Guest Editor
Journal Paper: Fractal and Fractional.
A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Geometry".

Wolfgang E. Lorenz und Matthias Kulcke
wolfgang.lorenz (at) tuwien.ac.at
www.dap.tuwien.ac.at

Keywords: visual complexity, fractal-like architecture, fractal analysis methods, Gestalt, fractal-based design, fractal-based applications

Deadline for manuscript submissions: 31 March 2023

Special Issue Information.

The Special Issue is aiming for contributions that deal with new methods of fractal based analysis and/or generative design suited to the field of architecture. With architecture in the center of interest, research in related fields, as well as interdisciplinary perspectives on this specific subject, is required, and architects, civil engineers, product designers, mathematicians, computer scientists, and others are invited to this Special Issue.

Since Mandelbrot's definition of fractal geometry, researchers have scrutinized possibilities of its application in the built environment across two general approaches—analysis and generation of architectural form in holistic shape and detail. Fractal methods that are implemented in analysis are box counting and its derivatives, as well as in generation, e.g., L-systems, DLA, and iteration function systems. These methods are vital for the development of (semi-)automated visual intelligence systems which aid in understanding and generating architecture, as well as in supporting complexity management within design tasks.

While seemingly simple techniques utilizing fractal driven generation procure very complex structures and processes, powerful analytic procedures aid the cognition of these underlying principles as they present themselves in highly detailed forms. In a number of fields, fractal-based procedures are recognized and routinely applied, in some cases close to standardized manner, while the professionals in architecture are still yearning for applicable algorithms that suit regular implementation.

Acceptance criteria are state of the art literature reviews, new and transparently explained methods, as well as innovative applications. The rules of the MDPI journal for submissions (including supplementary materials, data and software prototypes) according to www.mdpi.com/journal/fractalfract/instructions.

Journal Paper:
Fractal and Fractional, Vol. 5 (2021), 4; pp 1-25.

Wolfgang E. Lorenz und Matthias Kulcke
wolfgang.lorenz (at) tuwien.ac.at
www.dap.tuwien.ac.at

Keywords: architectural analysis, fractal analysis, visual complexity, box–counting, grasshopper, web application, redundancy, proportion, form and geometry, gradient analysis

Vienna/Hamburg; Austria/Germany; Nov., 2021

• DOI 10.3390/fractalfract5040244

Abstract.

This article contributes to clarifying the questions of whether and how fractal geometry, i.e., some of its main properties, are suitable to characterize architectural designs. This is done in reference to complexity–related aesthetic qualities in architecture, taking advantage of the measurability of one of them; the fractal dimension. Research in this area so far, has focused on 2–dimensional elevation plans. The authors present several methods to be used on a variety of source formats, among them a recent method to analyze pictures taken from buildings, i.e., 2.5–dimensional representations, to discuss the potential that lies within their combination. Color analysis methods will provide further information on the significance of a multilayered production and observation of results in this realm. In this publication results from the box–counting method are combined with a coordinate–based method for analyzing redundancy of proportions and their interrelations as well as the potential to include further layers of comparison are discussed. It presents a new area of boxcounting implementation, a methodologically redesigned gradient analysis and its new algorithm as well as the combination of both. This research shows that in future systems it will be crucial to integrate several strategies to measure balanced aesthetic complexity in architecture.

Talk and Proceeding: 38th eCAADe conference; Anthropologic - Architecture and Fabrication in the cognitive age, (Berlin, Germany, 2020 | virtual conference)

Wolfgang E. Lorenz und Gabriel Wurzer
{wolfgang.lorenz|gabriel.wurzer} (at) tuwien.ac.at
www.dap.tuwien.ac.at/
Vienna; Austria

Keywords: cell phone application, box counting, fractal dimension,
visual complexity, elevation analysis

Sept., 2020

• presentation (video)
• paper
• web application

Abstract.

