ject may not be accurate (see also 'Setting the Mar-
gin of Error' presented later in the paper). E.g. for a
tolerance coefficient of + / - 0.1° all angles inside this
range are counted as a repetition of this same angle.
In future work statistic interpretation of the tolerance
coefficient will be given further thought. In this stage
of the research, the current number of test-cases is
too low to provide significance in this regard.
Aims of Result Representation
Since the representations of the analysis have poten-
tially different recipients and user-contexts, allowing
the output to adopt different forms seemed useful.
Which output might be applied in specific cases has
to be decided accordingly.
To enable the design-professional to make an in-
formed decision on possible design-alteration of a
work in progress, a combination of separate angle-
and/or distance-representation with summarizing
numerical output may be sufficient, while a propo-
tion analysis of historic buildings could call for the
additional output of connections (and even distance
-representating circles) within the element, as well
as the listing of all angles and distances in an Excel
sheet. The latter should only be undertaken using
the tool with the precondition that such an analysis is
accompanied by further historical and other contex-
tual information. The problem of individual interpre-
tation and focus on certain parts of data can of course
not be solved by software.
As part of an evaluation process in an evolution-
ary algorithm for propotional optimization, the pro-
vision of data to be used to define a fitness-value
may be wished for, calling for the numerical out-
put in summary, utilizing the angle- and distance-
redundancy quotients.
LIMITATIONS OF UNDERLYING DATA AND
DATA ANALYSIS
In general the system shouldn't be asked to give con-
clusive answers considering design-quality or -value
as a whole. It will not enable its user to verify specula-
tions on specific thoughts of a designer or be of help
proving assumptions about which particular propor-
tion is especially aesthetically pleasing to people in
general. Its purpose is rather to give clues in addi-
tion to other methods of analysis and to generate
hypothesises; e.g. concerning the cognitive effort it
may take to read a design in correlation to its com-
plexity and by which geometric alteration this effort
could possibly be reduced - if this seems necessary or
desirable.
418 |eCAADe 33 - Shape, Form and Geometry - Concepts - Volume 1
Figure 3
Analysis of an
elevation
composed of more
elements, using
different margins of
error (a=red ... most
repetitions, b ...
second most
repetitions, c ...
third most
repetitions)
Shape, Form and Geometry - Concepts - Volume 1 - eCAADe 33 |419
Figure 4
Analysis of an
alteration of the
elevation in figure
3,using different
margins of error
(a=red ... most
repetitions, b ...
second most
repetitions, c ...
third most
repetitions)
420 |eCAADe 33 - Shape, Form and Geometry - Concepts - Volume 1
Setting the Margin of Error
As Wiemer and Wetzel (Wiemer and Wetzel, 1994)
pointed out (and lately Ostwald and Vaughan, 2013),
building CAD-data is often flawed e.g. because of
less than ideal execution or simply because of nec-
essary compromises regarding abstract representa-
tion. This makes it necessary to allow a margin of er-
ror in comparing values extracted from a CAD draw-
ing. As well as the decision about the degree of de-
tail and selection of the elements to be analyzed, the
extent of the margin of error influences the outcome
a great deal. These configurations and the motiva-
tions that led to their chosing should always be made
transparent; using a successive rise of the margin of
error and regarding the resulting output-series as a
whole is therefore advisable.
APPLICATIONOF THE METHOD USING DIF-
FERENT FACADES
For further testing a second artificially to the purpose
designed multi-storey elevation with more openings
has been used (figures 3 and 4). This time also repre-
sentations for window-frames have been added. In
order to represent significant tendencies to repeti-
tion more clearly, in figure 3 and 4 only those colors
representing the highest 10 levels of angle-repetition
are set to visible. As shown in Figure 3 this focuses
three main accumulations, using a tolerance of + / -
2°, around 88°, 64° and 47°; In contrast to that the ex-
ample of figure 4, which displays a different elevation
layout, only shows two main accumulations for the
same tolerance: around 88° and 54°; consequently
the angle-quotient of the example in figure 4 calcu-
lated for same tolerance is higher than that of the ex-
ample in figure 3, which is also true for the other cho-
sen tolerance coefficients in comparison, especially
regarding smaller tolerance-values. The complexity
of the object in figure 4 is at least slightly higher than
in figure 3 regarding the design aspect of proportion.
For the moment we must assume that the number
of accumulations does not necessarily influence the
angle-redundancy quotient, but may constitute an
additional information-layer for complexity-analysis.
Regular elements, like sets of staircases, window-
frame corners and the like, influence the outcome of
the analysis a great deal. These cases, with and with-
out window-frames and/or stairs, should be tested
seperately and included in any deduction based on
a gradient-analysis. In a last series facades from the
House Steiner by Adolf Loos have been tested (fig-
ures 5 and 6).
CONCLUSION AND OUTLOOK
The developed software and its underlying method
of gradient-analysis may be used to make repetitions
of proportions and distances visible in various forms
using different outputs for further processing or di-
rect support of an ongoing design. It remains crucial
to note: the gradient-analysis is in its core not about a
specific proportion (like e.g. the golden section), but
about the repetition (and the number of repetitions)
of any proportion within a design. Certain propor-
tions thus only become significant for analysis (and
possibly cognition by a viewer) because of the num-
ber of repetitions within an object under scrutiny by
the described method - leaving the aspect of certain
special proportions within cultural heritage solely to
the users interpretation of the provided data repre-
sentations.
The described method could be used in educa-
tion as part of a responsive CAD-system, with the
aim of giving feedback on proportional redundancy
overall in a given design, being of assistance if a dif-
ferent degree of entropie is wished for regarding its
measurement-relations.
Furthermore it will be a subject of future re-
search, to use the method as part of genetic algo-
rithms aiming at generating more balanced designs
of building elevations and/or other design objects;
the gradient-analysis could be applied in such a pro-
cess to determine the fitness-value of each successive
parent generation.
Shape, Form and Geometry - Concepts - Volume 1 - eCAADe 33 |421
Figure 5
Analysis House
Steiner, northern
elevation. Architect:
A. Loos (a=red ...
most repetitions, b
... second most
repetitions, c ...
third most
repetitions)
422 |eCAADe 33 - Shape, Form and Geometry - Concepts - Volume 1
Figure 6
Analysis House
Steiner, northern
elevation, outlines.
Architect: A. Loos
(a=red ... most
repetitions, b ...
second most
repetitions, c ...
third most
repetitions)
Shape, Form and Geometry - Concepts - Volume 1 - eCAADe 33 |423
REFERENCES
Bense, M 1971, Zeichen undDesign–SemiotischeÄsthetik,
Agis-Verlag, Baden-Baden
Birkhoff, GD 1933, Aesthetic Measure, Harvard University
Press, Cambridge, Massachusetts
von Cube, F 1965, Kybernetische Grundlagen des Lernens
und Lehrens, Ernst Klett Verlag, Stuttgart
Guski, R 1996, Wahrnehmen - ein Lehrbuch, Kohlhammer,
Stuttgart
Ostwald, MJ and Vaughan, J 2013, 'Representing archi-
tecture for fractal analysis: a framework for identi-
fying significant lines', Architectural Science Review,
56(3), p. 252ff
Wagner, FC 1981, Grundlagen der Gestaltung – plastis-
che und räumliche Gestaltungsmittel, Kohlhammer,
Stuttgart
Wiemer, W and Wetzel, G 1994, 'A report on analysis of
building geometry by computer', Journal of the Soci-
ety of Architectural Historians, 53(4), p. 442, 454
424 |eCAADe 33 - Shape, Form and Geometry - Concepts - Volume 1
