# Towards an Effective Architectural Form: The Composition of Squareness and Roundness Based on Scale Proportion—Evidence from the Yingxian Wooden Pagoda

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## Abstract

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## 1. Introduction

## 2. Theoretical Framework

#### 2.1. Theoretical Basis

#### 2.2. Technical Path

## 3. Methodology

#### 3.1. Case Study

#### 3.2. Data

**Plan Data:**Each floor of the wooden pagoda has an octagonal plan with an inner and outer ring of columns, one column at each angle of the inner groove, and one space per face; two additional columns are added between the two corner columns of the outer groove on each face, making three spaces per face, totaling thirty-two columns used. The planar column grid data involved in this paper follow previous conventions, which are based on the average values of the eight sides for each column’s top plan (Because the plan of the columns’ top of ancient wooden building is less affected by the structural deformation under the pulling knots of the Pu Pai Fang and lintel), including the side length, ${W}_{i}$; the middle bay span, ${WM}_{i}$; the bilateral bay span, ${WE}_{i}$; and the planar width, ${L}_{i}$ (Figure 3).

**Vertical Data:**These consist of storied heights and segmented heights. First, the area from the bottom of the first-floor eave column to just under the fifth-floor roof ridge is divided into five stories based on the upper edge of the Pu Pai Fang on the outer eave of each mezzanine story, with storied heights, ${H}_{i}$. Secondly, each story includes four sections: the mezzanine part, ${A}_{i}$, the column and lintel part, ${B}_{i}$, the bracket part, ${C}_{i}$, and the roof frame part, ${D}_{i}$, with the first floor lacking a mezzanine part. Additionally, the base height is ${t}_{0}$, the top height of the pagoda is ${H}_{6}$, and the total height of the pagoda is H (Figure 2).

#### 3.3. Methods

#### 3.3.1. Analysis of Architectural Scale

#### 3.3.2. Analysis of Architectural Modulus and Proportion

#### 3.3.3. Analysis Combined with Composition of Squareness and Roundness

## 4. Results

#### 4.1. Architectural Scale Constituent Characteristics

#### 4.1.1. Construction Ruler of Yingxian Wooden Pagoda

#### 4.1.2. Planar Scale

#### 4.1.3. Vertical Scale

#### 4.2. Modular System and Proportional Rules

#### 4.3. Control Principles of Squareness and Roundness Composition

#### 4.3.1. Analysis Results of the Overall Appearance

#### 4.3.2. Analysis Results of the Single Facade

#### 4.3.3. Summary of Squareness and Roundness Composition Principles

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) “Rounded-square and Squared-circle Diagrams” in the ancient Chinese astronomical and mathematical text “Zhou Bi Suan Jing”; (

**b**) “Rounded-square and Squared-circle Diagrams” in the Northern Song Dynasty architectural official book “Yingzao Fashi”.

**Figure 2.**(

**a**) Realistic view of Yingxian Wooden Pagoda; (

**b**) Schematic diagram of main data related to this paper.

**Figure 4.**Schematic diagram of the basic principle and operation steps for squareness and roundness composition controlling the form of multi-story buildings.

**Figure 5.**Modular grid analysis of the bracket set on columns of each exposed story of Yingxian Wooden Pagoda.

**Figure 6.**Modular grid analysis of architectural group layout in the general plane of the temple pagoda.

**Figure 10.**Squareness and roundness composition analysis of Yingxian Wooden Pagoda—overall appearance form.

**Figure 11.**Squareness and roundness composition analysis of Yingxian Wooden Pagoda—single facade form.

**Table 1.**Main scale constitution of the exposed story plan of Yingxian Wooden Pagoda (1 foot = 292 mm).

No. | Position | Data | ${\mathbf{W}\mathbf{M}}_{\mathbf{i}}$ | $2{\mathbf{W}\mathbf{E}}_{\mathbf{i}}$ | ${\mathbf{W}}_{\mathbf{i}}$ | ${\mathbf{L}}_{\mathbf{i}}$ |
---|---|---|---|---|---|---|