Fractal analysis helps explaining and understanding architectural quality, e.g., regarding visual complexity described by fractal (box counting) dimension. FRACAM, a cell phone application, uses fractal image analysis methods and takes into account the specific requirements of architectural purposes at the same time. It was developed by the authors to measure the fractal dimension of buildings; more precisely, to measure (color or grayscale) images of (street) views. This paper examines the results of various implemented algorithms for dependencies on camera settings and environmental factors. The main contribution of the authors deals with both an improved differential box counting mechanism applied to color images and a discussion about measurement results concerning influences on the algorithms presented.

Journal Paper, in Applied Mathematics & Information Sciences (AMIS)

Wolfgang E. Lorenz, Jan Andres und Georg Franck
wolfgang.lorenz (at) tuwien.ac.at
www.iemar.tuwien.ac.at/
Vienna; Austria

Keywords: Architectural analysis, design analysis, fractal geometry,
complexity, harmonic proportion, construct/constituent pairing, box-counting.

http://www.naturalspublishing.com/files/published/2z172l3c323t1r.pdf

June, 2017, Vienna & Olomouce

Abstract.

This paper deals with fractal aesthetics and proposes a new fractal analysis method for the perceptual study of architecture. The authors believe in the universality of formulas and aim to complement the architectural description in terms of proportion. Although a well established fractal analysis method to describe the complexity of facades across different scales already exists, box-counting is imprecise because of too many in?uences coming along with the method itself. The authors consider the self-similarity as an important part of aesthetic quality in architecture. This is due to the fact that it describes a concept of consistency that holds everything together from the whole to the smallest detail which refers to the classical meaning of the word symmetry. Hence, a new fractal analysis method is introduced which so far has been applied to quantitative linguistics. Basically, elements of different order, called construct/constituent pairs, are counted and related in a formula. In architecture the pairing consists of likewise elements belonging to different orders, from the overview, the fundamental elements to the smaller details. As a conjecture, some preferable fractal dimensions (from the aesthetical point of view) are proposed for architectural structures.

Talk and Proceeding: CAADRIA; Protocols, Flows and Glitches, (Suzhou, China, 2017)

Gabriel Wurzer und Wolfgang E. Lorenz
{wurzer|lorenz} (at) iemar.tuwien.ac.at
www.iemar.tuwien.ac.at/
Vienna; Austria

Keywords: Fractal analysis; Differential Box-counting; Fractal dimension;
Cell phone application.

March, 2017, Vienna

Abstract.

There are two kinds of algorithms: those that are ‘better’ with respect to accuracy and those that are ‘faster’. In the past, fractal analysis by means of box-counting - including both, binary and greyscale analysis - has been focused on the former. In our work, however, we want to aim at the second category: algorithms that are fast and easy to use, without losing view on significance. To this end we have devised a cell phone application which let users grasp and analyse photographs regarding the box-counting dimension of e.g. facades. The application includes two measurement methods for binary images, based on threshold conversions, and one for greyscale images. Accuracy has been tested on deterministic fractals with known fractal dimension. As a matter of fact we are able to produce what was formerly constraint to scientific implementations or discourse on every day’s hardware.

Talk and Proceeding: CAADence in Architecture; Back to command, (Budapest, Hungary, 2016)

Wolfgang E. Lorenz
wolfgang.lorenz (at) tuwien.ac.at
www.iemar.tuwien.ac.at/
Vienna; Austria

June, 2016, Vienna

Abstract.

Since Benoît Mandelbrot raised the question about the length of Britain’s coastline in 1967, it has become obvious that fractal geometry is appropriate for describing irregular forms. In 1996 Carl Bovill applied box-counting, a fractal analysis method, for the first time to architecture in order to quantify the characteristic visual complexity of facades. This paper presents an approach utilizing fractal analysis to provide another view on Le Corbusier’s architectural composition. Altogether 17 house designs are considered, 14 of them have been built between 1916 (Schwob Villa) and 1928 (Savoye House). Throughout this paper an implementation of the box-counting method written by the author is used. Besides discussing the results, the implementation itself with its advantages and disadvantages is explored.

