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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