# Relief Method: The Analysis of Architectonic Façades by Fractal Geometry

## Abstract

**:**

## 1. Introduction

## 2. Towards a Livable Built Environment

## 3. The Connectivity of Façades

## 4. A Brief Explanation of the Relief Method

- Patterns represented at each scale are quasi-self-similar, and
- The scales are defined by the layers of the given façade.

_{n}∙ d

_{n}≅ c,

_{H}

^{+}= ln(A

_{n}+ p

_{n}∙ d

_{n}) / ln√A

_{n},

_{n}stands for the total area of the relief on layer n seen as a 2D projection. The formula denotes that if, for instance, we chose 20 m as the base length of a square-shaped façade, we would not measure 400 square meters as its surface, as expected in 2D. A certain area (p

_{n}∙d

_{n}) would be added by the hidden depth of the layer; hence, we would measure the dimension of the relief as slightly more than two-dimensional.

_{H}must remain identical, no matter which layer we measure. Because, as stated before, p

_{n}∙d

_{n}needs to be more or less constant, while A

_{n}also has to be constant in order to satisfy the above equation. Realizing that every layer has to have the same projected area, we should also find that the possible number of layers are strictly limited, for they may overlap, but never fully cover each other inside the frame of the elevation. In addition, we may note that the number n does not have a further relevance for the constancy of D

_{H}. To encourage easy calculations, it is enough to take the first layer into consideration, thus the practical formula will slightly alter as far as n = 1.

_{H}

^{−}= ln(A

_{n}− V

_{n}) / ln√A

_{n},

_{n}signifies the total area of the voids on layer n. Finally, we need to raise the question whether the two types of Hausdorff dimensions could be tied together. The answer is yes if the layers of the two are congruent, and also when we match the common layers with the corresponding n. The unified formula would then demonstrate the dimension of the two fractals added together:

_{H}

^{±}= ln(A

_{n}+ p

_{n}∙ d

_{n}− V

_{n}) / ln√A

_{n}.

## 5. Building Materials and the Natural Scene (Results A)

_{H}

^{±}may be counted according to these less spectacular, either positive or negative, modifications, as the following example of a simple basalt-cladded wall demonstrates.

^{2}, then our equation becomes as simple as:

_{H}

^{±}= ln105 / ln10 ≅ 2.021

_{H}

^{±}depends on the exact area of the inspected surface, so the same ratio of unevenness will affect a greater plane less—which looks quite comprehensible from the viewport of an observer.

## 6. Two or Three Basic Design Trends (Results B)

_{H}

^{−}formula described above.

_{H}

^{−}and D

_{H}

^{±}) calculated from certain 19th and 20th century façades have substantial regressions compared to their predecessors. This is mostly due to the loss of the material surface in exchange for sunlight and transparency. Before this dramatic change, the differences between D

_{H}

^{+}and D

_{H}

^{−}were more or less balanced, whereas during the late Renaissance and Baroque, the plasticism or the materiality of buildings (D

_{H}

^{+}) clearly increased thanks to the voluminous walls and elaborate details characteristic of the epoch. It is important to note that the mixed dimension (D

_{H}

^{±}) has typically been over 2.0 until it dropped in the 19th century, which proves that Historicism was involved in modern trends already, before the turn of the millennium. As a parallel tendency, modern elevations were hard to identify as fractals anymore; yet, notably from the late 20th century [63], a certain revival of organic surfaces has begun, notwithstanding the losses of social connectivity. This tendency continues after the turn of the millennium as indicated by the calculated values of the MIT Simmons Hall.

## 7. A Guide to Reproduce Our Numeric Results in Table 1, Table 2 and Table 3

^{2}examined reference area. The visible side of the clinker elements have the dimensions 6.5 cm × 25 cm, which gives the area of 162.5 cm

^{2}each. We consider the calculated surface relatively smooth; however, the joints between the elements decrease the total area of the sample wall by 1 cm × (1 cm + 6.5 cm + 25 cm), that is, 32.5 cm

^{2}per brick. Consequently, the replicated area is 195 cm

^{2}, which is decreased by 32.5 cm

^{2}. This technically means that the surface decreases by 16.67%, but because of the softness of the bricks’ edges, we round this value up to 17%. Calculating with the value of the examined reference area, Equation (3) will result in 1.838 as the amount for D

_{H}

^{−}.

