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5.6.1 Material

Most of the indigenous buildings integrate part of nature through the available materials. Thinking of regions with soft rocks around, man-made dwellings but also freestanding protective walls often consist of these natural stones. No matter whether they are lying free on the surface or are roughcast, the character of the conical, rugged, baggy walls will be similar to the surroundings. The house itself has a certain similarity with the rugged mountains through steep roofs, massive rugged walls that are repeated in the detail - the wall itself.

The same is true for villages and indigenous buildings in areas with large forests. Again the character of a wood is taken up in the house itself, as in a log cabin, where the stem is moved from a vertical to a horizontal position, or by timberwork in the balcony and in the gable. Wooden tiles are also such identification with the surrounding area. So one source of organizing depth lies in the usage of local material, with the possible variations and results being nearly infinite. Through the material, the parts as well as the whole consist of a similar character and structure, it is the red thread of the object.

5.6.2 Indigenous Buildings

Indigenous buildings generally have a deeper scaling than modern houses. That means that coming closer to the surface, there will always be some new smaller interesting details, similar to the concept of the houses by Frank Lloyd Wright. The different kinds of rural houses depend on the specific region, the available materials, the size, the usage and inhabitants of the building.

In the comparison of two houses taken from "Alte Bauernhäuser in den Dolomiten"[01], one native-house in Gosaldo and one in Borca, the differences of the two regions are visualized. The elevation - the plan of the facade - of the house in Gosaldo seems to be smoother, which is pointed out by the box-counting dimension of the smaller scales:

D(10-20)=(log105-log42)/(log20-log10)=1.322 for Gosaldo
D(10-20)=(log208-log66)/(log20-log10)=1.656 for Borca

The more one zooms into the plan, the lower the dimension will be, which also means that the elevation then offers less additional information and fewer details respectively - the facade does not seem to be very rugged. The elevation of the house in Borca on the contrary offers more and more details the smaller the scale becomes. This is again shown in the fractal dimensions of the surface:

D(20-40)=(log283-log105)/(log40-log20)=1.430 for Gosaldo
D(20-40)=(log688-log208)/(log40-log20)=1.726 for Borca

In the case of Borca the dimension increases from the scale-range of 10boxes to 20boxes to the scale-range of 20boxes to 40boxes, in the example of Gosaldo it decreases. Comparing the dimensions of the plans of both facades on the same scale shows the difference in the overall view, the elevation of Borca has a box-counting dimension of e.g. D(40-80)=1.702, the one of Gosaldo D(40-80)=1.461. This is caused by the different structure of the elevations: the one of Gosaldo offers a smooth symmetrical front-facade with one balcony, symmetrically arranged windows and a small additional section on the left side. The front-elevation of Borca consists of an outdoor staircase, some balconies with wooden carved banisters, asymmetrically arranged windows, timberwork on the gable, a bigger additional part on the back left side and some parts have different heights. That points out that the box-counting dimension indicates the roughness and differentiation of the whole building - the plan respectively. The more rugged the elevation - picture of the facade - is the higher the dimension will be. Besides one can observe the decrease or increase in information when zooming closer, see picture 53.

picture 53: native houses

The following native houses are taken from "Alte Bauernhäuser in den Dolomiten" by Edoardo Gellner, (1989).

1st example: “native house in Gosaldo”

below: the results of the computer program "Benoit";

GS3 Gosaldo: measurement program "Benoit" 
width: 237; height: 144; side length of largest box: 36; coefficient of box size decrease: 1.3;
increment of grid rotation:dimension number of box sizes
measurement for 0° rotationDb=1.413SD=0.00512
measurement for 15° rotationDb=1.420SD=0.00411
measurement for 45° rotationDb=1.416SD=0.00512
GS3 Gosaldo
stageunity size "1/s"log 1/s"N" pieceslog(N)dimension
1s(1)=50.699N(1)=141.146 D
2s(2)=101.000N(2)=421.623D(s1-s2)=1.585
3s(3)=201.301N(3)=1052.021D(s2-s3)=1.322
4s(4)=401.602N(4)=2832.452D(s3-s4)=1.430
5s(5)=801.903N(5)=7792.892D(s4-s5)=1.461
6s(6)=1602.204N(6)=20143.304D(s5-s6)=1.370
  D(slope)=1.426

