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Urban Street Space Analysis with Spherical Box-

Counting

Holistic digital Gestalt analysis of architecture in urban space

Matthias Kulcke 1 and Wolfgang E. Lorenz 2

1 Hamburg University of Technology 2 TU Wien

1 matthias@kulcke.de 2 wolfgang.lorenz@tuwien.ac.at

Spherical box-counting of urban street spaces is a novel method developed and refined by

the authors to produce highly specific topological fractal fingerprinting of architecture in

relation to observer position and in the context of the accompanying surroundings. The

use of 360-degree spherical panoramas as input data and basis for fractal measurement

lies at the center of this method.

A holistic approach toward architectural and urban design, balancing between simplicity

and complexity of all Gestalt qualities, needs to take the influence of every (especially

man-made) object in view into consideration.

This research shows that Gestalt complexity is linked to the observer’s viewpoint as well

as the Gestalt complexity of all objects visible from the viewer’s position. This is another

decisive step toward holistic fractal and overall digital Gestalt analysis of urban spaces.

Keywords: Gestalt Analysis, Fractal Analysis, Box-Counting, Spherical Box-Counting

INTRODUCTION

Spherical box-counting is a recent new method

developed by the authors for fractal analysis of

urban street spaces as holistic environments

(Kulcke and Lorenz 2023), which has been

significantly widened in scope as part of the

research presented in this article.

By applying this new method, the paper

scrutinizes the influence of fractal complexity (see

e.g., Katona 2023 and Ostwald et al. 2016) of

objects near ground level observed from specific

viewpoints for the first time through a

comparison of the overall box-counting measures

of exemplary model street spaces (including e.g.,

street furniture, sidewalks, and vehicles). This

method belongs to the subject of fractal analysis

in architecture and to the field of digital analysis

of Gestalt quality in architectural design.

Aesthetic Quality Measures

Digital analysis of aesthetic quality serves to

quantify objectifiable aspects of Gestalt.

Proportions, for example, as a repetition of

dimensions of building parts (Symmetria

according to Vitruvius) as well as the repetition of

quotients of dimensions (Eurythmia according to

Vitruvius) are in part objectifiable. It can be

measured if such repetitions exist at all and how

many different instances of measurements and

measurement relations are present in one object.

It remains subjective which single proportion as a

measurement relation has significant receptive

qualities or distinct impact on observers in

comparison to others (see Kulcke et al. 2015). To

integrate the objectifiable aspects into

architectural design processes, the authors have

Volume 2 – Data-Driven Intelligence – eCAADe 42 | 567

developed and refined grid and gradient analyses

(see Kulcke et al. 2016, Kulcke 2019).

Quantitative aspects of aesthetic qualities can

be measured on several planes for analyses and

should be viewed in this layered context to bear

significance, since complexity on the layer of

proportion may be countered with simplicity on

another layer of self-similarity, e.g., as indicated

by box-counting. Following this insight, the

authors have developed multi-layered Gestalt

analysis approaches (Lorenz et al. 2021) and

discussed contexts of their application in their aim

to integrate digital analyses of proportion,

aspects of color, as well as sculptural

consequence into the design process.

Fractal Analysis

This paper is a further contribution to the

underlying hypothesis that architectural quality

and architectural features can be linked to fractal

features such as roughness, complexity, self-

similarity, and scale invariance, not least through

aesthetic rules associated with fractal theory

(Mandelbrot 1981, Mandelbrot 1982, Frank et al.

2008, Salingaros 2006). Fractal analysis and box-

counting in architecture have emerged based on

Benoît Mandelbrot (Mandelbrot 1975 and

Mandelbrot 1982) since 1996, when Carl Bovill

demonstrated possible architectural-specific

applications for the first time (Bovill 1996).

Spherical box-counting has been developed

by the authors to deal with the issue of the

deliberate choice of picture frame, to make

observer position a central aspect of analysis and

to take the aesthetic complexity of urban context

directly into account.

