IB MATHEMATICS SL

Phi and the Fibonacci Sequence in Art

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Table of Contents Introduction 2 Overview 2 The Fibonacci Numbers 3 The Fibonacci Numbers and the Golden Ratio 4 The Fibonacci Spiral 6 The Phi Matrix 9 Reflection 11

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INTRODUCTION

This investigation will be focused on exploring the Golden Ratio, also known as phi, and its correlation to the Fibonacci sequence, as well as its presence in famous architecture and artwork.

In order to explore the relationship of the Golden Ratio and the Fibonacci sequence, I will be using a variety of rounded ratios, functions, identities and geometric shapes to find my calculations. Once I have explored the correlation between the two, I will use this acquired knowledge to connect the mathematics with artistic renderings of the ratio. For example, this principle is commonly expressed to be found in famous artworks such as Leonardo Da Vinci's “The Last Supper”. I predict that the majority of appealing architecture and artwork that I select to analyze will use the Golden Ratio and Fibonacci sequence. The importance of the Golden Mean in art is significant to me as a scholar because I take HL Art and art is a subject I am particularly passionate about. It would also be interesting to learn how the mathematical techniques and compositions of famous artists could be used in order to make my own artwork even more successful. The aim of this investigation would also be to better my own artwork and understand what elements

OVERVIEW

The Golden Ratio is an “extreme mean and ratio” that can be found in organic forms in nature as well as in mathematical forms. The Golden Ratio is believed to produce the most satisfying and attractive shapes and compositions, which is why a large number of artists and architects have employed this principle in their creations. It is defined as the geometric relationship in which two quantities have a ratio equal to the ratio of their sum to the largest quantity. A common way to represent the ratio is by drawing a line segment and dividing it into two smaller segments labeled a and b, a being the longer portion. The ratio of a to b should be equal to a + b divided by a. This value, phi, is best known as approximately 1.618, however it is an irrational value that continues infinitely.

Similarly, the Fibonacci Sequence is an infinite sequence of integers that are calculated by finding the sum of the two numbers before it, with the exception of the first two integers. For example, 1,1, 2, 3, 5, 8 and 13 are the first 7 Fibonacci Numbers because 1+2=3, 2+3=5, 3+5=8 and so on. This sequence relates to the Golden Ratio because it can be presumed that the ratio of 5 to 8, is approximately equal to the ratio of 8 to 13. In other words, the Fibonacci Numbers continue to approach the Golden Ratio as their limit progresses. There are many ways to observe this relationship, such as algebra, geometry, precalculus and calculus, some of which will be present and analyzed in this assessment.

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The Fibonacci Numbers The Fibonacci Sequence, as described above, is commonly represented using the first 10

or so numbers, (0,1,1,2,3,5,8,13,21,34) because the sequence begins to escalate very quickly. In order to calculate the rest of the numbers more efficiently, there are multiple equations that can be used. It is commonly observed that as the term grows larger, the ratio of a fibonacci number to its predecessor approaches phi closer and closer.

so (n) f = 51/2P hi n (1) .723592 f = 51/2

1.6181= 0 ≈ 1

This equation uses n to find the nth term of the sequence, where phi is equal to 1.618. However, when n=0, the equation will be equal to 0, which would mean that 0 is not a term in the sequence. In order to consider 0 as a term in the sequence, a modified equation (below) must be used.

(n) f = P hinP hi +2   

In this equation, if n=1, f(1)=0. It is important to consider 0 as a term because the sequence otherwise starts as 1,1,2 but 0+1=1, and 1 can’t spontaneously create a duplicate of itself. Consequently, the 2nd and 3rd terms should be 1, and the 4th term is 2, which is proved below.

so (2) f = P hi4P hi +2 (2) f = 1.6182

1.618 +2 f (2) .72358 = 0 ≈ 1

so (3) f = P hi3P hi +2 (3) f = 1.6183

1.618 +2 (3) .17076 f = 1 ≈ 1

so (4) f = P hi4P hi +2 (4) f = 1.6184

1.618 +2 (4) .89429 f = 1 ≈ 2 Below are the first 50 terms in the fibonacci sequence. After calculating them, they will be used to show their increasing approach to phi, 1.618033988749895. TABLE 1: The First 50 Fibonacci Numbers

1 0 11 55 21 6,765 31 832,040 41 102,334,155

2 1 12 89 22 10,946 32 1,346,269 42 165,580,141

3 1 13 144 23 17,711 33 2,178,309 43 267,914,296

4 2 14 233 24 28,657 34 3,524,578 44 433,494,437

5 3 15 377 25 46,368 35 5,702,887 45 701,408,733

6 5 16 610 26 75,025 36 9,227,465 46 1,134,903,170

7 8 17 987 27 121,393 37 14,930,352 47 1,836,311,903

8 13 18 1,597 28 196,418 38 24,157,817 48 2,971,215,073

9 21 19 2,584 29 317,811 39 39,088,169 49 4,807,526,976

10 34 20 4,181 30 514,229 40 63,245,986 50 7,778,742,049

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Fibonacci Numbers and the Golden Ratio