2nd example: "native house in Borca"

below: the results of the computer program "Benoit";

BC Borca: measurement program "Benoit" 
width: 301; height: 230; side length of largest box: 57; coefficient of box size decrease: 1.3;
increment of grid rotation:dimension number of box sizes
measurement for 0° rotationDb=1.691SD=0.00511
measurement for 15° rotationDb=1.694SD=0.00511
measurement for 45° rotationDb=1.691SD=0.00511
BC Borca
stageunity size "1/s"log 1/s"N" pieceslog(N)dimension
1s(1)=50.699N(1)=201.301 D
2s(2)=101.000N(2)=661.820D(s1-s2)=1.722
3s(3)=201.301N(3)=2082.318D(s2-s3)=1.656
4s(4)=401.602N(4)=6882.318D(s3-s4)=1.726
5s(5)=801.903N(5)=22383.350D(s4-s5)=1.702
6s(6)=1602.204N(6)=69793.844D(s5-s6)=1.641
  D(slope)=1.692

3rd example: "native house"

below: the results of measuring by hand;

FZ 10
stageunity size "1/s"log 1/s"N" pieceslog(N)dimension
1s(1)=50.699N(1)=191.279 D
2s(2)=101.000N(2)=601.778D(s1-s2)=1.659
3s(3)=201.301N(3)=2152.332D(s2-s3)=1.841
4s(4)=401.602N(4)=6982.844D(s3-s4)=1.699
5s(5)=801.903N(5)=23183.365D(s4-s5)=1.732
6s(6)=1602.204N(6)=75493.878D(s5-s6)=1.703
  D(slope)=1.734

4th example: "native-house in Laggio"

below: the results of measuring by hand;

VC4
stageunity size "1/s"log 1/s"N" pieceslog(N)dimension
1s(1)=40.602N(1)=161.20 D
2s(2)=80.903N(2)=581.76D(s1-s2)=1.858
3s(3)=161.204N(3)=1172.23D(s2-s3)=1.551
4s(4)=321.505N(4)=3802.58D(s3-s4)=1.161
5s(5)=641.806N(5)=9702.99D(s4-s5)=1.352
6s(6)=1282.107N(6)=27153.43D(s5-s6)=1.485
  D(slope)=1.361

5th example: "native house"

below: the results of measuring by hand;

SA8
stageunity size "1/s"log 1/s"N" pieceslog(N)dimension
1s(1)=40.602N(1)=121.08 D
2s(2)=80.903N(2)=441.64D(s1-s2)=1.875
3s(3)=161.204N(3)=1442.16D(s2-s3)=1.711
4s(4)=321.505N(4)=4962.69D(s3-s4)=1.781
5s(5)=641.806N(5)=18463.27D(s4-s5)=1.899
6s(6)=1282.107N(6)=46593.67D(s5-s6)=1.336
  D(slope)=1.713

The fractal dimension of the smoother elevation of the 1st example decreases down to the 5th stage from 1.585 to 1.370. The same holds true for the 4th example in which the fractal dimension between the 3rd and 4th stage only amounts to 1.16. This results from the circumstance that between these two scales the number of new interesting details also decreases.

The fractal dimension of the slope of the 1st example is about 1.426, that of the 4th 1.36, which is quite similar. This means that the elevations of both facades - the representation through their plans - do not offer many interesting details, that is to say much information. This stands in contrast to the 2nd example of the elevation of a rougher facade. For this native building in Borca the fractal dimension remains high from stage to stage, which derives from the fact that each scale has its "own" details. The wooden gable for example can only be differentiated at the 6th stage, which is also true for the railing of the balcony. These characteristics can also be found in the 3rd example. The elevations of both buildings are similar not only in the fractal dimension of the slope, which is 1.692 for the 2nd and 1.734 for the 3rd example, but also in the visual characteristics. The 2nd and 3rd example display wooden-work balcony, wooden gables, saddlebacks, both are equipped with little additions and their size is nearly the same.