Talk and Proceeding: The 32nd International Conference on Education and research in Computer Aided Architectural Design in Europe; Fusion, (Newcastle upon Tyne 2014)

Wolfgang E. Lorenz
wolfgang.lorenz (at) tuwien.ac.at
www.iemar.tuwien.ac.at/
Vienna; Austria

September, 2014, Vienna

Abstract.

As evidence from recent years has demonstrated, box-counting provides an objective fractal analytical method to evaluate the visual complexity of architecture. This paper for the first time explores the potential of box-counting with regard to the work of the Viennese architect Adolf Loos (1870-1933). Loos is seen as the pioneer of modern architecture, as someone who anticipated the International Style. This impression derives from his resentments towards the ornament, expressed especially in his texts. However, Loos did not reject ornamentation in general. Thus, the group of smooth plastered facades provides a narrowed view on his overall architectural concept. A more differentiated view on Loos' oeuvre is not new; however, the author further develops the possibilities of describing facades geometrically by using an implementation of the fractal analytical method, especially created for facades. This paper not only focuses on the possibility of grouping facades with similar characteristic values, but considers other aspects of Loos' design such as space as well.

Fraktalähnliche Architektur - Einteilung und Messbarkeit: Ein Programm in VBA für AutoCAD

Wolfgang E. Lorenz
wolfgang.lorenz (at) tuwien.ac.at
www.iemar.tuwien.ac.at/
Vienna; Austria

Begutachter/in(nen): G. Franck, H. Pottmann; E259.1 Institut für Architekturwissenschaften; Digitale Architektur und Raumplanung, 2014; Rigorosum: 14.01.2014.

also see: Publication Database of the TU Vienna

January, 2014, Vienna

Abstract.

This thesis investigates to what extent the fractal geometry is suitable for characterizing architecture. It describes the benefits of such an approach as opposed to a description by means of Euclidean geometry (which we have been familiar with for more than two thousand years). In this context, buildings of different epochs are examined to determine whether the characteristics of fractals occur and subject to which restrictions such a point of view is possible at all. The focus lies on the International Modern (International Style) that is in most cases called 'smooth` and a fractal-like organic architecture. The final analysis of facades is carried out by means of the so-called box-counting method. This method has been implemented by the author in a CAD program. The studies indicate a possible categorization of architecture on the basis of fractal geometry. In addition to the box-counting dimension this comprises a definite range of scales and the coefficient of determination.

Talk and Proceeding: The 31st International Conference on Education and research in Computer Aided Architectural Design in Europe; Computation and Performance, (Delft 2013)

Wolfgang E. Lorenz
wolfgang.lorenz (at) tuwien.ac.at
www.iemar.tuwien.ac.at/
Vienna; Austria

September, 2013, Vienna

Abstract.

When Benoît Mandelbrot raised the question about the length of Britain’s coastline in 1967, this was a major step towards formulating the theory of fractals, which also led to a new understanding of irregularity in nature. Since then it has become obvious that fractal geometry is more appropriate for describing complex forms than traditional Euclidean geometry (not only with regard to natural systems but also in architecture). This paper provides another view on architectural composition, following the utilization of fractal analysis. The procedure concerning the exploration of a façade design is demonstrated step by step on the Roman temple front of the Pantheon by Appolodorus and its re-interpretation – in the particular case the entrance front of Il Redentore, a Renaissance church by Palladio. Their level of complexity and range of scales that offer coherence are visualized by the specific measurement method of boxcounting.

Talk and Proceeding: The 30th International Conference on Education and research in Computer Aided Architectural Design in Europe; Digital Physicality | Physical Digitality, (Prague 2012)

Wolfgang E. Lorenz
wolfgang.lorenz (at) tuwien.ac.at
www.iemar.tuwien.ac.at/
Vienna; Austria

Keywords: Box-Counting Method; Range Analysis; Hurst-Exponent; Analyzing Architecture; Scalebound and Scaling objects.

September, 2012, Vienna

Abstract.