Leonardo Fibonacci discovered this sequence of numbers in 1202 AD and deemed it to be an extremely important foundation for the relationship behind the golden ratio. In order to calculate phi, you simply take any term of the sequence and divide it by the number before it. It is important to note that the ratio of the first few terms won’t be as close to 1.618 as greater terms because they must begin to gradually approach the limit. This could be represented by the equation below.

so f (n)f (n+1) .61803f (14)

f (15) = 233377 = 1

TABLE 2: The Fibonacci Numbers in Relation to Phi

Position in the sequence

Corresponding Fibonacci Number

Ratio of a Fibonacci number to its Previous Number

The difference of the Fibonacci Number from Phi (1.618033988749890)

1 0 0 0

2 1 0 1.618033988749890

3 1 1.00000000000000 0.618033988749890

4 2 2.00000000000000 -0.381966011250110

5 3 1.50000000000000 0.118033988749890

6 5 1.66666666666667 -0.048632677916777

7 8 1.60000000000000 0.018033988749890

8 13 1.62500000000000 -0.006966011250110

9 21 1.615384615384620 0.002649373365275

10 34 1.619047619047620 -0.001013630297729

11 55 1.617647058823530 0.000386929926361

12 89 1.618181818181820 -0.000147829431928

13 144 1.617977528089890 0.000056460660002

14 233 1.618055555555560 -0.000021566805666

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15 377 1.618025751072960 0.000008237676929

16 610 1.618037135278510 -0.000003146528625

17 987 1.618032786885250 0.000001201864644

18 1,597 1.618034447821680 -0.000000459071792

19 2,584 1.618033813400130 0.000000175349765

20 4,181 1.618034055727550 -0.000000066977664

21 6,765 1.618033963166710 0.000000025583184

22 10,946 1.618033998521800 -0.000000009771913

23 17,711 1.618033985017360 0.000000003732532

24 28,657 1.618033990175600 -0.000000001425707

25 46,368 1.618033988205320 0.000000000544565

26 75,025 1.618033988957900 -0.000000000208012

27 121,393 1.618033988670440 0.000000000079447

28 196,418 1.618033988780240 -0.000000000030353

29 317,811 1.618033988738300 0.000000000011587

30 514,229 1.618033988754320 -0.000000000004432

31 832,040 1.618033988748200 0.000000000001686

32 1,346,269 1.618033988750540 -0.000000000000651

33 2,178,309 1.618033988749650 0.000000000000242

34 3,524,578 1.618033988749990 -0.000000000000099

35 5,702,887 1.618033988749860 0.000000000000031

36 9,227,465 1.618033988749910 -0.000000000000019

37 14,930,352 1.618033988749890 0.000000000000000

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Fibonacci Numbers in Art The Fibonacci Spiral

The Fibonacci Spiral is a unique logarithmic spiral because every turn of is in exact2π

increments of 1.618, thus creating a perfect golden ratio form. This spiral is commonly composed of squares with side lengths of the fibonacci numbers (1,2,3,5,8,13,21,etc.) and can be found in nature and in a variety of famous artworks. In art, the composition of a painting is extremely important to the success of the overall piece, which is why many artists try to use the golden ratio to ensure effective subject placement. An example of this can be seen below as well as in a still life by George Weissbort entitled The White Coffee Pot with Fruit and Wine, where the wine glass and pears fit perfectly into the spiral.

In order to further understand the characteristics of the Fibonacci spiral (prior to applying it to any artwork), I analyzed the dimensions of a template that is provided on the official site of the

golden ratio by putting it into a graphing program and plotting the points. Like the picture above, the rectangle that the spiral depends on is composed of squares with side lengths that correspond to numbers in the sequence. Each element of the rectangle needs to follow the

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golden ratio in order for the spiral to be perfect, since it is a logarithmic spiral with a growth factor of phi, 1.618. However, using the rectangles and squares with the golden ratio proportions to form the spiral only gives an almost accurate representation since the actual function would need to be graphed. Assuming that the picture is slightly distorted, the square DEFA has relative side lengths of around 7.4 cm, which is close enough to the 7th value of the series, 8. The slightly smaller square, MKBF, has side lengths that can be rounded up to 5, which is the 6th Fibonacci number. Continuing with this pattern, the smaller square, GLKJ, has side lengths slightly under 3, square EGHI has side lengths at approximately 2 and so on. However, the actual property of the golden ratio can be identified when the length/width of the overall rectangle is divided by the side lengths of the squares that share its perimeter. For example, the rectangle’s side length AB,12, is segmented by the side length FB, 4.57 cm, and AF, 7.43 cm. According to the divine proportion, the ratio of a (being the longer segment: 7.43) to b (4.57), should be equal to the ratio of the entire side length (a+b= 12) to a, which is almost exactly true.

