Mathematical exploration: The mathematical aspect of simultaneously liking and disliking works of the same artist

Introduction

As a student of the sciences with a deep interest in programming and mathematics, my natural instinct has been to quantify the world and reduce everything into a formula or an algorithm. A computer is just a combination of 0s and 1s; it is easy to explain and it is easy to understand. But art is not. Even though the world of numbers and data seems easier to understand for me, I am not completely robotic and do house an interest for art and painting. When I present my paintings in front of my family and friends, I always have a conception about them. Sometimes I love them and sometimes I am just not satisfied enough. If asked why, my only answer is "I don't know." I have never liked having such an answer for anything. Feelings, emotions, art, beauty and general human psychology are all things that have too many conditions and too many exceptions. They very often do not let us provide calculated answers to questions. At least, that is what I have believed my whole life. But Galileo Galilei once said "Measure what can be measured, and make measurable what cannot be measured." This made me think: what if the world has limited itself from developing actual mathematical expressions for everything? What if there is some concept that could help me finally explain why I do not like looking at something I made myself? And what if mathematics could, to a certain extent, nullify emotions and psychological conditions? The world is still, after all, a system that could be broken down into a number. Hence, piece by piece, I'd like to break this world down mathematically. To start with, art and painting, one of my areas of pleasure, is a good topic to quantify. This investigation would focus on using mathematical concepts like the Golden Ratio and Proportion to explain beauty and aesthetics of chosen paintings and apply the understood concepts to all forms of art. In this research paper I will try to establish a mathematical expression for the likability of a given painting. This investigation would aim at finding the factors which have helped make the creative works of some people increasingly popular.

Proportion

Proportion is a concept in math which talks about the equality of two ratios.

𝑎: 𝑏 ∷ 𝑐: 𝑑

It is the mathematical comparison between two numbers and is usually referred to as a part, or share, or a number in the comparative relation to a whole. Proportion is used varyingly in many different concepts, but in every such place it is used to describe the relation between two objects. Proportion has played an important role in art even without any actual mathematical calculations being involved. Often when drawing figures or objects, the object length to breadth proportion, object area to page area proportion and many other proportions are considered. In every drawing, the different parts must be "proportionate" to each other. This essentially means that if one is drawing a typical human figure, the body cannot have 10 cm long hands and 5 cm long legs. The relative size of each object or figure drawn has to be taken into consideration. Colours are also an important part of art. The world consists of a plethora of colours and they are all said to be made simply by the combinations of different proportions of the three basic colours: red, blue, green. The infinite combinations of colours in varying quantities play a very crucial role in art.

Golden Ratio

The golden ratio is an irrational number with the value

and it is denoted by the Greek symbol  (phi). Two quantities are said to be in the golden ratio if the ratio of the larger number to the smaller number is equal to the ratio of the sum of the two values to the larger value.

This value is also called the Divine Proportion. This number is often referred to as "The Most Beautiful Number." It is considered this special because the golden ratio can be found in countless objects around us. More than once, the existence of the golden ratio has been associated with beauty and aesthetics. According to a study by Professor Adrian Bejan from Duke Pratt School of Engineering, the aesthetic appeal of the golden ratio exists because the eye is able to most efficiently scan the scene in the shape of the golden rectangle. "When you look at what so many people have been drawing and building, you see these proportions everywhere," Bejan said. "It is well known that the eyes take in information more efficiently when they scan side-to-side, as opposed to up and down."

A Golden Ratio in Geometric Figures i. The Golden Rectangle

B The Golden Ratio and The Fibonacci Series

The Fibonacci series is a series of numbers formed when each term of the series is a sum of the previous two terms. It is represented by the given formula Hence, we get the series: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…. The Fibonacci sequence is considered an important sequence because of its appearance in nature. The Fibonacci series is used by financial and technical analysts to make predictions and to understand the trading trends. The series is also used to optimize and create many algorithms such as the computational run-time analysis of Euclid's theorem, number generator and the binary tree.

Using the series, a figure called the golden spiral can be created (see III.C). This spiral occurs in many forms of nature.

It is observed that when the terms of the series (3 and onwards) are divided by the term preceding them, we get a value close to the golden ratio. From this series we can get the expression for :

Fibonacci series is an important concept to note when talking about beauty and art. This series can give us different patterns with an interesting incorporation of symmetry. It is always noticed that symmetrical visuals are easier on the eyes. Hence, Fibonacci series can be applied to make art work more attractive.