The spherical image as a basis for box-

counting

In the past, box-counting in architecture was

applied solely to 2D imagery based on

architectural plans (mostly elevations, but also

floor plans and sections). Later on, steps toward

images that reflect on 3D qualities were taken,

e.g., by resorting to photographic material that

corresponded to straight line vanishing point

perspective architectural representation using

advanced measuring techniques within the

FRACAM framework (Lorenz et al. 2020). FRACAM

is a web application that calculates the box-

counting dimension (𝐷) of black and white,

grayscale and color images, e.g., taken with a cell

phone camera (where 𝐷 is equivalent to the

fractal dimension, which is the ratio between the

change in detail, and the respective scale; see

figure 1). In addition to the value of 𝐷, an image

is then characterized by the range in which a clear

dependency between detail change and scale

exists (as a straight line in the double-logarithmic

graph). FRACAM may also be used to calculate

the box-counting dimension of spherical imagery.

Although this approach opened up new

possibilities in fractal analysis in architecture,

several problems remained, like the deliberate

choice of picture boundaries for the imagery to

undergo box-counting analysis.

Figure 1

FRACAM:

measuring

buildings in their

environment.

On the right in

each case: double

logarithmic graph

with detail versus

scale (RGB

calculations).

Top: Frank Lloyd

Wright, Robie

House, Chicago’s

Hyde Park

(𝐷 2.873).

Bottom: Santiago

Calatrava,

Milwaukee Art

Museum

(𝐷 2.474).

568 | eCAADe 42 – Volume 2 – Data-Driven Intelligence

Especially this problem of deliberate picture

frame choice was tackled by the new approach of

using 360° spherical panoramas for evaluating

architecture in urban space. Thus, integrating the

natural as well as the man-made environment

into box-counting analysis became a lot easier,

since the question of how much of the

surroundings of an architecture is to be included

in the picture frame was no longer an issue.

Thus, spherical box-counting has been

introduced as a special method in box-counting

analysis (Kulcke et al. 2023).

MEASUREMENT METHOD

The term fractal was coined in the year 1975 by

Benoît Mandelbrot (Mandelbrot 1975) to describe

none-smooth and irregular curves with seemingly

strange or contradictory properties (such as

infinite length within a finite space). What is

nowadays well known as fractal geometry also

maps the irregularity of nature – including

mountains, coastlines, and clouds; and in the form

of fractal fluctuations indicating a healthy human

heartbeat – better than Euclidean geometry ever

could (Mandelbrot 1982 and Peitgen et al. 1992).

In urban and architectural science, researchers

have used fractal geometry or the analytical

methods derived from it to describe, for example,

urban growth (Batty et al. 1994) or to provide

assistance in the aesthetic analysis of buildings

(Lorenz et al. 2017).

Just as Mandelbrot explained that nature is

not smooth ("Clouds are not spheres, mountains

are not cones, coastlines are not circles, and bark is

not smooth, nor does lightning travel in a straight

line.", (Mandelbrot 1982)), architecture in a certain

sense cannot be regarded as smooth either. A

façade consists of openings, design elements

such as window sills and cornices, but also

constructive necessities like window frames; all of

which bring a roughness to the façade that leads

to a measurable fractal characteristic such as

fractal dimension, at least on a limited scale range.

Urban space as well is not perfectly smooth and

thus sufficiently describable by Euclidean

geometry. For example, due to changes in heights

(e.g., as a difference in level between the street

and sidewalks) or the profile of the built

environment (e.g., protruding and retracted

buildings or building parts, like bay windows), and

fixtures such as street lamps give the street space

as a whole a characteristic measurable overall

detectible roughness.