ba .62 4.57

7.43 = 1 aa+b .61512

7.43 = 1 This also occurs in the width of the rectangle as well:

.65 .60LKKB = 2.79

4.61 = 1 LBKB = 7.4

4.61 = 1 And also within the smaller rectangles:

.62 .62GLEG = 1.75

2.84 = 1 GLEL = 2.84 + 1.75 = 2.84

4.59 = 1 These simple ratios prove that the spiral in the template does correspond with the divine proportions.

It is commonly believed that the golden spiral is actually hardwired into our brains and that we are always subconsciously using it to determine the beauty of a person or an object. The closer something is to being proportional to the spiral, the more visually appealing it is to the eye. In fact, the golden spiral is actually extremely prevalent in nature and therefore would probably be most successful in judging the proportions of organic form. Examples of this would be shells, human body parts, and other naturally occurring shapes. Since this internal assessment focuses on the presence of phi in art, I wanted to put one of my own IB artworks to the test.

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For my IB Visual Art exhibition, I was inspired by the rhythm of repetition and created this piece which focuses on the unique shape of the human ear. During the process, I had not even thought about the golden ratio or its applications to a composition, but since the subject is a naturally occurring form, it is likely that the golden spiral can be identified in my artwork. This piece was so successful that it ended up receiving a superior rating at a state art competition. Surely, its visual appeal could be attributed to an underlying mathematical foundation.

I put this image into the graphing software that I had been using to analyze the golden spiral and positioned the template over my artwork. I found that the spiral fits perfectly into the lower crevice of the purple ear hole, when the outside curve is aligned with the helix. The bottom line of the rectangle is also almost exactly lined up with the bottom of the other two ear holes and the curve of the ears seems to follow the outermost curve of the spiral. Also notable are the yellow lines that I graphed where I saw significant elements in the artwork. The horizontal line at approximately y= 6.8 is the point at which the ear lobes blend into each other and the colors meet, while the vertical line at about x=13.7 is an extension of the length of the square in the template. Interestingly enough, if the measurements of the segments these lines created are taken and put into a ratio, the quotient is extremely close to the golden ratio. This reinforces the fact that the artwork successfully correlates to the golden spiral and extreme mean, which could explain why it was such a strong piece.

The ratio of the width to the lower point where the colors merge: .617JL

LA = 10.9817.75 = 1

The ratio of the width to the square of the spiral template:

.66AOAD = 13.47

22.37 = 1

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The Phi Matrix The phi matrix is a rectangle composed of many different parts which all adhere to the divine proportions. This diagram is commonly used to determine the focal point of something in relation to the rest of the composition. Like the golden spiral, I graphed the points of the sections of the phi matrix and took measurements to understand how exactly the golden ratio is involved in this template. It became

clear that, once again, every line and line segment ratio ended up being approximately equivalent to phi. In addition, the area of the rectangles or squares could be rounded to get a number in the fibonacci sequence such as: NRBF, AEMQ, PLGJ, and DCOK have an area of 5 cm2, KLNM has an area of 2 cm2, and PLKO, LGFN, DKME and MNRQ have an area of 3 cm2 . This creates an interesting pattern of values that follows the progression of the Fibonacci numbers from the center outwards. Also, the center portions

are measured to be the same distance from each side, which allows for there to be two different ratios for every series of segments that should be equivalent to each other. Width: .63 .64 .62 .61 EA

CE = 1.712.78 = 1 AD

DC = 1.702.79 = 1 CA

CE = 2.784.49 = 1 DA

CA = 2.794.49 = 1

All the line segments are extremely close in value, representing how the golden ratio and symmetry are used in the matrix. Length: .64 .61 .61 .62 P J

CP = 2.754.51 = 1 JO

OC = 2.784.48 = 1 CJ

CP = 4.517.26 = 1 JO

CJ = 4.487.26 = 1

Since the phi matrix seems to be centered around the rectangle KLNM, it can be easily inferred that a most successful composition would have a focal point or a large portion of the subject within that area. Upon initial research, one of the artworks that is said to utilize very clear divine proportions is Salvador Dali’s The Sacrament of the Last Supper. Not only did Dali actually

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frame his artwork in a golden rectangle, he also is said to have positioned the table perfectly to create the extreme mean, and used a large “dodecahedron” in the background because of its consistency with phi. As for my analysis, I lined up the lower line of the matrix to the table and measured the distances to create my own scaled down ratios. It is also clear in this graph that Jesus, the focal point, is entirely within the center section of the matrix, which explains why the eye is automatically drawn to him. This famous painting is pleasing for a multitude of other reasons not related to mathematics, but the golden ratio is undoubtedly present here, and was likely used on purpose to symbolize the perfection of the religious scene.