01 | Bottom Porch | Actual Value/mm | 4452 | 8034 | 12,486 | 29,965 |

Converted Value/feet | 15.246 | 27.514 | 42.760 | 102.619 | ||

Restored Value/feet | 15.25 | 27.5 | 42.75 | 102.6 | ||

Conformity | 99.98% | 99.95% | 99.97% | 99.98% | ||

02 | First floor | Actual Value/mm | 4228 | 5336 | 9564 | 23,075 |

Converted Value/feet | 14.479 | 18.273 | 32.753 | 79.024 | ||

Restored Value/feet | 14.5 | 18.25 | 32.75 | 79 | ||

Conformity | 99.86% | 99.87% | 99.99% | 99.97% | ||

03 | Second floor | Actual Value/mm | 4162 | 5020 | 9182 | 22,074 |

Converted Value/feet | 14.253 | 17.192 | 31.445 | 75.596 | ||

Restored Value/feet | 14.25 | 17.25 | 31.5 | 76 | ||

Conformity | 99.98% | 99.66% | 99.83% | 99.47% | ||

04 | Third floor | Actual Value/mm | 3798 | 5034 | 8832 | 21,317 |

Converted Value/feet | 13.007 | 17.240 | 30.247 | 73.003 | ||

Restored Value/feet | 13 | 17.25 | 30.25 | 73 | ||

Conformity | 99.95% | 99.94% | 99.99% | 99.99% | ||

05 | Fourth floor | Actual Value/mm | 3725 | 4738 | 8463 | 20,451 |

Converted Value/feet | 12.757 | 16.226 | 28.983 | 70.038 | ||

Restored Value/feet | 12.75 | 16.25 | 29 | 70 | ||

Conformity | 99.95% | 99.85% | 99.94% | 99.95% | ||

06 | Fifth floor | Actual Value/mm | 3649 | 4349 | 7998 | 19,276 |

Converted Value/feet | 12.503 | 14.894 | 27.390 | 66.014 | ||

Restored Value/feet | 12.5 | 14.9 | 27.4 | 66 | ||

Conformity | 99.98% | 99.96% | 99.97% | 99.98% |

No. | Story | Segment | Height Value | |||||
---|---|---|---|---|---|---|---|---|

Actual Value/mm | Converted Value/Feet | Story Height/Feet | Basic Module M = 5 Feet | |||||

01 | $\mathrm{Bottom}\mathrm{porch}{\mathrm{H}}_{0}$ | $\mathrm{Column}\mathrm{and}\mathrm{Lintel}\mathrm{Part}{\mathrm{B}}_{0}$ | 4377 | 14.99 | H_{0} ≈ 30(Conformity 99.94%) | 3 M | 3 M | 6 M |

$\mathrm{Bracket}\mathrm{Part}{\mathrm{C}}_{0}$ | 1023 | 3.5 | 0.7 M | 3 M | ||||

$\mathrm{Roof}\mathrm{Frame}\mathrm{Part}{\mathrm{D}}_{0}$ | 3365 | 11.52 | 2.3 M | |||||

02 | $\mathrm{First}\mathrm{floor}{\mathrm{H}}_{1}$ | $\mathrm{Columnand}\mathrm{Lintel}\mathrm{Part}{\mathrm{B}}_{1}$ | 8726 | 29.88 | H_{1} ≈ 45(Conformity 99.58%) | 6 M | 6 M | 9 M |

$\mathrm{Bracket}\mathrm{Part}{\mathrm{C}}_{1}$ | 1463 | 5.01 | 1 M | 3 M | ||||

$\mathrm{Roof}\mathrm{Frame}\mathrm{Part}{\mathrm{D}}_{1}$ | 2896 | 9.92 | 2 M | |||||

03 | $\mathrm{Sec}\mathrm{ond}\mathrm{floor}{\mathrm{H}}_{2}$ | $\mathrm{Mezzanine}\mathrm{Part}{\mathrm{A}}_{2}$ | 1475 | 5.05 | H_{2} ≈ 30(Conformity 99.77%) | 1 M | 3 M | 6 M |

$\mathrm{Column}\mathrm{and}\mathrm{Lintel}\mathrm{Part}{\mathrm{B}}_{2}$ | 2907 | 9.96 | 2 M | |||||

$\mathrm{Bracket}\mathrm{Part}{\mathrm{C}}_{2}$ | 1468 | 5.03 | 1 M | 3 M | ||||

$\mathrm{Roof}\mathrm{Frame}\mathrm{Part}{\mathrm{D}}_{2}$ | 2930 | 10.03 | 2 M | |||||

04 | $\mathrm{Third}\mathrm{floor}{\mathrm{H}}_{3}$ | $\mathrm{Mezzanine}\mathrm{Part}{\mathrm{A}}_{3}$ | 1473 | 5.04 | H_{3} ≈ 30(Conformity 99.43%) | 1 M | 3 M | 6 M |