5.6.2.a Translation

Through the factor of time, a random process is brought into the building, as it is found in nature with mountains, trees, clouds, coastlines and rivers[02]. If the clustered random cascade in nature is fractal, then the cascade of rhythm and detail in indigenous buildings may also be fractal and the measured fractal dimensions of the environment, elevation and detail will be similar[03]. Beside that, when planning in the surrounding of any other existing buildings, their character can be analyzed and interpreted as a sort of planning-aid. The structure includes the widths and heights-diversity, similarity of the characteristics of bases, middle parts and tops, window and door-types and their sizes, ledges and setbacks. These parts can be given by a fractal-dimension range and translated into a fractal rhythm. This "new" rhythm can then be used as a planning tool for setbacks, window sizes and distribution, construction grid and more, see picture 54.

picture 54: Surrounding - translation/Fractal rhythm

The picture below is taken from "Alte Bauernhäuser in den Dolomiten" by Edoardo Gellner.

The fractal dimension of the rhythm of the vertical axes is D=1.63 - the calculation method, the rescaled range analysis being explained at pictures 58-60. The box-counting dimension, calculated with the computer program “Benoit”, is Db=1.55.

below: bar chart of the rhythms of the elevation;

below: sequence of elements based on the Van Der Laan scale - see picture 60;

below: fractal curve with a Hurst component of 0.37 - D=1.63 - as a source for the fractal variation of the setbacks and widths of row houses and the sequence of windows respectively;

5.6.3 Landscape and Building

Certain surroundings demand certain fractal dimensions of the building and by that the fractal dimension can be used as a critical tool to find out whether certain buildings or groups of houses fit into their environment or not. Short time ago this could not be measured. But looking at some examples of the measurements by Carl Bovill will make us believe that harmony between man-made objects, such as buildings, and the surroundings can be indicated by their similar fractal dimension. The reasons often lie in the harmony of the materials used and the factor of time.

5.6.3.a Kind of Surrounding

If the environment is very rugged it asks for a house with greater detail-richness - which is automatically brought into native-buildings by the use of traditional, available materials. In contrast to that, a house with a much smoother elevation may fit better in flat, uniform surroundings. Nevertheless these two different possibilities also work the other way round. So it is possible to use different fractal dimensions of building and environment to attract attention. These possibilities hold true for a natural surrounding as well as for an artificial environment. In this way the fractal rhythms can be used to produce planning grids that utilize the rhythms of nature as a source of layout inspiration. For example the fractal dimension of a mountain ridge behind an architectural project could be measured and used to guide the fractal rhythms of the project design. The project design and the site background would then have similar rhythmic characteristics. Time and local materials are important factors for this process as well. The composition of indigenous buildings has a random cluster, which is similar to the clustered randomness of mountains, clouds and trees.

In this sense the fractal dimension of surroundings can be used on bigger scales like mountain-ridges, forests and typical landscapes for calculating the dimension of the site plan and overall view, or on smaller scales like single trees and immediately surrounding landscape for elevation or floor plan of a building.

5.6.3.b Two Different Regions

There is an example in "Fractal Geometry"[04], where Carl Bovill compares two different houses with their typical environment. One example is the sketch of the condominium building at Sea Ranch, California. The coastline there has a self-similar structure in form of a very rough edge from the scale of the bay to the scale of pebbles, which calls for complex shapes of the building. In general many of the buildings in this region and especial the condominium building by Moore - built in the sixties - offer the complexity of the coastline, which means that they are like a mirror of the environment. The forms of the surrounding rocks of the coastline and the building itself have similar characteristics. Nantucket, Massachusetts in contrast has a very smooth coastline. This smoothness is also included in the texture of the waves washing up on the beach, which calls for simple basic shapes. The natural concept is not only shown in the forms of the coastline but also in form of the site-plan and the elevations of the indigenous settlements, see picture 55.

picture 55: Sea Ranch and Main Street

"Sea Ranch, California" and "Main Street in Nantucket"

The picture below shows a drawing of three houses along Main Street in "Nantucket", Massachusetts. The “smooth” environment cries for smooth facades.