Regardless of whether nature, cities,

streetscapes, or façades are under scrutiny, the

relation of all of the above to fractal geometry

may be grasped and measured as irregularity

across several scale levels. The characteristic

element within irregularity arises from fractals'

inherent (statistical) self-similarity, which emerges

as a specific degree of complexity. From this

perspective, buildings are irregular shapes that do

not possess smooth surfaces with some designs

offering a relationship to fractal geometry

displayed in large and small cut-outs, protrusions,

and retractions on different scales (see also Bovill

1996, Mandelbrot 1982, Vaughan et al. 2010). If a

similar characterization of irregularity across a

range of scales exists, this is indicated by a strong

correlation between change in details and

respective scale. Continuity across scales means

that the degree of irregularity of a building under

scrutiny observed from a greater distance

corresponds to that of a smaller section, which

may be regarded as a consequence of at least

statistical self-similarity. This is an important

property of fractals. It is one significant layer of

architectural quality in which the whole is

reflected even in the smallest details – if not

exactly, then at least statistically as bearing same

visual characteristic – which promotes one central

aspect of harmony (see as an example Robie

House (Hoffmann 1984); for more detail see

Lorenz et al. 2017, Lorenz et al. 2021). In addition,

self-similarity on different scales corresponds to

phenomena of human visual cognition, i.e. object

detection is adjusted to human scale (Salingaros

2006) as a whole and in the increase of visibility

Volume 2 – Data-Driven Intelligence – eCAADe 42 | 569

𝑛𝑖,

𝑗

𝑐𝑒𝑖𝑙𝑙𝑘1

𝑠

𝑖𝑓 𝑙𝑘

1 𝑖𝑓 𝑙𝑘

(2)

𝐷𝑙𝑖𝑚

→𝑙𝑜𝑔𝑁

𝑙𝑜𝑔

1

𝑠

(1)

𝑛𝑖,

𝑗

𝑐𝑒𝑖𝑙𝑙𝑘1

𝑠

∙𝑆,

𝑚∙𝑛

(3)

of detail as the observer approaches. The degree

of irregularity (also how much space is occupied

by a contour) is expressed by its fractal dimension

(which exceeds the topological dimension

(Mandelbrot 1982 and Batty et al. 1994)). Its value

increases as does the irregularity, whereby the

fractal dimension of a fractal curve lies between

one (a straight line) and two (a surface),

depending on how much of the surface is filled.

This is where the fractal-based method of box-

counting comes into play.

Like Bovill (1996) and other researchers who

followed him thereafter (Ostwald et al. 2016, Capo

2004, Md Rian et al. 2007) have shown, roughness

or irregularities on façades can be described by

the fractal dimension, which in turn can be easily

calculated by using the box-counting method.

Basics of the box-counting analysis

The box-counting method has found its way into

many areas (including medicine (Zebari et al.

2021), image compression (Huang et al. 2022),

architecture (Bovill 1996, Ostwald 2016), and

urban studies (Batty et al. 1994, Cooper et al.

2013)), and has continued to be developed and

sometimes adapted to the different requirements

of several disciplines. One of the latter is

architecture and the architecture-specific

extensions of the calculation method, which is

usually based on pixel graphics but also bears the

possibility of dealing with a vector-based input as

a calculation directly based on the geometrical

data provided in a CAAD program (Lorenz, 2012).

With the standard box-counting method, a

grid with a specific box size is placed over a black-

and-white raster graphic. Thereby, the number of

boxes in a row defines the scale of the grid,

whereby four boxes have proven to be a useful

starting size (Foroutan-pour et al. 1999). Now all

the boxes that contain black pixels are counted,

i.e. the boxes that cover the image. In the next

step, the scale of the grid is reduced (usually by

one half) and the covering boxes are again

counted. The box-counting dimension, which

corresponds to the fractal dimension (Falconer

2003), for a certain scale range finally results from

the slope of the replacement line in a double-

logarithmic graph with grid size versus covering

boxes:

with s being the side length of a single box that

defines the scale of the grid. The deviations of the

data points from the replacement line in turn

show graphically how strongly the two axis values

depend on each other.

