Fractal Aesthetics in Architecture

Journal paper, in Applied Mathematics & Information Sciences. (article)

Wolfgang E. Lorenz, Jan Andres und Georg Franck. 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. ...

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SimAUD 2017

A Building Database for Simulations Requiring Schemata. (book)

Gabriel Wurzer, Jelena Djordjic, Wolfgang E. Lorenz und Vahid Poursaeed.
Obtaining spatial representations of existing buildings for use in simulation is challenging: To begin with, getting permission to access submitted construction plans can take a long time.. ...

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Steuerberater Kanzlei

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V Fractals and Architecture

If architecture stands for continuing the development from the protecting caves over the fallen down tree as a first shelter to buildings made of timber or stones and up to modern interpretations of nature like Frank Lloyd Wright’s examples, then architecture, natural materials, time and the structure of nature may still be an unity. In this way, nature, as we have seen in the sections before, offers characteristics of fractal geometry rather than such of Euclidean geometry. So maybe because of similarities between nature and architecture, with regard to material and structure, some of the fractal attributes can also be found in buildings, in their elevations and ground floors. Thinking of self-similarity for example this is not a new aspect in architecture, as there are similar forms on different scales e.g. in the Gothic style. But also Frank Lloyd Wright used variations of form on different scales as a concept of his buildings and he did not copy nature as it is offered in trees, but was looking for the underlying structure of their organization[01]:

"Quite a different form may serve for another, but from one basic idea all the formal elements of design are in each case derived and held well together in scale and character[02].

The true mathematical fractals offer such ideas where complex forms have simple underlying algorithms.

5.1 Visual Perception and Fractal Range

Architectural composition is concerned with the progression of interesting forms from the distant view of the facade to the intimate details. As one approaches and enters a building, there should be another smaller scale, interesting detail that expresses the overall intent of the composition, which is the fractal conception. Thus fractal geometry is the formal study of this progression of self-similar detail from large to small scales.

5.2 Topology

In mathematics there is one branch that analyzes form, which is called topology - in architecture we also know this branch of topology that deals with forms. Topology, which is also-called the geometry of the position, for example says that all pots with two handles have the same form, because under the presumption of infinite flexibility and compression they can be formed continually into each other without producing new holes or closing them. It also says that all coastlines have the same form because they are topologically identical with a circle - both have a topological dimension of one. Adding islands off the coastline only means that the new object is topologically identical with many circles.

Benoit Mandelbrot looked for other aspects of form in nature than this topological one. He pointed out that though topology does not differentiate between different coastlines, they nevertheless have different fractal dimensions. These different dimensions express the differences in a non-topological aspect of the form, which he calls the fractal form[03].

5.2.1 Information and Memory

From a physiological point of view man can take up a total set of information of  bit per second out of the rich perception offered.  can be dealt with in our consciousness, but only one bit per second is saved in our memory[04]!

So what does that mean to the elevation of a building? It means that a lot of information is taken up by observing a building, of which only a little part is dealt with in our consciousness, where it is compared with information we have already in our memory. E.g. if one does not know what a church "normally" looks like, that is if someone has never seen any, or even does not know what it is for, the form including signs like the cross is an absolutely new information for this person. Almost all people from our cultural background already have an idea of a "typical" appearance of a church, which may include the cross as a sign, a church-tower, a higher middle section, rose-windows, certain proportions and positions of windows, through which even more modern interpretations will be recognized as a church building. That means we have seen and learned about different form-types of churches that are in some way similar, which also means that we will have information about details of the elevation in our memory.

The more we generally know, the more we can process certain details. That is because the first impression confirms our picture in our memory and so we can concentrate on the second impression, on a smaller scale. If it is not like that, we feel insecure; the church has a lot of new information for us to be dealt with. This is then quite similar to information we get in a language we do not understand. But on the contrary if a church only consists of the most typical parts of a simple reduced "church-form" we do not learn anything new, we do not get any information. The part in between is interesting for us, including what we learn if we come closer to the building, that is if there is any new information coming up or not. The two extreme cases of too much and too low information is quite similar to the two extremes in chaos-theory between total confusion and determinism, but also to the complexity generated by simple algorithms of fractal geometry and fractal dimension as the measurement of a mixture of order and surprise.

5.3 The Box-Counting Dimension and Architecture

Throughout the following sections some fractal dimensions of elevations or ground plans are calculated for analysis and comparison. The measurement method used for these calculations is the so-called box counting method as described in chapter "4.2.3 Box-Counting Dimension ‘Db’". For better comparison of these fractal dimensions of buildings, it is important to give the scale of the map or plan that is used, which can be interpreted as the distance e.g. to the elevation. As mentioned before in chapter "5.1.3 The Scale-Range of an Elevation" the distance again influences the dimension through the recognizability of details - the smaller the scale of the plan the more details can be offered. Because in general only those boxes are counted in which some "interesting" details can be found, which may be window-frames and edges of window-strips, doors, edges of walls, the structure of the facade, but even the surroundings or, on lower scales, door knobs, the importance of certain scales and its information can be measured and compared.