The concept of describing and analyzing architecture from a fractal point of view, on which this paper is based, can be traced back to Benoît Mandelbrot (1981) and Carl Bovill (1996) to a considerable extent. In particular, this includes the distinction between scalebound (offering a limited number of characteristic elements) and scaling objects (offering many characteristic elements of scale) made by B. Mandelbrot (1981). In the fi rst place such a differentiation is based upon a visual description. This paper explores the possibility of assistance by two measurement methods, fi rst time introduced to architecture by C. Bovill (1996). While the box-counting method measures or more precisely estimates the box-counting dimension D b of objects (e.g. facades), range analysis examines the rhythm of a design. As CAD programs are familiar to architects during design processes, the author implemented both methods in AutoCAD using the scripting language VBA. First measurements indicate promising results for indicating the distinction between what B. Mandelbrot called scalebound and scaling buildings.

in "Biomimetics -- Materials, Structures and Processes: Examples, Ideas and Case Studies"; Gruber, P.; Bruckner, D.; Hellmich, C.; Schmiedmayer, H.-B.; Stachelberger, H.; Gebeshuber, I.C. (Eds.); Springer; (2011); S.179-200; herausgegeben von BMVIT

Wolfgang E. Lorenz
wolfgang.lorenz (at) tuwien.ac.at
www.iemar.tuwien.ac.at/
Vienna; Austria

June, 2011, Vienna

Abstract.

In Fractals smaller parts and the whole are linked together. Fractals are self-similar, as those parts are, at least approximately, scaled-down copies of the rough whole. In architecture, such a concept has also been known for a long time. Not only architects of the twentieth century called for an overall idea that is mirrored in every single detail, but also Gothic cathedrals and Indian temples offer self-similarity. This study mainly focuses upon the question whether this concept of self-similarity makes architecture with fractal properties more diverse and interesting than Euclidean Modern architecture. The first part gives an introduction and explains Fractal properties in various natural and architectural objects, presenting the underlying structure by computer programmed renderings. In this connection, differences between the fractal, architectural concept and true, mathematical Fractals are worked out to become aware of limits. This is the basis for dealing with the problem whether fractal-like architecture, particularly facades, can be measured so that different designs can be compared with each other under the aspect of fractal properties. Finally the usability of the Box-Counting Method, an easy-to-use measurement method of Fractal Dimension is analyzed with regard to architecture.

Talk and Proceeding: 27th eCAADe CONFERENCE; Computation: The new Realm of Architectural Design, (Istanbul 2009)

Wolfgang E. Lorenz
wolfgang.lorenz (at) tuwien.ac.at
www.iemar.tuwien.ac.at/
Vienna; Austria

Keywords: Fractal Architecture; Box-Counting Dimensions of Façades; Visual Perception; Implementation in a CAD-software.

October, 2009, Vienna

Abstract.

The author describes the basic principles for measuring architecture from the point of view of Fractal Geometry outlining the principle connections between Fractal Geometry and architecture, giving some examples and explaining the Box-Counting Method, which is an easily manageable method that can be applied to elevations. The paper not only deals with problems arising from using the Box-Counting Method but also with its relation to visual perception. It shows how the Box-Counting Dimension DB of façades can be measured with the help of a software program that was written by the author and has been implemented into AutoCAD. Finally, results of different configurations are given for the Koch curve and Robie House by Frank Lloyd Wright, showing the accuracy of this measurement method.

in "Bionik: Innovation & Qualifikation", (2010); S.92; herausgegeben von BMVIT

Wolfgang E. Lorenz
wolfgang.lorenz (at) tuwien.ac.at
www.iemar.tuwien.ac.at/
Vienna; Austria

December 7th, 2008, Vienna

Initial Point – Fractional Architecture.

Wenn von fraktaler Architektur gesprochen wird, so sind bestimmte, den Fraktalen inhärente Eigenschaften gemeint, die sich in eingeschränkter Form auch an Gebäuden erschließen. Die uns umgebende Natur besteht nicht aus glatten Elementen, wie sie die euklidische Geometrie kennt, sondern aus unterschiedlich stark zerklüfteten Objekten. In ähnlicher Form sind Gebäude auf Grund verschiedener Aus- und Einschnitte, Überlagerung von Bauteilen aber auch bedingt durch Materialoberflächen nicht glatt, sie weisen vielmehr unterschiedliche Komplexität auf. ...

