A parametric analysis of the “digitally-derived geometric design” of the façade of the Macau St. Dominic’s Church

ABSTRACT The Macau St. Dominic’s Church was built in 1588 and renovated in 1828. This ancient building in Macau’s historical town is listed in UNESCO’s World Heritage List. “Digitally-derived Geometric Design” is an important implicit logic of the correlation between numbers and shapes in Western Classic Architecture, and the method of parametric analysis is highly effective in the interpretation of “Digitally-derived Geometric Design”. However, there are limited digital studies on Western Classic Buildings in the Far East and even less conducted from a parametric and detailed perspective. With the façade of the Macau St. Dominic’s Church as the research subject, this study applied the methods of historical document research, field measurement and digital analysis, especially parametric analysis with the assistance of the modeling software of Rhinoceros 6.0 and the software plug-in of Grasshopper, and analyzed the design features of the façade: its upward “successive decrease by equal difference” in stratification, modularization and golden ratio. The “digitally-derived” design pattern of the Macau St. Dominic’s Church is of significant historical value.


Introduction
Macau St. Dominic's Church (also known as Ban Zhang Tang in Chinese) was built in 1588 (Lam 1982, 18) (Domingos and Ka-tseung 1982a) ( Figure 1) and renovated in 1828, laying the foundation for today's church (Valente 1993a) (Figure 2). Most of the existing Western-classical buildings in China were built after the Opium War in 1840. The Macau St. Dominic's Church is a rare Western Baroque style building built before the Opium War (1840) in China. Included in the UNESCO's World Heritage List, this building in Macau's historic town is of significant conservation and research value. Geometrical composition and rational order are beloved and common factors in Western Classic Architecture, while the rule of proportion is an important one in "digitally-derived" design. "Digitallyderived Geometric Design" is an important implicit logic of the correlation between numbers and shapes in Western Classic Buildings, and the origin of the profound "grammar" and "rational composition" behind western classic building. Christianity plays a significant role in Western civilization. In the construction of Western classical architecture in Macao, especially churches, people proficient in mathematics are always chosen to take charge of the planning and design. The favor for geometry comes from the favor for "number" (Ambrosio 1628). However, in the study of Western classical architecture, it's almost impossible to analyze the implicit logic of the correlation between numbers and shapes without useful analytical method. Digital analysis and especially parametric analysis are highly effective in the interpretation of "Digitallyderived Geometric Design".
Professor (Suzuki 1998) had discussed the characteristics of buildings of Baroque Style. However, there are limited digital studies on Western Classic Buildings in the Far East (Ge 2005) and even less conducted from a parametric perspective (Tang 2018(Tang , 2021.

Research objectives and methods
Previous research has found that Macao's Baroque style buildings (churches) are characterized by upward successive decrease in stratification (Tang 2018). However, there is no definite mathematical pattern behind this successive decrease. With the façade of the Macau St. Dominic's Church as the research subject (Figures 2 and 3), this study applied the methods of historical document research, field measurement and digital analysis, especially parametric analysis, and analyzed the design patterns of the "digitally-derived geometric" of the modular, especially the feature of "Upward Successive Decrease by Equal Different" and Golden section and its combination of Macau St. Dominic's Church.
The research methods are listed as follows: b. Field Measurements and Parametric Analyses: this study used aerial photography by unmanned aerial vehicles (DJI Mavic Pro) to generate threedimensional point cloud and manual measurement (accurate to the millimeter) to measure Macau St. Dominic's Church and digitalized it with the assistance of the software AutoCAD while revising them with reference to data acquired from related architectural drawings. This study further conducted a parametric analysis of the design patterns of the Golden Ratio (nestings and combinations), modular, equal partition and successive decrease in the façade with the assistance of the modeling software of Rhinoceros 6.0 and the software plug-in of Grasshopper (using Rhinoceros 6.0 to place the façade and axes used in drawing analyses and using Grasshopper to write programs that composes datamapping diagram).