As mentioned before the fractal dimension is the identification or better to say the quantifiable measure of the mixture of order and surprise, which also holds true in design and architecture. So an extremely high dimension would indicate that the elements, components of the composition do not correspond to each other, which means there is a high degree of surprise. The reason for that may be that they do not stand in a certain relation to each other and to the whole, which means that the viewer cannot identify the overall concept or its parts. But as the chapter "9 Statistics" will show, this is not true for all measured objects. There are examples where high dimensions result from the roughness of the surface, e.g. from timber-paneling which nevertheless may have a high order in their distribution and the elevation may display high symmetry with regard to rhythmical distribution of windows, which means high order and simple information of the surface. This example leads to the conclusion that the cuts in the facade like windows and doors are not the only parts that influence the dimension and that they are not of main importance for the structure. From this follows that high dimensions result rather from the roughness of the facade and by that from the "character" of the used material than from order by a symmetrical distribution of elements, see picture 29.

picture 29: Comparison of elevations

The following four examples point out the differences between certain materials used at an elevation and the influence of symmetry, expressed through their certain box-counting dimensions. Preceding from the simple symmetric elevation of the 1st example the distribution of the windows has been changed in the 2nd, the used material in the 3rd and 4th respectively. It turns out that the distribution has hardly any influence on the dimension - "Db"symmetric=1.348, "Db"asymmetric=1.344 -, that is because the number and length of the lines of the image, representing the elevation, remain the same, which means that the degree of roughness of the elevations is equal. Something different happens to the other two examples because in these cases the roughness of the elevation changes, in the 3rd image through the brickwork and in the 4th through the timberwork, which both increase the box-counting dimension - from 1.348 to 1.746 and to 1.617 respectively. 

As mentioned at picture 28 every object and every detail has its own "scale", which means that certain elements can only be realized from a certain distance onward. The brickwork for example can only be differentiated at the 5th stage, before that all boxes of the wall are counted, which means that up to this scale the wall is perceived as a two dimensional plane - the whole image remains at a high dimension: 1.643 - 1-784.

Looking at the development of the box-counting dimension from stage to stage it turns out that the timberwork offers similar dimensions for the 2nd and 3rd stage - "Db"from 1.797 to 1.859 -, which means that between these sizes the roughness keeps nearly the same. At the 4th stage the dimension falls to 1.589 and finally at the 5th stage to 1.116, which tells us that from then on the timberwork does not offer additional information to us. At this or some later stage other details, like the veining of the timberwork, windowsills, door- and window-frames will come into attention, which lie below the scale of the image.


elevation; grid size = 1/28; grid size = 1/56

symmetric (measurement)
stage unity size "1/s" log 1/s "N" pieces log(N) dimension
1 s(1)= 14 1.146 N(1)= 99 1.996   D
2 s(2)= 28 1.447 N(2)= 296 2.741 D(s1-s2)= 1.580
3 s(3)= 56 1.748 N(3)= 776 2.890 D(s2-s3)= 1.390
4 s(4)= 112 2.049 N(4)= 1897 3.278 D(s3-s4)= 1.290
5 s(5)= 224 2.350 N(5)= 4186 3.622 D(s4-s5)= 1.142
    slope= 1.348


elevation: grid size = 1/28; grid size = 1/56

asymmetric (measurement)
stage unity size "1/s" log 1/s "N" pieces log(N) dimension
1 s(1)= 14 1.146 N(1)= 103 2.019   D
2 s(2)= 28 1.447 N(2)= 280 2.447 D(s1-s2)= 1.443
3 s(3)= 56 1.748 N(3)= 736 2.867 D(s2-s3)= 1.394
4 s(4)= 112 2.049 N(4)= 1898 3.278 D(s3-s4)= 1.367
5 s(5)= 224 2.350 N(5)= 4175 3.621 D(s4-s5)= 1.137
    slope= 1.344


elevation: grid size = 1/28; grid size = 1/56

brick wall (measurement)
stage unity size "1/s" log 1/s "N" pieces log(N) dimension
1 s(1)= 14 1.146 N(1)= 116 2.064   D
2 s(2)= 28 1.447 N(2)= 388 2.589 D(s1-s2)= 1.742
3 s(3)= 56 1.748 N(3)= 1336 3.126 D(s2-s3)= 1.784
4 s(4)= 112 2.049 N(4)= 4589 3.662 D(s3-s4)= 1.780
5 s(5)= 224 2.350 N(5)= 14330 4.156 D(s4-s5)= 1.643
    slope= 1.746


elevation: grid size = 1/28; grid size = 1/56

timberwork (measurement)
stage unity size "1/s" log 1/s "N" pieces log(N) dimension
1 s(1)= 14 1.146 N(1)= 116 2.064   D
2 s(2)= 28 1.447 N(2)= 403 2.605 D(s1-s2)= 1.797
3 s(3)= 56 1.748 N(3)= 1462 3.165 D(s2-s3)= 1.859
4 s(4)= 112 2.049 N(4)= 4397 3.643 D(s3-s4)= 1.589
5 s(5)= 224 2.350 N(5)= 9531 3.979 D(s4-s5)= 1.116
    slope= 1.617

5.4 Symmetry

Architects have used symmetry in the form of bilateral symmetry, where one side of an image is a mirror of the other, in elevations but also in ground plans of buildings for centuries. In nature such a bilateral symmetry can also be observed, e.g. in the human body. But also symmetry developed into rhythm can be found, which means that an image is repeated along a line, as it is true for the fingers, or that it is rotated around a point - see picture30[05]..