Since this standard box-counting method

only examines graphics in black-and-white, it is

not suitable for grayscale images. In order to be

able to measure grayscale images as well, the so-

called differential box-counting method is used

(Sarker et al. 1994). With this method, the

respective grayscale value of a certain pixel is

interpreted as a z-coordinate. Instead of placing a

grid over the two-dimensional image, stacks of

cubes of the size 𝑠𝑠𝑠 – with 𝑠 being the base

size and 𝑠 the height – are used where each stack

at position 𝑖,𝑗 includes a minimum (darkest)

grayscale value 𝑘 and a maximum (lightest)

grayscale value 𝑙. For a squared image the

difference 𝑛𝑖,𝑗 for a stack at 𝑖,𝑗 and the

scale 𝜀 is then determined as (Lee et al. 2014):

(see figure 2). Extended to rectangular images, the

formula is modified as follows:

with 𝑠 being the height of a single box (calculated

as the total number of grayscale levels 𝐺 divided

by the scale of the grid), 𝑚,𝑛 the base size of

the standard stack in x- and y-direction, and 𝑆,

the base area of the stack 𝑖,𝑗 (Long et al.).

570 | eCAADe 42 – Volume 2 – Data-Driven Intelligence

𝑎𝑟𝑐𝑐𝑜𝑠𝑝

𝑎𝑟𝑐𝑐𝑜𝑠𝑐𝑜𝑠2𝛼

1𝑠𝑖𝑛2𝛼∙𝑐𝑜𝑠2𝜑

(4)

For every grid scale (size of cubes) the total

number of differences is calculated, and similar to

the standard method, the regression lines

determine the fractal dimension for a certain scale

range (for a more detailed description of the

method, its improvements, such as shifted

stacked boxes, and the concrete implementation

as a web application see Lorenz et al. 2020).

Spherical box-counting

Spherical box-counting (Kulcke et al. 2023) is a

recent new method developed by the authors for

fractal analysis of urban street spaces as holistic

environments which has been significantly

widened in scope as part of the research

presented in this article. By applying this new

method, the paper scrutinizes the influence of

fractal complexity (see e.g. Katona 2023 and

Ostwald et al. 2016) of objects on street level for

the first time through a comparison of the overall

box-counting measures achieved by spherical

box-counting of exemplary model street spaces.

For this purpose, the complexity of the space

between the buildings of the model test

cases – i.e., the number of constructive edges of

sidewalks, streets, and outside objects – is

increased successively and measured from several

observer positions.

The basis for spherical box-counting is

provided by 360° panoramic imagery based on

the spherical grid described by the formula (4)

(compare Kulcke 2019):

with 𝛼 being the angle of deviation from the

viewing angle axis in vertical and 𝜑 in horizontal

direction for each point constituting all horizontal

lines of a grid projected onto a sphere to be rolled

out torus-like as well as cylindrically, and with the

radiant 𝑎𝑟𝑐𝑐𝑜𝑠𝑝 as the thus spherically projected

y-coordinate of each point. The special aspects of

spherical box-counting of urban street space are

described in the following paragraphs.

Spherical box-counting of urban street

space

A holistic approach toward architectural and

urban design, balancing between simplicity and

complexity (see e.g., Birkhoff 1933) of all Gestalt

qualities, needs to take the influence of every

(especially manmade) object in view into

consideration.

The analyses conducted as part of this

research consecutively changed the complexity of

the objects (in number, arrangement, and

proportion of edges, openings, etc.) within one

series while maintaining the view-point. This was

followed by the comparison of different observer

position series (see figure 3).

A PARAMETRIC STREET SPACE MODEL

The parametric model street space constructed in

Rhinoceros®/Grasshopper® allows for a

parametric manipulation of the building façades,

the design of the street space in terms of

sidewalks, street surface additions like pavement

Figure 2

Red colored (8x8

Pixels) stack of

“boxes” on a

grayscale image

with (k) being

the vertical box

index containing

the darkest

grayscale value

and (l)

that of the

lightest.

