Mathematics Analysis and Approaches Internal Assessment Standard Level

Investigating the Presence of the Golden Ratio in Architectural Landmarks

Research question: To what extent does the design of architectural landmarks align with the Golden Ratio?

1. Introduction

Context and Relevance

The Golden Ratio of 1.618 is widely associated with beauty and harmony,presenting the unique visual enjoyment in nature. After finding out the mystery of Golden ratio, more designers apply this principle in their works like architecture and painting.

As an IB Visual Arts student, I compared the paintings of artists who specialised in using  the Golden Ratio to create beauty in their compositions. I was inspired by the Golden Ratio and combined it with mathematical principles to bring out how it has influenced art and architectural design. It has been historically associated with both natural phenomena and man-made structures. From the Parthenon in Ancient Greece to the Taj Mahal in India, the use of the Golden Ratio has been the subject of debate. Due   to its abstract nature, some scholars have argued that its appearance is a purposeful design choice, while others insist that its appearance is a mathematical coincidence   or a man-made explanation. By examining these claims, this study reveals the connection between mathematics and art in world-famous architecture.

Objective

This investigation explores whether famous architectural structures exhibit

dimensions approximating the Golden Ratio ( ϕ ) and whether this alignment is

intentional or coincidental. By analyzing key dimensions in several landmarks, this paper is aimed to provide geometrical insights into the historical and aesthetic role of typical architectural design.

Mathematical Tools

Using ratios of the architectures to show the proportions of the line segments in the buildings. GeoGebra were used for geometric analysis, while spreadsheets were employed for ratio calculations and statistical analysis. This investigation measures and compares the dimensions from selected landmarks to ϕ, the alignment and intentionality are measured in terms of deviations from ϕ .

Background theory

2.1 Definition of the Golden Ratio

The Golden Ratio is defined as:

The golden ratio (ϕ), also known as Phi, is derived from the Fibonacci sequence and represents a unique proportionality where the ratio of larger to smaller parts is the same as the ratio of the whole to the larger parts.Where a and b are lengths such that a > b > 0. By deducing the formula,  ϕ can be defined as the terms of itself^[1].

Table 1 Calculation of ϕ

The ratio ϕ is unique because of its self-similarity and its role in the Fibonacci  sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55..., where the ratio of successive terms converges to ϕ. As larger numbers are used, the ratio between two number is more close to 1.618,for instance 21/13=1.615,55/34=1.618.

When the golden ratio is extended to two dimensional representations (xy, xz, or yz), the golden rectangle is obtained. A rectangle is considered a golden rectangle if the ratio of its long side to its short side is equal to φ. More geometric shapes are created,like the Golden Triangle, Golden Rectangle and Golden Spiral^[2]. A golden rectangle refers to a rectangle with side lengths in the golden ratio, namely one to φ (phi), approximately in the ratio of 1:1.618.

The golden rectangle has never been utilized to analyze the generative two dimensional elements of the three dimensional forms of buildings. Nearly all analyses in this field were centered on the overall dimensions of the elevation or the arrangements of the two dimensional elements, with the third dimension being neglected. Golden Spiral is based on the proportion of the golden ratio,this is made by doubling/having the square. The irrational number combined with the symmetry, so that the diagonal of the halves will be further produce a new square  by extension in the ratio 0.618 to the original width of the previous square. This results in the rectangle which the ratio of width and ratio is 1:1.618^[1].

2.2 Applications in Architecture

Throughout history, architects, philosophers, and mathematicians also used and recommended other proportional systems. For instance, Plato attached great significance to two geometrical ratios: the double progression of 1:2:4:8 and the triple progression of 1:3:9:27, which are applied in musical proportions.

Architectural design often reflects proportional systems, with the Golden Ratio being a popular candidate for aesthetic harmony. Since the golden section were  shown in many nature beauty such as flowers,seashells as well as some fruits, some artists may even refer to the nature to design. In many architecture design, proportional ratios are included to describe the insights since the qualities and quantities are perceived indirectly.

Theoretically, ϕ can be applied to facades, room dimensions, and structural layouts. It can be easily found the irrational number and the 5:3 or 8:5 or 4:9 in most buildings[4]. Historically, Renaissance architects like Leonardo da Vinci and Palladio explored ϕ in their work. This investigation examines whether such proportions persist in selected landmarks.For example, the gigantic pyramids of the ancient Egypt, one of the seven wonders in the world, is also followed the golden ratio.It is designed that the Khafre’s pyramid is perfectly corresponds to the right triangle with the side ratios. Artists can change the number, scale, and location of the golden rectangles to generate the new 3-D shapes with more various changes. The two dimensional shapes can be deduced from these golden rectangles to create the composition.