$\mathrm{Column}\mathrm{and}\mathrm{Lintel}\mathrm{Part}{\mathrm{B}}_{3}$ | 2891 | 9.90 | 2 M | |||||

$\mathrm{Bracket}\mathrm{Part}{\mathrm{C}}_{3}$ | 1457 | 4.99 | 1 M | 3 M | ||||

$\mathrm{Roof}\mathrm{Frame}\mathrm{Part}{\mathrm{D}}_{3}$ | 2889 | 9.89 | 2 M | |||||

05 | $\mathrm{Fourth}\mathrm{floor}{\mathrm{H}}_{4}$ | $\mathrm{Mezzanine}\mathrm{Part}{\mathrm{A}}_{4}$ | 1489 | 5.10 | H_{4} ≈ 28(Conformity 99.87%) | 1 M | 3 M | 5.6 M |

$\mathrm{Column}\mathrm{and}\mathrm{Lintel}\mathrm{Part}{\mathrm{B}}_{4}$ | 2876 | 9.85 | 2 M | |||||

$\mathrm{Bracket}\mathrm{Part}{\mathrm{C}}_{4}$ | 1028 | 3.52 | 0.7 M | 2.6 M | ||||

$\mathrm{Roof}\mathrm{Frame}\mathrm{Part}{\mathrm{D}}_{4}$ | 2772 | 9.49 | 1.9 M | |||||

06 | Fifth floor ${\mathrm{H}}_{5}$ | $\mathrm{Mezzanine}\mathrm{Part}{\mathrm{A}}_{5}$ | 1175 | 4.02 | H_{5} ≈ 37(Conformity 99.92%) | 0.8 M | 2.7 M | 7.4 M |

$\mathrm{Column}\mathrm{and}\mathrm{Lintel}\mathrm{Part}{\mathrm{B}}_{5}$ | 2763 | 9.46 | 1.9 M | |||||

$\mathrm{Bracket}\mathrm{Part}{\mathrm{C}}_{5}$ | 892 | 3.05 | 0.6 M | 4.7 M | ||||

$\mathrm{Roof}\mathrm{Frame}\mathrm{Part}{\mathrm{D}}_{5}$ | 5965 | 20.43 | 4.1 M | |||||

07 | Top of the pagoda ${\mathrm{H}}_{6}$ | $\mathrm{Roof}\mathrm{Ridge}{\mathrm{t}}_{1}$ | 1170 | 4.01 | H_{6} ≈ 10.5(Conformity 99.54%) | 0.8 M | 2.1 M | 2 M |

$\mathrm{Pinnacle}\mathrm{Base}{\mathrm{t}}_{2}$ | 1910 | 6.54 | 1.3 M | |||||

$\mathrm{Pagoda}\mathrm{Pinnacle}{\mathrm{t}}_{3}$ | 9520 | t_{3} ≈ 32.5 (Conformity 99.68%) | 6.5 M | 9 M | ||||

08 | Others | $\mathrm{Pagoda}\mathrm{Base}{\mathrm{t}}_{0}$ | 3635 | h_{0} ≈ 12.5 (Conformity 99.59%) | 2.5 M | |||

09 | Total Height of the Pagoda H | 65,770 | H ≈ 225 (Conformity 99.89%) | 45 M |

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## Share and Cite

**MDPI and ACS Style**

Shi, L.; Chen, X.; Xu, Y.; Gao, X.; Lai, J.; Wang, S.
Towards an Effective Architectural Form: The Composition of Squareness and Roundness Based on Scale Proportion—Evidence from the Yingxian Wooden Pagoda. *Buildings* **2024**, *14*, 1472.
https://doi.org/10.3390/buildings14051472

**AMA Style**

Shi L, Chen X, Xu Y, Gao X, Lai J, Wang S.
Towards an Effective Architectural Form: The Composition of Squareness and Roundness Based on Scale Proportion—Evidence from the Yingxian Wooden Pagoda. *Buildings*. 2024; 14(5):1472.
https://doi.org/10.3390/buildings14051472

**Chicago/Turabian Style**

Shi, Lu, Xu Chen, Yuqian Xu, Xing Gao, Jialong Lai, and Shusheng Wang.
2024. "Towards an Effective Architectural Form: The Composition of Squareness and Roundness Based on Scale Proportion—Evidence from the Yingxian Wooden Pagoda" *Buildings* 14, no. 5: 1472.
https://doi.org/10.3390/buildings14051472