Some examples of Fibonacci based patterns are given below:

III. c The Golden Spiral

The Golden Spiral is constructed by adjoining squares where the sides have the values of consecutive Fibonacci numbers. The ratio of the size of squares to each other would be powers of the golden ratio.

The golden spiral is found in the head of a sunflower, in snails shells and nautilus shells, in hurricanes, and in spiral galaxies.

III. d The Golden Triangle

The Golden Triangle is an isosceles triangle in which the ratio of length of the common side to the base is the golden ratio. III.e The Golden Ratio in the World As discussed earlier, the golden ratio is observed in several places naturally. Science has taught me that the world is not just a coincident combination of elements. There is a reason the world works the way it does and every component has something to contribute to its functionality. Hence, the repeated appearance of golden ratio suggests that the number must have some significance beyond simple coincidence. To further explore how the golden ratio is omni present and how it could be contributing to beauty, four people were randomly selected and their facial features were measured.

IV. Experiment to Understand Relation between Art and Mathematics

Proportion, golden ratio and Fibonacci series seem to have significance in aesthetic and visual beauty. These different mathematical concepts that point to beauty have been discussed in the topics before. To analyse and prove the dependence of visual beauty on mathematics and incorporate people's perspective in the discussion, four paintings made by the same artist are selected: Null Hypothesis H0: Beauty of the painting is independent of the ranking given. Alternate Hypothesis H1: Beauty of the painting is not independent of the ranking given. Degree of Freedom:

Hence, null hypothesis is rejected and alternate hypothesis is accepted. This means that the ranking of the paintings is dependent on beauty. Now that it is established that the rankings given to the painting by the people in the survey has a relation with its inherit beauty, I would like to mathematically analyse all paintings to see if it gives results parallel with the results of the survey. For each of the four paintings, I will try to identify special mathematical shapes and calculate general ratios of lengths of different parts with each other.

V. Mathematical Inspection of The Subject paintings V.A Buddha

V.E Analysis

Analysing paintings from the point of view of geometric designs has shown that works of art that are considered beautiful have unintentionally incorporated forms of the golden ratio. While making these paintings, I did not have any focus on incorporating the golden ratio or the special shapes. However, they are present nonetheless. When the average of ratios of different parts were taken with each other, it was found that the general proportion for all paintings came out to be close to the golden ratio .

From the survey, it is apparent that Painting III (Fig 13) Painting III, as the most liked painting, had the least ∆𝑥 and the highest n. Painting II was the second most liked. However, its ∆𝑥 was more than Painting I and Painting IV while its n was greater than both the other paintings. This could suggest that presence of mathematical shapes has more impact on likability than general proportion. The results of the survey are parallel to our assumptions and so it is reasonable to believe that the presence of the golden ratio and geometric shapes enhances the beauty of a painting

VI. Qualitative Analysis

The chi square test and analysis of the paintings has proved that beauty and ranking are dependent on mathematical factors. However, art is an abstract and subjective concept. There are a hundred different techniques in art including crayons, pencil sketching, water painting, acrylic painting, oil painting, knife painting, and much more. Every method and technique has its own range of appeal. Moreover, the appeal may also differ from person to person. Hence, mathematics cannot be the only contributing factor. This leads to the question about the other factors that appeal to the spectators. In order to answer this question, a qualitative analysis using rank correlation will be done.

Painting III, the painting liked the most by the people in the survey, was sent to two artists: Mrs. Vandana Garg and Mr. Siddhant Rai. They were asked to give the painting marks out of 10 on the basis of different qualitative factors. Since |𝑟| < 0.5, the two artists have weak correlation about the qualitative factors of the painting. This means that judgment and critique from all artists is related even though they may have varying perspectives.

VII. Limitations of Research

1. There are various forms of art and different mediums for art. For this research, all paintings selected were oil paintings. Hence, the art of acrylic and other mediums was not taken into consideration. Moreover, only one artist's work was analysed. If there was more diversity in the kind of art selected, the results would be more accurate. 2. All measurements were taken using a scale with least count 1mm. Even so, the measurements would not be completely accurate due to human error.

VIII. Conclusion

The exploration was successful in proving that mathematics has an impact on the likability of paintings. The research was able to identify some key shapes in the paintings that seemed to have been contributing to the beauty of the painting. It was seen that the assumptions made about golden ratio and proportion making the paintings more attractive were true. Hence, the aim of the exploration was achieved.