Volume 2 – Data-Driven Intelligence – eCAADe 42 | 571

markings, and parking spaces including cars, as

well as the addition of further elements of urban

space design like urban furniture.

Façade specifications

The façade of each building within the entire row

of houses facing the street space may be

manipulated in terms of height, width, attica,

cornices, façade openings with several underlying

grids, number and height of floors, number of

openings, the proportion of openings, and a

separate design of the street level area (see

figure 4 top). Roof parameters can be chosen by

type and slope (from three to eighty degrees),

ridge orientation, and height. The number of and

the height variation of floors, as well as variations

in the proportion of the openings, have not been

scrutinized in the conducted experiments. The

model provides the basis for undertaking such

measures in future research.

Street space specifications

The street space is especially characterized by

variations of sidewalks and curbs. Parking spaces

may be arranged in different manners, e.g., they

may be running parallel, flanking the street, or at

an angle to the street direction. In addition to

these installments, street lamps and elements of

urban furniture can be added e.g., to mask

irregularities along street edges to guide spatial

view toward the vista, which can improve urban

visibility (Oppong et al. 2016) (see figure 4 center).

Vehicle specifications

The degree of measurable complexity is

influenced by a variety of factors regarding

vehicles in street space, i.e. color, individual form

according to brand and design, orientation and

number of vehicles present, influenced by day-

time and available parking space.

The consistent degree of abstraction is

decisive for the model to work and to produce

meaningful measures regarding the fractal

dimension. This goes especially for the

representation of cars. It is important to note that

color, shadow, and brand specificity are not

integrated into the model following the degree of

abstraction observed in coordination with the

architectural façades, streets, and curbs, as well as

the urban furniture. To address the aspect of

complexity through car typology, four different

types of vehicle models (small, standard, family

car, and transporter) simplified according to the

overall degree of image abstraction are

disseminated randomly in each image rendering

(see figure 4 bottom).

Figure 3

Viewer position

and position-

specific spherical

perspective

(background:

straight line

vanishing point

perspective; on

top left: spherical

perspective)

Figure 4

Increasing street

space complexity

(from top to

bottom)

572 | eCAADe 42 – Volume 2 – Data-Driven Intelligence

D

B

R²

Simple façade 2.468 0.999731

+ openings 2.505 0.999753

+ detail 2.621 0.999977

D

B

R²

Street view 2.413 0.999749

+ pavement 2.443 0.999719

+ street furniture 2.449 0.999744

+ cars 2.468 0.999731

D

B

R²

original pos 2.413 0.999749

changed pos A 2.403 0.999715

+ pavement 2.439 0.999631

+ street furniture 2.445 0.999652

+ cars 2.459 0.999734

changed pos B 2.353 0.999863

+ pavement 2.373 0.999754

+ street furniture 2.371 0.999789

+ cars 2.407 0.999783

Digital workflow

For the generation of each model, graphic macro

programming in Rhinoceros®/Grasshopper®

was used. After instantiation, the generated

model was exported as an .fbx-file and imported

into Blender® for the rendering of 1024 x 2048px

360-degree panoramic pictures (see figure 5). For

this second step Archicad® may also be used.

MEASUREMENT

The fractal analysis engines used for the spherical

box-counting method presented here are

available at http://128.130.52.15/processviz/

fracam/ (see figure 6). Of the three main

methods – the standard box-counting method

(black and white), the improved differential box-

counting method (grayscale), and the improved

differential box-counting method (color

RGB) – the first one was used in this paper as it

proved to be correspondent to the level of

abstraction of the digital model and image.

RESULTS

The level of abstraction that each model and

image thereof is subjected to has a crucial

influence on measured results when applying the

spherical box-counting method.

1. The comparison of the same façades with

increasing complexity of objects and

edges defining the built street space

shows that the fractal analysis of the

spherical image reveals this as an overall

increase in complexity (table 1).

