# Multilayered Complexity Analysis in Architectural Design: Two Measurement Methods Evaluating Self-Similarity and Complexity

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## Abstract

**:**

## 1. Introduction

#### 1.1. Fractal Geometry and Architecture

#### 1.2. Scalebound and Scaling Shapes

#### 1.3. The Leitmotif

#### 1.4. The Whole and Its Parts

#### 1.5. Complexity and Irregularity

## 2. Materials and Methods

#### 2.1. Fractal Dimension

- allows the quantitative measurement of mixture between order and surprise in a structure [3]
- gives the visual complexity a quantitative value [3]
- allows to analyze and to compare geometric properties [22]
- enables statements about harmonic relation between the whole and it parts [8]
- enables the comparison of different design solutions [8]

#### 2.2. The Box-Counting Method Applied to 2-Dimensional Line-Graphics

#### 2.2.1. Influence Factors

#### 2.2.2. Implementation

^{®}for Rhinoceros

^{®}[28]. This time, too, the possibility of setting various parameters considers the influencing factors (see Figure 1). The main result of a measurement series consists of:

- The box-counting dimension ${D}_{B}$ as a result of the minimum number of covering boxes and the corresponding coefficient of determination ${R}^{2}$ (which serves as a measure of coherence)
- The median ${D}_{B}$ (central value of a data series) of all results visualized in a box-plot, with the interquartile range describing the accuracy of the entire set
- The average ${D}_{B}$ of all results

^{®}program, the user selects the object under consideration in Rhinoceros

^{®}and assigns it. For performance reasons, the authors recommend grouping all relevant objects. With the definition of the bounding box, the program also calculates the shortest side. Then follows, depending on the user selection (see (e) in Figure 1), the subdivision of the smallest side or of the x-parallel side into the specified number of boxes (“numBoxesSide” (d) in Figure 1). The subdivision of the other side results from the approximation of square boxes. Regarding the following iterations (number of divisions; “iterations” (a) in Figure 1), exactly dividing each box into quadrants means recursively considering only those boxes that have already covered a piece of the object. This speeds up the algorithm as smaller grid sizes do not examine empty boxes anymore. Dividing the distance between the two largest box sizes by the number of “steps in between” (see (f) in Figure 1) results in the additional starting grid sizes. To achieve better accuracy of the measurement, the program adds an empty space around the object in a further step. The empty space corresponds to the difference resulting from a multiplication of the smallest side of the bounding box with the factor “enlargement” (see (b,g) in Figure 1). This also enables different starting positions of the grid, while at the same time maintaining the absolute size and completely covering the object. With only two offsets (see (c) in Figure 1), the lower-left starting point results from the fact that the grid still has to cover the upper right corner of the object (its bounding box). The same applies to the other three corner points. Consequently, two offsets result in four (2 by 2)-starting positions. In the visualization, the iterations are in the horizontal direction while the offsets and intermediate steps are in the vertical direction. In addition to the before-mentioned parameters, the user defines the worksheet number of an open Excel sheet into which to write the result.

#### 2.3. The Box-Counting Method Applied to Photographs

#### 2.4. Redundancy of Proportions, Gradient Analysis and Grid Analysis

#### Implementation of the Gradient Analysis

- rounding the values of a given set to represent coordinates on an architecturally relevant scale;
- calculating all points on an orthogonal intersectional grid derived from the given set;
- repositioning of the intersectional grid on the x- and y-axis according to a chosen vertical and horizontal tolerance value (as a percentage of maximum x- and y- distance within the given set);
- calculating all gradients of all possible connections between points;
- calculating local maxima of the gradients according to a chosen tolerance value;
- assigning neighboring values within that chosen range to the gradients representing local maxima;
- listing the remaining gradients and their weight according to the local maxima calculation;
- counting the remaining gradients and forming the quotients of the number of remaining gradients and the possible number of connections based on the original coordinate list as well as the list of all points on the intersectional grid.

## 3. Results and Discussion

#### 3.1. Box-Counting: Verification of the Data

#### 3.2. Box-Counting Applied to Fractal Curves

#### 3.3. Optimized Settings

- Enlargement factor: 1.15
- Number of boxes for the shortest side: 4 (here the authors made a compromise between the two test series of 3 and 5 starting boxes; during the verification phase it turned out that the former worked well for less complex curves, while the latter gave better results for more complex curves)
- Offsets: 5 × 5 (the authors reduced this number due to performance reasons)
- Steps in between: 5 (this value divides the box size in the x-direction between the largest and the next smaller one, whereby the latter is not allowed to be undercut; hence, it defines a maximum value that can change depending on the proportion of the bounding box; e.g., for a square shape and 4 starting boxes for the smallest side, the next smaller grid size equals 8; accordingly, there are only 3 intermediate steps possible; such an adjustment is automatically done by the program)