^{2}new surface area. Comparing with 195 cm

^{2}, the theoretical surface increase is 16.15%, but because of certain irregularities—e.g., the curvature of the mortar, or the varying position of the bricks versus the ideal vertical plane—this value grows at least by 2%. Following Equation (2), the rounded 18% of surface increase finally makes D

_{H}

^{+}= 2.144. From the unified Equation (4), the value 2.009 for D

_{H}

^{±}can be easily calculated. We can apply the same scheme to reproduce the Hausdorff dimensions of all the other building materials listed in Table 1. Furthermore, we may examine new ones with more detailed specifications.

_{H}

^{+}, whereas all the proportion of ‘missing areas’ will affect D

_{H}

^{−}. In order to obtain the proper data for the latter, a reference plane must be set with an adjustable level of depth tolerance. The sum of the empty areas on the plane will decrease the overall surface of the processed image, so the percentage of decrease can be calculated.

_{H}

^{+}. To avoid this complication, we ‘compress’ the forest’s space into a relief, so that a vertical reference plane—with a certain depth tolerance—could be defined. The forest between the plane and the viewer is considered a layer composed of plastic elements, while the nature behind the plane is looked at as an image. Both D

_{H}

^{+}and D

_{H}

^{−}are calculated relative to the reference plane, which implies the comparability of this layer to that of an architectonic façade.

_{H}

^{+}to reach the amount of 2.093, whereas the openings associated with the same layer take only 20% area away. Due to this slighter decrease, the joined Hausdorff dimension calculated from layer 1 equals to 2.048. It exceeds the amount of 2, which indicates the overall plasticity of the selected façade. We underline that D

_{H}

^{+}can only be associated with D

_{H}

^{−}if we relate them to the same reference plane—most often the plane of the window frames—and calculate them by measuring the elements on the same scale, thus the same layer only. Our surface measurements of the buildings listed in Table 3 were made accordingly through the analysis of architectural plans, photography and Google Earth data. Photography and Google data was necessary to check if the actual buildings differed from the architectural plans.

## 8. Conclusions

_{H}in this case is a possible numeric determination of the genius loci—the ‘atmospheric’ phenomenon—which is our main purpose behind the comparison of the data findings listed in Table 1, Table 2 and Table 3. These tables provide information about the mathematical similarities of natural scenes and building materials, and grant us a comparison between them and the architectonic façades (see Figure 5).

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The essence of tectonic planning is the coherence between the load bearing structure and the building’s membrane.

**Figure 2.**The rearranged layers of Palazzo Valmarana in Vicenza, Italy. The left column represents the relief layers, while the right represents the void layers. The layers in the same rows display the same scales (n = 1 to 3), or the levels of magnification. The layers on the same levels may be united in order to calculate their combined Hausdorff dimension.

**Figure 3.**Voids of different scales define different layers. The windows and joints of Palazzo Strozzi in Florence, Italy.

**Figure 4.**A perspective drawing of Steven Holl’s Simmons Hall at the MIT Campus in Cambridge, Massachusetts. A new epoch of fractal architecture has already started.

**Figure 5.**The comparability of the patterns from different sources—architectonic façades, natural scenes, and textures of building materials—through their Hausdorff dimensions.

**Table 1.**The Hausdorff dimensions of walls or claddings built from some well-known materials. Surface measurements were made through the use of digital photography, image processing and metric data provided by manufacturers.

Type of Building Material | Examined Reference Area (m^{2}) | Surface Increase (%) | Surface Decrease (%) | D_{H}^{+} | D_{H}^{−} | D_{H}^{±} |
---|---|---|---|---|---|---|