2. The comparison of increasingly complex

façades with fixed complexity of objects

and edges defining the built street space

shows that the fractal analysis of the

spherical image reveals this as an overall

increase in complexity (table 2).

3. Points 1. and 2. show that the overall

visual complexity may be influenced by

street space design as well as the

building design (see figure 7).

4. Changes in observer position produce

different results when applying the

spherical box-counting method for the

same model complexity (table 3).

Figure 5

Digital workflow

Figure 6

Image section of

the web

application

Table 1

Standard street

view (position in

the middle of the

street) with

increasing street

space complexity

Table 2

Calculations of

standard street

view (position in

the middle of the

street) with

increasing façade

complexity for

most complex

street space

Table 3

Calculations of

standard street

view (position in

the middle of the

street) with

increasing façade

complexity for

most complex

street space

Volume 2 – Data-Driven Intelligence – eCAADe 42 | 573

5. Each measure in a certain observer

position produces a topological

fingerprint in regards to an observer

position dependent complexity measure.

DISCUSSION

There appears to be a threshold for the influence

of the complexity of the façades on the overall

complexity, which, if reached, lessens the impact

of the complexity of street space objects as

measurable through the fractal dimension.

This shows that the method must be regarded

and applied as part of a larger tool-set to judge

Gestalt quality of architecture in street space. So,

while providing a measure that objectifies, if used

in a sensible manner, statements on the visual

complexity of urban spaces, its integration in

design processes calls for comparable imagery

and/or modelling in regards to rules of

abstraction and object inclusion.

Nonetheless the overall design complexity of

the urban street space, including all constructive

edges as well as all surfaces visible from a specific

observer position, can be measured using

spherical box-counting.

It is feasible to use the presented method as

a basis to also scrutinize panorama-based

imagery that is restricted by a defined central

orientation of the observer's viewing angle

(monocular roughly 120°); the latter then serves

as a boundary to restrict the image submitted as

the basis for a box-counting analysis. To do this in

a sensible way viewer orientation must be

supported by argument (special aesthetic

attractions in the urban space) and quantitative

research (e.g., eye-tracking and quantitative

analysis thereof).

In regards to the spherical imagery (as is the

case for any imagery used for box-counting

analysis), it is crucial for the method to observe a

consistent degree of abstraction, i.e., the relation

between map and territory. Increasingly difficult

questions of logical typing arise as more

dimensions of visual complexity are added to the

360° panoramic imagery serving as the basis for

spherical box-counting. For example, if light and

structure are taken into account, are the shadows

cast by architecture and objects on the same

logical level? How about reflections on glass and

metal? These questions are desiderata for future

research.

OUTLOOK

This research shows that urban Gestalt complexity

can be linked to the observer’s viewpoint as well

as to the Gestalt complexities of all objects visible

from the viewer’s position (if free observer

orientation is permitted).

To increase the visibility of complexity

changes in the model via the spherical box-

counting method, the use of several grid sizes in

one image is a viable option to be examined.

Figure 7

Increasing façade

complexit

y

574 | eCAADe 42 – Volume 2 – Data-Driven Intelligence

While the influence of detailed alignment and

specific proportion correlation between overall

façade measurements, openings, and street space

objects on the fractal measure needs further

study, it is safe to say that spherical box-counting

of urban street space models and imagery is

another decisive step toward holistic fractal and

overall digitally enhanced Gestalt analysis (Lorenz

et al. 2021) of urban spaces.

This method is suitable for integration into

complex digital algorithms and so-called AI

systems optimized for urban space analyses (see

e.g. Yonder et al. 2023) but can be used as well in

combined analog and digital processes in urban

design. It paves the way for the use of fractal

Gestalt measures as a guideline for urban designs

that are similar in their simplicity and complexity

characteristics while potentially widening the

solution space of individual design decisions on

the part of architectural and urban planners who

still face extensive and detailed restrictions

through contemporary building regulations.

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