3 Methodology

3.1 Selection of Structures

Below are the chosen buildings to be investigated around the world at different stages of time, which are The Parthenon (Greece), The Taj Mahal (India) and the The Guggenheim Museum (USA)

The Parthenon is one of the well-known ancient geometric example of the golden spiral and  it is also the typical building presenting the Greek temples. Thales and Pythagoras introduce the basic theory of mathematics and geometry, which better promote the application of the Golden section^[1].

The Taj Mahal, a mausoleum made of white marble, was constructed from 1631 to 1648. It was constructed because  Mughal emperor Shah Jahan in remembrance of his beloved wife. It is a 22 feet high and 313 feet square platform. with corner minarets 137 feet tall and 81 feet high and 58 feet in diameter central inner dome. The main inner chamber is an octagon with 7.3-metre (24 ft) sides, with the design allowing for entry from each face with the main door facing the garden to the south.

The Guggenheim Museum in USA is one of the famous architecture applied the Golden Ratio. It is a triangular gallery inside with six floors.And what is intersting is that the its lengthe and the width will increase as the floor goes up. For example, the width increases from 25 feets to 32 feets from the lowest floor to the highest one(wiki). It provides a significant contrast with its surrounding buildings because of the spiral form. with good combination of triangles, circles and squares which correspond to the concept of organic architecture.

Table 2 Chosen architectures

3.2 Data Collection

After identifying the specific building for study, I began to obtain measurements of salient architectural details (e.g., elevation dimensions, column ratios, and window ratios) from building plans, scaled images, and credible online references. If specific dimensions were not available, approximate measurements were taken using scaling tool software.

3.3 Analysis Method:

The analysis method is to collect the database of the building as soon as possible ,so that some measurements can be calculated or deduced. By identifying key dimensions (a and b) for each structure,for example, the length and the width or the heights .Calculate the ratio b/a and compare it to \phi . Using the formula below to quantify deviations to the Golden ratio 1.618.

This method can be further used to plot the histogram and the pictures using more mathematical tools.

4. Results

Below are some Golden Rectangles with the length to width of ϕ using Desmos. Similarly, when designing the three-dimension draft of the building, calculate the size at different scale factor considering the harmony  with the the symmetry and geometry would be an efficient way to design.

Figure 1 Golden Rectangle generated by Desmos

Figure 2 The Golden Spiral and the expand

In addition, the Golden Spiral can be created via the formula and the ratio in the Golden Rectangle. The golden spiral can be drawn with squares. By drawing squares with side lengths following the Fibonacci sequence and connecting the adjacent corners of consecutive squares with quarter arcs. When more squares and arcs are added, the golden spiral were shown.

4.1 Parthenon

Most of its proportions followed a ratio of 9:4(≈1.44). Its height, from the level of the stylobate to the top of the pediments was 13.72 m.Dimension of the Stylobate of another ancient temple at the same period were found: 30.88 m x 69.50 m，and the axial spacing external columns are 4.29 m fronts (3.68m corners) and 4.29m flanks (3.69m corners).The lower diameter exterior columns are 1.91m(1.95m corners) and the height exterior columns are 10.43m; height entablature are 3.30m. The deviation can be calculated as below.

These two pictures show the fitted Golden Spiral and Golden rectangle presented in the building, not only the part and the whole indicates the proportions but also the vertical and horizontal elements combined to present the geometric myth. The width of the frontal columns is 1.9m while the spacing between two columns is 4.3 m, leading to a ratio of 4.3/1.9 = 2.26, which is very close to 9/4^[3].

https://www.goldennumber.net/parthenon-phi-golden-ratio/

The modeling of Parthenon are shown below. Some coordinate are picked to model the dimensions of the building not on the scale, which are (0,140),(0,200) (0,278),(253,278),(175,278),then can calculate the length ratio

Figure 3 Length ratio modeling of Parthenon

Using the spiral equation to model fit in Desmos,

by adjusting the parameters, part of the building can be fitted in the the equation r = — 4.7ebθ, and the b= , in which the is about 1.674.The  deviation(%) = = 3.4%.

Figure 4 Spiral modeling of Parthenon

Following draft shows how the technique is used to design the Parthenon by Leonardis^[1], showing how the square changed by the sides and create the Golden Section. In his works, y doing the calculation and interpretation, the dimension of the stylobate of 69.503m dinsmoor field and 68.911m in dinsmoor calculation. It is clear that the width of the temple inside is one half to square of the square on the stylobate. Besides, the setbacks on the gable of the room are almost one fourth square of the 1/2 square on the krepidoma.

Figure 5 Skteching deducing of the ancient temple