Historical research
According to A Brief History of Macao, written in 1751, "the original Ban Zhang Tang was said to be low and small, the poor foreigners used planks to build it (Yin and Zhang 1992 (Bry 2000), Macau St. Dominic's Church was composed of three two-storey buildings with gable roof. In front of those buildings was an open square, the St. Dominic's Square. There was a huge cross standing in the center of the square (Figure 1). Macau St. Dominic's Church was rebuilt with bricks and stones in 1721. The following renovation was conducted in 1828 and was hosted by a Spanish priest excel in architecture. This renovation laid the foundation for today's church and the facade of 24 meters high (Valente 1993a

Obtaining the main controlling line in the horizontal direction (Axis ①-⑪) and hereby deducing symmetry, three-segment in the horizontal direction and modular
1) The overall width of the facade: the overall width of Axis ①-⑪ ( Figure 3, Table 1) is 17,100 mm and it is bilaterally symmetric along the Axis ⑥. The facade is also divided into three parts by Axis ④ and Axis ⑧: the left part, the middle part and the right part. If the width of these three parts is respectively: a, b, a (a = 4,500 mm, b = 8,100 mm, a + b + a = 17,100 mm), their geometrical relationship would be (unit: mm, the same below):

Obtaining the main controlling line in the vertical direction (Axis Ⓐ-Ⓝ) and hereby deducing five-segment in the vertical direction and upward "decrease by equal difference" in stratification
1) The Cross Storey (Axis Ⓛ-Ⓝ):  , 1944-1945. 2 The peak of the cross is 22,350 mm high and drawing a horizontal line through this point brings about Axis Ⓝ. Shifting Axis Ⓝ downward by 1,350 mm brings about the Axis Ⓜ, which is the bottom edge of the cross. Shifting Axis Ⓜ downward by 800 mm is Axis Ⓛ, which is the peak of the triangle pediment. The height h5 of the Cross Storey is 1,350 mm + 800 mm = 2,150 mm.
2) The Triangle Pediment Storey (Axis Ⓚ-Ⓛ): Shifting Axis Ⓛ downward by 3,250 mm (the height h4 of the Pediment Storey) brings about the bottom edge of the pediment, Axis Ⓚ. The triangle pediment is bilaterally symmetric along the Axis ⑥. Connecting K3 and L6 as well as K9 and L6 brings about the slopes of the pediment rooftop. The slope gradient is 1.677, close to the Golden Ratio (the height is 3,250 mm and the bottom line is 5,450 mm × 2).
3) The third Storey (Axis Ⓖ-Ⓚ): Shifting Axis Ⓚ downward by 2,250 mm brings about the central line of the oval decoration, Axis Ⓙ, which almost equally divided the third storey in the vertical direction. Shifting Axis Ⓙ downward by 1,240 mm brings about the top edge of the column base in the third storey, Axis Ⓗ. Shifting Axis Ⓗ downward by 1,060 mm (the height of the column base in the third storey, less than the one in the second storey, which is 1,090 mm) brings about the separating line of the Second and Third Storey, Axis Ⓖ. The height h3 of the Third Storey is 2,250 mm + 1,240 mm + 1,060 mm = 4,550 mm.
4) The second Storey (Axis Ⓓ-Ⓖ): Shifting Axis Ⓖ downward by 650 mm brings about Axis Ⓕ, the horizontal line of the bottom edge of the eaves in the second storey. Shifting Axis Ⓕ downward by 3,910 mm brings about Axis Ⓔ, the top edge of the column base in the second storey. Shifting Axis Ⓔ downward by 1,090 mm (the height of the column base in the second storey, less than the one in the first storey, which is 1,380 mm) brings about Axis Ⓓ, the separating line of the First and Second Storey. The height h2 of the Second Storey is 650 mm + 3,910 mm + 1,090 mm = 5,650 mm.
5) The First Storey (Axis Ⓐ-Ⓓ) Shifting Axis Ⓓ downward by 670 mm brings about Axis Ⓒ, the horizontal line of the bottom edge of the eaves in the first storey. Shifting Axis Ⓒ downward by 4,700 mm brings about the Axis Ⓑ, the top edge of the column base in the first storey. Shifting Axis Ⓑ downward by 1,380 mm brings about Axis Ⓐ, the Ground. The height h1 of the First Storey is 670 mm + 4,700 mm + 1,380 mm = 6,750 mm.  Thus, all the axes in the vertical direction, Axis Ⓐ-Ⓝ, and all the axes in the horizontal direction, Axis ①-⑪ are obtained (Table 1).