Talk and Proceeding: First International Conference on Fractal Foundations for 21st Century Architecture and Environmental Design, Madrid; 25.03.2004 - 27.03.2004; in: "First International Conference on Fractal Foundor 21st Century Architecture and Environmental Designations f", (Madrid 2004)

Wolfgang E. Lorenz
wolfgang.lorenz (at) tuwien.ac.at
www.iemar.tuwien.ac.at/
Vienna; Austria

January 13th, 2004, Vienna

Abstract.

My paper deals with the question if there is a connection between architectonic quality and characteristics of architectonic form which can be described by fractal geometry – taking up and continuing Carl Bovill’s ideas on the topic.
This will be investigated thoroughly in two respects. First the existence of fractal characteristics (self-similarity, ruggedness and iteration) is  studied in works of architecture that are commonly regarded as outstanding examples such as buildings by Frank Lloyd Wright, Bruce Goff, Gerrit Rietveld or Steven Holl. In the course of that process, various distances and levels of scale are examined.
The second approach is based on measuring the fractal dimension of architectural drawings by using the ‘box-counting method’ on different levels of scale. Factors influencing the different parameters of this method of measurement are studied.
The hypothesis is put forward that a connection between the fractal characteristics of form and quality exists, which determines architectural rank. The paper also deals with the difficulties of verifying such a hypothesis and if esthetical quality can be described by formal methods at all.

Talk and Poster: archdiploma 2003, Kunsthalle Project Space, Wien; 07.10.2003; in: "archdiploma 2003", (Vienna 2003), S. 94 - 95

Wolfgang E. Lorenz
wolfgang.lorenz (at) tuwien.ac.at
www.iemar.tuwien.ac.at/
Vienna; Austria

February, 2003, Vienna

Fractals and Fractal Architecture.

Euclidean geometry with its perfect "clinical" shapes of cones, pyramids, cubes and spheres, is not the best way to describe natural objects. Fractal geometry, as opposed to Euclidean, offers better methods for description or for producing similar nature-like objects. The language expressing it is called algorithms. Complex objects can be reduced to simpler formulas or transformation rules. The "new" geometry may help us to understand and analyse complexity that can be found in medieval towns but also in cathedrals and other contemporary man-made objects. It may also help us to transfer this complexity, which also arises from the development over time, to newly planned cities and buildings.

Fractal attributes can be found anywhere from coastlines to clouds, mountains, trees, plants, ... and in architecture. Fractals are used as a tool in many fields ranging from medicine to economy.

Master Thesis

Wolfgang E. Lorenz
wolfgang.lorenz (at) tuwien.ac.at
master-thesis
Vienna; Austria

Betreuer: G. Franck; E259-1, 2003

"... Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth ..." [Mandelbrot Benoit B., Dr. Zähle Ulrich (editor of the german edition), Die fraktale Geometrie der Natur (1991), Birkhäuser Verlag Berlin, p.13.]

January, 2003, Vienna

Introduction

This quotation by Mandelbrot shows that the Euclidean geometry - the perfect “clinical” shapes of cones, pyramids, cubes and spheres - is not the best way to describe natural objects. Clouds, mountains, coastlines and bark are all in contrast to Euclidean figures not smooth but rugged and they offer the same irregularity in smaller scales, which are some important characteristics of fractals - see chapter “2.2 Characteristics”. As the following pages indicate, fractal geometry, in opposition to Euclidean geometry, offers better methods for description or for producing similar natural-like objects respectively. The language in which it is expressed is called “algorithms”, by which complex objects like a fern or a cloud can be reduced to simpler formulas or transformation rules respectively. Fractals can be found everywhere from coastlines, border-lines and other natural rough lines to clouds, mountains, trees, plants, ... and maybe also in architecture. The following chapters explain what a fractal is in general and how fractals can be used for architectural analysis and in the stage of planning. Fractals are used as a helping tool for explanation in many fields ranging from medicine to economy. From this point of view fractals should not be excluded from architecture.

master thesis