- Koch curve (${D}_{Median}$= 1.2639, ${D}_{s}$= 1.2619, 0.16%)
- Koch curve with 40° (${D}_{Median}$= 1.0963, ${D}_{s}$= 1.0986, −0.21%)
- Koch curve with 80° (${D}_{Median}$= 1.6041, ${D}_{s}$= 1.6247, −1.27%)
- Minkowski curve (${D}_{Median}$= 1.4782, ${D}_{s}$= 1.50, −1.45%)
- Sierpinski gasket (${D}_{Median}$= 1.5745, ${D}_{s}$= 1.5850, −0.66%)
- Hilbert curve (${D}_{Median}$= 1.9256, ${D}_{s}$= 2.0, −3.72%)
- Peano curve (${D}_{Median}$= 1.9238, ${D}_{s}$= 2.0, −3.81%)

#### 3.4. Box-Counting Applied to Test Cases

#### 3.5. Box-Counting Applied to Iconic Architecture

#### 3.5.1. Settings

- (1)
- All data points are very close to the regression line, i.e., there is a clear relationship between box size and number of covered boxes over the entire range under consideration.
- (2)
- There are two clearly separable intersecting straight segments, with different gradients. Accordingly, a range with larger boxes can lead to higher fractal dimensions, followed by a section with a smaller fractal dimension.
- (3)
- The data points show a continuous curve without straight segments.

^{®}). Furthermore, the use of Grasshopper

^{®}for an architectural environment is “state of the art” considering the practical introduction of such analyses into the design process.

#### 3.5.2. Measurements

#### 3.6. Gradient Analysis Applied to Test Cases

#### 3.7. Gradient Analysis Applied to Iconic Architecture

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**(

**a**) Hilbert Curve without offsets; (

**b**) Hilbert Curve with 3 offsets; (

**c**) Koch Curve without offsets; (

**d**) Koch Curve with 3 offsets.

**Figure 8.**Results with pre-defined settings showing the difference for 1.1, 1.15 and 1.2% enlargement factor, with (

**a**) showing the medians and (

**b**) showing the results for only smallest number of covering boxes.

**Figure 12.**Analyzed buildings without scale: (

**a**) Robie House, street view with and without window design, FL Wright, 1908, (

**b**) House Steiner, north, east, south, west view, Loos, 1910, (

**c**) House Scheu, north, east, south view, Loos, 1913, (

**d**) Villa Schwob, north-east, north-west, southeast, south-west, Le Corbusier, 1916, (

**e**) House Mandl, Loos, north, east, south, west view, 1916, (

**f**) Villa Savoye, north, east, south, west view, Le Corbusier, 1928.

**Figure 13.**Kink in the data curve between the range of larger and smaller box sizes for House Mandl.

**Figure 14.**Box-plot of measurement series using values of straight data curve. The value in the box plot gives the median, the percentage at the bottom gives the interquartile range in relation to the value two.

**Figure 16.**Comparison of previous measurements (Lorenz 2014) of House Mandl top: (

**a**,

**b**) with current method, bottom: (

**c**,

**d**), with (

**a**,

**c**) giving the box plot of ${D}_{B}$ and (

**b**,

**d**) giving the ranges of box sizes in relation to the building height.

**Figure 18.**Box plot with median of the differential box-counting method of 23 exterior images, 17 detail images and 31 interior images of Robie House.

**Table 1.**Calculation of the box sizes depending on the building height and the distance for the north view of House Steiner.

House Steiner: North Elevation | Angles: | 18° | 27° | 45° |
---|---|---|---|---|

Approximate distances: | 40.50 m | 25.80 m | 13.20 m | |

Distance corresponds to: | Overview | Overview with environment | Single parts | |

0°10′ | 11.80 cm | 7.50 cm | 3.85 cm | |

1°00′ | 0.85 m | 0.50 m | 0.25 m | |

7°26′ | 6.35 m | 3.60 m | 1.75 m | |

10°00′ | 8.55 m | 4.85 m | 2.35 m |

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**MDPI and ACS Style**

Lorenz, W.E.; Kulcke, M.
Multilayered Complexity Analysis in Architectural Design: Two Measurement Methods Evaluating Self-Similarity and Complexity. *Fractal Fract.* **2021**, *5*, 244.
https://doi.org/10.3390/fractalfract5040244

**AMA Style**

Lorenz WE, Kulcke M.
Multilayered Complexity Analysis in Architectural Design: Two Measurement Methods Evaluating Self-Similarity and Complexity. *Fractal and Fractional*. 2021; 5(4):244.
https://doi.org/10.3390/fractalfract5040244

**Chicago/Turabian Style**

Lorenz, Wolfgang E., and Matthias Kulcke.
2021. "Multilayered Complexity Analysis in Architectural Design: Two Measurement Methods Evaluating Self-Similarity and Complexity" *Fractal and Fractional* 5, no. 4: 244.
https://doi.org/10.3390/fractalfract5040244