The partition patterns in the overall dimension of the facade
In the horizontal direction, Axis ④ and ⑧ divide the facade into three parts: the left, middle and right part. The facade is bilaterally symmetric along Axis ⑥. The height and width of the middle doorways in each storey are bigger than those in the left and right one. The width of the double-column in the middle (Modular e) is bigger than the ones in the right and left part (Modular c).
The horizontal width of the facade narrows from "a + b + a" in the Third Storey to "a" through the circular arcs on the two sides and continued to narrow in the Pediment Storey by the slope of the Golden Ratio until the Cross Storey. More spectacularly, the facade is stratified into five parts in the vertical direction while the height of each part is upward successively decreased by equal difference. The height is successively decreased by the equal difference of 1,100 mm in the First/Second/ Third storey (the height h1 of the First Storey minus 1,100 mm is equal to the height h2 of the Second Storey; the height h2 of the Second storey minus 1,100 mm is equal to the height h3 of the Third storey). The difference between the height h3 of the Third storey and the height h4 of the Pediment Story is 1,300 mm (the height h3 of the Third Storey minus 1,300 mm is equal to the height h4 of the Pediment Storey). The difference between the Cross Story and the Pediment Story returns to 1,100 mm (the height h4 of the Pediment Storey minus 1,100 mm is equal to the height h5 of the Cross Story).
After the analyses of the partitions of the facade in the horizontal and vertical direction (Axes) ( Table 1), this study continues to conduct a parametric analysis of the diagonal lines (gradient of slopes) and successive partitions (spiral lines).
[The golden rectangle of none-partition is marked by one solid diagonal line; the golden rectangle of onetime partition is marked by two diagonal lines (dotted lines); the golden rectangle of two-time partition is marked by golden spiral lines (dotted lines)].

Assessment of "Golden Rectangles+Squares" or "Root Rectangles" in the composition of the facade
There are mainly "Golden Rectangles + Squares" and "Root Rectangles" in the proportion partition in Western Classic Architecture.
Golden Section and Fibonacci Sequence (Figure 4): A Golden Rectangle could be divided into a subordinate golden rectangle and a square in succession, that is to say "A main golden rectangle = a subordinate golden rectangle + a square". Such divisions bring about a series of mutually perpendicular/ mutually paralleled sidelines, diagonals and golden spirals. The mathematical expression of Golden Ratio is ffi ffi 5 p þ1 2 ¼ 2 ffi ffi 5 p À 1 � 1:618. The Fibonacci Sequence, also known as the Golden Section Sequence, is: 1, 1, 2, 3, 5, 8, 13, 21, 34 . . . . . .
In actual construction, architects usually take round number in measurement so the Golden Ratio applied in actual construction is usually a range rather than an exact number like 1.618. In this study, the author takes the range between 3:2 (1.5) and 5:3 (1.67) in the Fibonacci Sequence as the range of the Golden Ratio (Range Value: 0.17).
Root Rectangle Composition ( Figure 5): Root Rectangle Section is one of the partition methods. They are mainly ffi ffi ffi 2 p ; ffi ffi ffi 3 p ; ffi ffi ffi 4 p ; ffi ffi ffi 5 p . . . rectangle composition.
In this study, the author imports the facade into the Rhino 6.0 software and writes parametric programs with the assistance of the software plug-in of Grasshopper. Based on the analysis of the slope k of   , this study finds that the nodes and sides of golden rectangles have the largest numbers and coverage area and are usually the main controlling lines of the division of the facade, and that these golden rectangles bring about golden spirals through successive partition.
Accordingly, this study concludes that Golden Ratio is an important foundation of the partition of the facade of the Macau St. Dominic's Church.