Plank | 10 | 17 | 6 | 2.136 | 1.946 | 2.091 |

Shingle | 10 | 23 | 7 | 2.180 | 1.937 | 2.129 |

Wattle-work | 10 | 41 | 25 | 2.298 | 1.750 | 2.129 |

Rock | 10 | 39 | 29 | 2.286 | 1.703 | 2.083 |

Rough stone | 10 | 16 | 13 | 2.129 | 1.879 | 2.026 |

Fine-cut stone | 10 | 8 | 3 | 2.067 | 1.974 | 2.042 |

Brick | 10 | 20 | 17 | 2.158 | 1.838 | 2.026 |

Clinker | 10 | 18 | 17 | 2.144 | 1.838 | 2.009 |

Plaster | 10 | 5 | 0 | 2.042 | 2.000 | 2.042 |

Rough concrete | 10 | 27 | 5 | 2.208 | 1.955 | 2.173 |

Shuttered concrete | 10 | 17 | 8 | 2.136 | 1.928 | 2.075 |

Precast concrete | 10 | 2 | 2 | 2.017 | 1.982 | 2.000 |

Steel | 10 | 0 | 1 | 2.000 | 1.991 | 1.991 |

Weathering steel | 10 | 3 | 1 | 2.026 | 1.991 | 2.017 |

**Table 2.**The Hausdorff dimensions of natural scenes surrounding a building or an object. Surface estimates were made through the use of digital photography, image processing and Google Earth data.

Type of Natural Scene | Extrapolated Reference Area (m^{2}) | Surface Increase (%) | Surface Decrease (%) | D_{H}^{+} | D_{H}^{−} | D_{H}^{±} |
---|---|---|---|---|---|---|

Clay wall | 1000 | 23 | 5 | 2.060 | 1.985 | 2.048 |

Basalt wall | 1000 | 45 | 8 | 2.108 | 1.976 | 2.091 |

Sedimentary cliffs | 1000 | 65 | 13 | 2.145 | 1.960 | 2.121 |

Reed marsh | 1000 | 45 | 23 | 2.108 | 1.924 | 2.058 |

Temperate forest | 1000 | 55 | 18 | 2.127 | 1.943 | 2.091 |

**Table 3.**The Hausdorff dimensions of the façades of some historical and contemporary buildings. Surface measurements were made through architectural plans, photography, and Google Earth data.

Building and Location | Century of Origin | Location of Reference Layer | C. Area ^{1} of Reference Layer (m^{2}) | Surf. Incr. (%) | Surf. Decr. (%) | D_{H}^{+} | D_{H}^{−} | D_{H}^{±} |
---|---|---|---|---|---|---|---|---|

Diocletian’s Pal., Split, Croatia | 3th | sea front | 3000 | 13 | 14 | 2.031 | 1.962 | 1.997 |

Durham Cath., Durham, UK | 11th | north aisle | 800 | 19 | 17 | 2.052 | 1.944 | 2.006 |

Pal. Strozzi, Florence, Italy | 15th | east elevation | 1000 | 16 | 13 | 2.043 | 1.960 | 2.009 |

Basilica Palladiana, Vicenza, Italy | 16th | south elevation | 700 | 33 | 16 | 2.087 | 1.947 | 2.048 |

Pal. Valmarana, Vicenza, Italy | 16th | west elevation | 450 | 33 | 20 | 2.093 | 1.927 | 2.040 |

Pal. Carignano, Turin, Italy | 17th | west elevation | 2000 | 39 | 19 | 2.087 | 1.945 | 2.048 |

Bauakademie, Berlin, Germany | 19th | north elevation | 950 | 15 | 18 | 2.041 | 1.942 | 1.991 |

Wainwright Bldg., St. Luis, MO | 19th | south elevation | 800 | 18 | 43 | 2.050 | 1.832 | 1.914 |

National Library, Ljubljana, Slovenia | 20th | east elevation | 900 | 23 | 11 | 2.061 | 1.966 | 2.033 |

Seagram Bldg., New York, NY | 20th | west elevation | 6500 | 29 | 63 | 2.058 | 1.774 | 1.905 |

MIT Simmons Hall, Cambridge, MA | 21st | south elevation | 2500 | 67 | 44 | 2.131 | 1.852 | 2.053 |

^{1}The projected area of the reference layer does not necessarily equal to the total projected area of the façade.

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**MDPI and ACS Style**

Katona, V.
Relief Method: The Analysis of Architectonic Façades by Fractal Geometry. *Buildings* **2021**, *11*, 16.
https://doi.org/10.3390/buildings11010016

**AMA Style**

Katona V.
Relief Method: The Analysis of Architectonic Façades by Fractal Geometry. *Buildings*. 2021; 11(1):16.
https://doi.org/10.3390/buildings11010016

**Chicago/Turabian Style**

Katona, Vilmos.
2021. "Relief Method: The Analysis of Architectonic Façades by Fractal Geometry" *Buildings* 11, no. 1: 16.
https://doi.org/10.3390/buildings11010016