The actual canvas is 166.7 cm x 267 cm, while this image is 11.71 cm x 18 cm.The width is scaled by a factor of 14.236 and the length is scaled by a factor of 14.833. Ratio of table to the area above table to the table:

( ) .64 IAAG = 4.43

7.28 = 1 Ratio of area above table to entire canvas: 4.236 ( ) .61 IA

IB = 1 7.2811.71 = 1

The Sacrament of the Last Supper is an artwork that is frequently used as an example to represent the golden ratio in artwork, but I wanted to see if other famous artworks not present employ this design as well. It is safe to assume that, if an artwork is well known and loved, and if the golden ratio is extremely attractive to the eye, the artwork may in turn contain phi. This painting, however, is also pleasing because of the realist style, but can an abstract piece be just as visually appealing? This artwork, Composition VII, by Wassily Kandinsky is actually deemed to be one of the most influential artworks in the history, since it completely changed the perception of abstraction. Although there are a great deal of vibrant shapes and chaos, the painting has a very clear focal point which captures the attention of the audience. In my graph, I have plotted points around this form.

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The amorphous shape that is shaded here in green seems to have the most emphasis since it has such a distinct black outline and is almost a different style than the rest of the forms. Instead of applying the template to the artwork, I used the square I plotted and graphed two horizontal lines similar to the ones in the matrix. These ended up being at about y=5.9 and y=3.3. Next, I repeated the same process and took the measurements of the segments produced by the lines and the canvas. Although roughly estimated and obviously not symmetrical, the segments I created actually proved to create overall canvas ratios with quotients relatively close to 1.618.

.51 or .67BFBN = 6.44

9.72 = 1 BFMF = 5.82

9.72 = 1 From these observations, the presence of the golden ratio can be identified in even this abstract piece with no real subject matter, and is likely responsible for why it is so satisfying to view. It is also probable that many other famous artworks not used as examples have some similarities to the divine proportion in either the golden spiral or phi matrix. While there are many other ways to identify the extreme mean in design and art, I found that these two diagrams seemed to be the most common and effective in comparing composition. Reflection

This investigation was inspired by my love for art as well as my interest in the golden ratio, which is a concept I had never fully understood despite my art teacher constantly advising us to use it. Although there is not a lot of complex math involved in understanding and locating the ratios, the concept can be difficult to grasp, so it was definitely beneficial to have done so

much research. I found it extremely interesting that one of my most successful artworks works so well with the golden spiral and I had not even known that I was using it. The power of the mind is that I can subconsciously use mathematics in something I thought was purely emotional, and

that it is responsible for a portion of the aesthetic beauty I was able to produce. Although phi and the golden ratio isn’t completely understood despite being around and studied for centuries, this investigation proved to be extremely interesting and will be beneficial even after its completion.

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Works Cited

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50 Most popular paintings. (n.d.). Retrieved March 27, 2018, from

http://www.wassilykandinsky.net/painting1896-1944.php

Bourne, M. (n.d.). Golden Spiral. Retrieved March 27, 2018, from

https://www.intmath.com/blog/mathematics/golden-spiral-6512

Famous Abstract Paintings That Changed The Way We Perceive Art. (n.d.). Retrieved March

27, 2018, from

http://www.ideelart.com/module/csblog/post/94-1-famous-abstract-paintings.html

GeoGebra Graphing Calculator. (n.d.). Retrieved March 27, 2018, from

https://www.geogebra.org/graphing

Kangas, B. (2014, November 15). Exploring Dali's Sacrament of the Last Supper. Retrieved

March 27, 2018, from

http://www.patheos.com/blogs/billykangas/2014/11/exploring-dalis-sacrament-of-the-last-

supper.html

Knott, R. (n.d.). Retrieved March 27, 2018, from

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibInArt.html#arch

Most Memorable Golden Ratio Examples in Modern Art. (n.d.). Retrieved March 27, 2018,

from https://www.widewalls.ch/golden-ratio-examples-art-architecture-music/

The Fibonacci Sequence in Artistic Composition. (n.d.). Retrieved March 27, 2018, from

https://www.markmitchellpaintings.com/blog/the-fibonacci-sequence-in-artistic-compositi

on/