A parametric analysis of the façade: the complicated combination of rectangles and spirals controlled by golden ratio
(1) Golden Rectangles and Spirals from the Twotimes Partition ( Table 2) 1) Main Rectangle 1 = Sub-rectangle 1 + Square 1; Sub-rectangle 1 = Sub-rectangle 1.1 + Square 1.1 Main Rectangle 1 (A1-A7-K7-K1): Line A1-A7 is 10,450 mm long and Line A1-K1 is 16,950 mm long. Their proportion is 1.62, approximate to the Golden Ratio, that is to say that the proportion of the distance (a + e + 200 + f + 200) between the left side line of the facade and the left side line of the double column on the right side of the middle doorway to the combined height (h1+ h2+ h3, three times the height of the Second Storey, 3× h2) of the First, Second and Third Storey is in accordance with the Golden Ratio, which brings about Main Golden Rectangle 1.
Sub-rectangle1 (A1-A7-D7-D1): Line A1-A7 is 10,450 mm long and Line A1-D1 is 6,750 mm long. Their proportion is 1.55, approximate to the Golden Ratio, that is to say that the proportion of the distance between the left side line of the facade and the left side line of the double column on the right side of the middle doorway to the height h1 of the First Storey is in accordance with the Golden Ratio, which brings Sub-golden-rectangle 1.
Square 1 (D1-D7-K7-K1): Line D1-D7 is 10,450 mm long and Line D1-K1 is 10,200 mm long. Their proportion is approximate to 1:1, the one of a square, that is to say that the proportion of the distance between the left side line of the facade and the left line of the double column on the right side of the middle doorway to the combined height (h2+ h3) of the Second and Third Storey is in accordance with 1:1, which brings about Square 1.
Sub-rectangle 1.1 (A5-A7-D7-D5): Line A5-A7 is 3,800 mm long and Line A5-D5 is 6,750 mm long. Their proportion is 1.776, approximate to the Golden Ratio, that is to say that the proportion of the distance (200+ the width f of the middle doorway+200) between the double columns on both sides of the middle doorway to the height h1 of the First Storey is in accordance with the Golden Ratio, which brings about Sub-golden-rectangle. 1.1.
Square 1.1 (A1-A5-D5-D1): Line A1-A5 is 6,650 mm long and Line A1-D1 is 6,750 mm long. Their proportion is approximate to 1:1, the one of a square, that is to say that the proportion of the distance (a + e) between the left side line of the facade and the right side line of the double column on the left side of the middle doorway to the height h1 of the First Storey is in accordance with 1:1, which brings about Square 1.1. (2) Golden Rectangles from the One-time Partition ( Table 2) 1) Main Rectangle 2 = Sub-rectangle 2 + Square 2 Main Rectangle 2 (A1-A7-D7-D1): Line A1-A7 is 10,450 mm long and Line A1-D1 is 6,750 mm long. Their proportion is 1.55, approximate to the Golden Ratio, that is to say that the proportion of the distance (a + e + 200 + f + 200) between the left side line of the facade and the left side line of the double column on the right side of the middle doorway to the height h1 of the First Storey is in accordance with the Golden Ratio, which brings about Main Golden Rectangle 2.
Sub-rectangle 2 (A1-A4-D4-D1): Line A1-A4 is 4,500 mm long and Line A1-D1 is 6,750 mm long. Their proportion is 1.50, approximate to the Golden Ratio, that is to say that the proportion of the width a of the left part of the facade (the distance between the left side line of the facade and the left side line of left double column of middle doorway) to the height h1 of the First Storey is in accordance with the Golden Ratio, which brings about Sub-golden-rectangle 2.
Square 2 (A4-A7-D7-D4): Line A4-A7 is 5,950 mm long and Line A4-D4 is 6,750 mm long. Their proportion is approximate to 1:1, that is to say that the proportion of the distance (e + 200 + f + 200) between the left side line of the double column on the left side of the middle doorway and the left side line of the double column on the right side of the middle doorway to the height h1 of the First Storey is in accordance with 1:1, which brings about Square 2.
2) Main Rectangle 3 = Sub-rectangle 3 + Square 3 Main Rectangle 3 (D1-D11-K11-K1): Line D1-D11 is 17,100 mm long and Line D1-K1 is 10,200 mm long. Their proportion is 1.676 (5:3), approximate to the Golden Ratio, that is to say that the proportion of the overall width (a + b + a) of the facade to the combined height (h2+ h3) of the Second and Third Storey is in accordance with the Golden Ratio, which brings about Main Golden Rectangle 3. Sub-rectangle 3 (D7-D11-K11-K7): Line D7-D11 is 6,650 mm long and Line D7-K7 is 10,200 mm long. Their proportion is 1.53, approximate to the Golden Ratio, that is to say that the proportion of the distance (e + a) between the left side line of the right double column of middle doorway and the right side line of the facade to the combined height (h2+ h3) of the Second and Third Storey is in accordance with the Golden Ratio, which brings about Sub-golden-rectangle 3.
Square 3 (D1-D7-K7-K1): Line D1-D7 is 10,450 mm long and Line D1-K1 is 10,200 mm long. Their proportion is approximate to 1:1, the one of a square, that is to say that the proportion of the distance (a + e + 200 + f + 200) between the left side line of the facade and the left side line of the double column on the right side of the middle doorway to the combined height (h2+ h3) of the Second and Third Storey is in accordance with 1:1, which brings about Square 3.
3) The Main Rectangle 4 = Sub-rectangle 4 + Square 4 Main Rectangle 4 (D6-D11-M11-M6): Line D6-D11 is 8,500 mm long and Line D6-M6 is 14,250 mm long. Their proportion is 1.67 (5:3), approximate to the Golden Ratio, that is to say that the proportion of the half of the overall width of the facade to the combined height of the Second, Third and Pediment Storey as well as the base of the Cross is in accordance with the Golden Ratio, which brings about Main Golden Rectangle 4.
Sub-rectangle 4 (D6-D11-G11-G6): Line D6-D11 is 8,550 mm long and Line D6-G6 is 5,650 mm long. Their proportion is 1.51, approximate to the Golden Ratio, that is to say that the proportion of the half of the overall width of the facade to the height h2 of the Second Storey is in accordance with the Golden Ratio, which brings about Sub-golden-rectangle 4.
Square 4 (G6-G11-M11-M6): Line G6-G11 is 8,550 mm long and Line G6-M6 is 8,600 mm long. Their proportion is approximate to 1:1, the one of a square, that is to say that the proportion of the half of the overall width of the facade to the combined height of the Third and Pediment Storey as well as the base of the Cross is in accordance with 1:1, which brings about Square 4.
4) The Main Rectangle 5 = Sub-rectangle 5 + Square 5 Main Rectangle 5 (D4-D9-G9-G4): Line D4-D9 is 9,500 mm long and Line D9-G9 is 5,650 mm long. Their proportion is 1.68, approximate to the Golden Ratio, that is to say that the proportion of the distance between the left side line of double column on the left side of middle doorway and the center line of the right doorway to the height h2 of the Second Storey is in accordance with the Golden Ratio, which brings about Main Golden Rectangle 5.
Sub-rectangle 5 (D7-D9-G9-G7): Line D7-D9 is 3,550 mm long and Line D9-G9 is 5,650 mm long. Their proportion is 1.59, approximate to the Golden Ratio, that is to say that the proportion of the distance between the left side line of the right column of the middle doorway and the central line of the right doorway to the height h2 of the Second Storey is in accordance with the Golden Ratio, which brings about Subgolden-rectangle 5.
Square 5 (D4-D7-G7-G4): Line D4-D7 is 5,950 mm long and Line D4-G4 is 5,650 mm long. Their proportion is approximate to 1:1, the one of a square, that is to say that the proportion of the distance (e + 200 + f + 200) between the left side line of the double column on the left side of middle doorway and the left side line of the double column on the right side of middle doorway to the height h2 of the Second Storey is in accordance with 1:1, which brings about Square 5.
The golden rectangles, spirals and their partitions are bilaterally symmetric along the Medial Axis with another correspondent set of golden rectangles, spirals and partitions. Accordingly, it could be concluded that the Golden Ratio is one of the important underlying basis of the design drawing and lines and number-rounding adjustments of the controlling points.