Charles Moore-William Turnbull - “Sea Ranch, California”, 1966: a rough facade in a rough environment.

5.6.3.c Including the whole Environment

For showing how buildings may correspond with their surroundings, Carl Bovill analyzed the example of a row of traditional houses in Amasya in Turkey. He found out that for this example the fractal dimensions, determined by the box-counting method, of the environment such as the mountain and the river, were very close to dimension of the site layout and elevations of the buildings, see picture 56.

picture 56: traditional houses

Elevation of a typical row of traditional houses in Amasya, Turkey.

Elevation: D(13-26)=1.69; D(26-52)=1.62; D(52-104)=1.33;
the results of the measurement by Bovill: D(6-3m)=1.67; D(3-1.5m)=1.67; D(1.5-0.75m)=1.70;

Site plan of the row houses: D(13-26)=1.544; D(26-52)=1.63; D(52-104)=1.59; D(104-208)=1.176;
D(52-104)=(log978-log325)/(log104-log52)=1.59:
the results of the measurement by Bovill: D(50-25m)=1.63; D(25-12.5m)=1.70; D(3.13-1.56m)=1.10;

Mountain ridge behind the traditional row houses:
D(13-26)=1.585; D(26-52)=1.526; D(52-104)=1.234; D(104-208)=1.086; 
the results of the measurement by Bovill: D(100-50m)=1.63; D(50-25m)=1.67; D(25-12.5m)=1.40;

The cluster of traditional dwellings was established in the 3rd century B.C. and is situated along a river at the foot of the dominant mountain ridge. The measurement of the fractal dimension of the row, which is represented as a clustered random cascade of rhythm in height and width, results in 1.67 for a scale-range from 6 to 3 meters and rises to 1.83 on a scale-range from 0.75 to 0.38 meters. The site-plan, that includes the bends of the river, results in a fractal dimension of 1.63 for a scale-range from 50 to 25 meters and 1.1 for 3.13 to 1.56 meters. The mountain ridge, which is reduced to a sketch of the characteristic edges, has a fractal dimension of 1.63 for a scale-range from 100 to 50 meters and 1.4 for 25 to 12.5 meters. Carl Bovill then interprets this data material. Thus it turns out that on a scale-range of 6 to 3 meters the site layout runs out of interesting details, which is the same scale at which the details of the elevation of the houses becomes the attraction.

Two important findings result from this example. First the degree of abstraction of the mountain-ridge influences the results of measurement, as this is true for the elevation of the building, which is represented in a certain characteristic abstraction. This leads us to the second finding, namely that the scale of the plan has to fit to the scales that are measured by the box counting method. This means that the smallest boxes that are measured have to be of a certain size and have to include the scale that is represented on the plan.

5.6.3.d Summary

Reasons for the similarity of fractal dimensions between buildings, environment and site-plans are the surroundings in form of available material for buildings, which may be timber, stone, clay, but also in form of defense, demonstration of power, and the form of the environment itself. Flat or rough mountain landscapes give the people who have been living there for hundreds of years a specific character and behavior, in the same way as light, i.e. daylight because of north-south distribution, weather, religion and society do. Buildings in their function as mirrors reflect those different characters. The kind of characteristic features may, however, be different because of outer influences, such as the contact with other cultures, adaptations or the different importance of certain factors.

Footnotes

[01] Edoardo Gellner, Alte Bauernhäuser in den Dolomiten, Verlag Georg D. W. Callwey (1989), ISBN 3-7667-0946-1.
[02] Time is added because the family-background or money changes, which leads to additions or other space-distributions. Then the existing house is the factor which determines that the changes remain under the same concept. In Honkong there are high-rise houses where - illegal - additions are built like bird's-nests, which look similar because of constructional limitations of depth, width and the use of timber. In this case the reason lies in the very small space which the dwellings occupy.
[03] Bovill Carl, Fractal Geometry in Architecture and Design (1996), Birkhäuser Bosten, ISBN 3-7643-3795-8, p.145.
[04] Bovill Carl, Fractal Geometry in Architecture and Design (1996), Birkhäuser Bosten, ISBN 3-7643-3795-8, p.180.