But on closer observation it turns out that the symmetry and rhythm is not that strict in nature and that there exists complex diversity - as it is also true for architecture. By that every object is different from the other, which means for example that not all human beings are the same size and shape, and even both sides of the bilateral symmetry need not really be identical, that is the right and left side of the human body may have little differences. Self-similarity as an important characteristic of fractal geometry describes nature’s underlying diversity and shows invariance with respect to size as a new kind of symmetry[06].

picture 30: Symmetry

5.5 Fractal Characteristics in the History of Architecture

Architecture as a mirror of society is also a kind of public image, which is promoted by our time and by the culture in which we are building. The architect translates and interprets the conscious and unconscious thoughts of society. This also means that the architect has to face history and the present[07], with fractal geometry belonging to the present and so it should therefore be included in the one or other way. On the following pages the question will be dealt with if fractal geometry was present in the architecture of the past anyway.

Benoit Mandelbrot contrasts architecture that is rich in self-similarity and posses different scales of length - see chapter "5.1.2 The Scale-Range of a Wall" -, such as Gothic and Beaux Arts, with the curtain-walled lifelessness of modern architecture. Modern architecture, the International Style and the Machine Aesthetic, usually have no focus, no climax above the entrance, middle or top[08].

5.6 The Influence of the Surrounding

Our natural surroundings, our society, our life consist of differentiation but nevertheless the whole always looks like a unity. That is because in all the parts, sections of this unity we will nevertheless always find similarities to the whole - Charles Jencks called this phenomenon the organizing depth. In a society based on agriculture not only the economy is adjusted to agriculture, but also lifestyle, men, buildings. This circumstance is reflected by the landscape as if it were a mirror - it is formed by typical courses of roads, farmhouses and villages.

As the previous pages have illustrated, the box-counting dimension ofelevations and ground plans can be measured and then compared with each other or with its surroundings. But there might arise some difficulties from the measurement itself, which are dealt with in chapter "7 Problems with Measuring".

In chapter "9 Statistics" some of the comparisons of different house types, such as double houses, one family houses, dwelling houses or farmhouses, under the usage of different parameters, such as including tiles and furniture or not - concerning ground plans - or including the surface material or not - concerning elevations - are mentioned. One question arising from that is: whether different types of houses have different ranges of fractal dimensions or not. Because of  different scales and qualities of plans no unequivocal results but some tendencies can be given. In this sense elevations of smaller architectural tasks seem to have a lower average fractal dimension - around 1.48 - in comparison to bigger dwelling houses and public buildings - around 1.55. But the dimension of the latter mentioned group might also result from the generally smaller scale of plans used or of too little data used.

In any case, what is to be measured is the roughness or smoothness of the ground plan or elevation. This differentiation is then worked out with the examples of farmhouses in chapter "9 Statistics".

5.8 Helping Tool in the Stage of Planning

In the previous pages fractal geometry was mainly used as an analyzing tool in form of calculating the fractal dimension with the box-counting method or as a means of finding out whether buildings offer any fractal characteristics or not and if self-similarity and higher fractal dimensions are the reasons for greater acceptance of certain styles in contrast to the modern flat style. Now, in the following chapter, fractal geometry is used as a helping tool in the stage of planning by getting grid-layers and possible distributions.


[01] Frank Lloyd Wright used the vocabulary of nature not just by forms rebuilt in an organic matter but by translating the natural structure which mostly belongs to fractal geometry.
[02] Bovill Carl, Fractal Geometry in Architecture and Design (1996), Birkhäuser Bosten, ISBN 3-7643-3795-8, p.128.
[03] Mandelbrot Benoit B., Dr. Zähle Ulrich (editor of the German edition), Die fraktale Geometrie der Natur (1991) einmalige Sonderausgabe, Birkhäuser Verlag Berlin, ISBN 3-7643-2646-8, p.28.
[04] Bit is the shortened form of binary digit, a unit of computer information equivalent to the result of the choice between two alternatives as yes/no or on/off. 
[05] Bovill Carl, Fractal Geometry in Architecture and Design (1996), Birkhäuser Bosten, ISBN 3-7643-3795-8, p.1.
[06] ;Bovill Carl, Fractal Geometry in Architecture and Design (1996), Birkhäuser Bosten, ISBN 3-7643-3795-8, p.3. Self-similarity is the symmetry that is scale-independent, where the smaller parts, elements are similar in the one or other way to larger parts and to the whole.
[07] Borcherdt Helmut, Architekten - Begegnungen 1956-1986 (1988), Georg Müller Verlag GmbH, ISBN 3-78844-2181-4, p.133.
[08] Jencks Charles, The Architecture of the Jumping Universe, (1995), academy editions, ISBN 1 85490 406 X, p. 43.