Conclusions
According to previous studies, the characteristic of upward successive-subtraction in stratification also exists in other Baroque-style buildings such as the Church of St. Paul in Macau (Tang, 2018) the same as (Tang 2018). Moreover, the upward "Successive Decrease by Equal Difference" in Stratification in the facade of Macau St. Dominic's Church has embodied high-level mathematical design. The underlying aesthetics-mathematics pattern of "digitally-derived geometric design" is of unparalleled historical value: 1) Its designer applied three-section upward narrowing in the horizontal directions and especially five-section stratification upward successive decrease by equal difference in stratification in height in the overall dimension of the façade. The height of the First, Second, Third, Pediment and Cross Storey have successive decrease by equal difference in stratification in sequence; the subtraction only deviates between the Pediment and Third Storey.
2) Its designer applied modularization (modular) design and combination in details of the columns and doorways.
3) Parametric analyses with the assistance of the modeling software of Rhinoceros 6.0 and the software plug-in of Grasshopper found that the (successive) partition of the façade is closer to the "Golden Section" than "Root Rectangle Partition", and further analyses lead to Golden Rectangle (one-time division) and Golden Spiral (successive division) in the façade.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work is funded by the National Social Science Fund of China [Grants No. 20BZJ026]. Gratitude should also be extended to Dufeng Yu; National Office of Philosophy and Social Science of China [The National Social Science Fund of China 20BZJ026].