Jamie Coltart – Fibonacci Magazine

THE F I B ONAC C I QUA RT E R LY A paper issued every quarter devoted soley to the Fibonacci sequence

#1

The first physical release after 60 online issues

Robb Jannoff An in-depth interview with the designer behind the iconic apple logo

An introduction to the Fibonacci Sequence An in-depth look at the origin of the Fibonacci sequence and history of the pioneer now known as Leonardo Fibonacci

The use of the Fibonacci Sequence in modern design An insight into what modern designers actually think of the sequence including iconic pieces from Le Corbusier

Psycology of the sequence A look at experiments carried out on the psycology of the golden section

Speculation of the sequence Ex-sequence enthusiast Keith Devlin and mathematician Robb Enzmann give their insight and speculation over the Fibonacci sequence

I N T RODU C T ION

This newspaper was designed applying the Fibonacci sequence where possible below is a spread layout I used with a key explaining how it was applied 3

3

3

2 2

2

5

5

KEY

3

43 POINT

22 Point Size 16 POINT SIZE

14 Point – 10, 11, 12, 14, 17, 22, 30, 43, 64

These lines represent columns used on

The lines above represent the gutter

The mumbers on the spread layout

Above is a scale example of type size I used in the

the spread – 2, 3, 9 and 13 column

on the spread – the gutter size used

show what prortions are used for the

design of the spreads – though not Fibonacci numbers

grids are used throughout

throughout is 5mm

margins of the spreads

they are proportioned using the ratio of the sequence

04 06 08 10 12 14 16 18 20 22 23

C ON T E N T S OR IG I N OF T H E SE QU E N C E Dating back 2500B.C. we look into the history of the sequence

W HAT I S T H E F I B ONAC C I SE QU E N C E ?

A deeper look into what exactly the sequence is

T H E G R E AT DE BAT E

A look at the controversy surrounding the sequence

A T I M E L I N E OF F I B ONAC C I

A timeline of where the seqeunce has arguably been applied

T H E E X P ORT S OF A RT

An infographic surrounding the exports of art

W H Y D OE S T H E SE QU E N C E WOR K ?

Looking into the use of the sequence in nature

T H E P SYC HOLO G Y

Psychology experiments explaining why the sequence works

T H E SE QU E N C E I N MODE R N DE SIG N Looking at who is using the sequence in modern design

R OB B JA N OF F – I N T E RV I EW An interview with the designer behind the iconic Apple logo

A P P L E L O G O DE C ON ST RU C T E D A breakdown of the Apple logo to see if it conforms to the Fibonacci sequence

L E C OR BU SI E R Insight to Swiss designer Le Corbusier’s use of the Fibonacci sequence

T H E OR IG I N OF T H E F I B ONAC C I SE QU E N C E

With Fibonacci theories dating back earlier than 2500B.C. where exactly did the famous Fibonacci sequence originate from?

THE LIFE OF FIBONACCI Leonardo Pisano Bigollo, called Fibonacci was born in Pisa around 1170. His father, Guglielmo dei Bonacci, a wealthy Pisan merchant and representative of the merchants of the Republic of Pisa in the area of Bugia in Cabilia, after 1192 took his son with him, because he wanted Leonardo to become a merchant.

He thus got Leonardo to study, under the guidance of a Muslim teacher, who guided him in learning calculation techniques, especially those concerning Indo-Arabic numbers, which had not yet been introduced in Europe. Fibonacci’s eduction started in Bejaia and continued in Egypt, Syria and Greece, places he visited with his father along the trade routes, before returning permanently to Pisa around 1200.

For the next 25 years, Fibonacci dedicated himself to writing mathematical manuscripts: of these, Liber Abaci (1202) – which is resposible for Europe becoming aware of Indo-Arabic numbers, Practica Geometriae (1220), Flos (1225) and Liber Quadratorum (1225) are what are known to us today.

Leonardo’s reputation as a mathematician became so great that the Emperor of Italy, Federico II, honored him with salary for his service he gave to Pisa. After 1228, not much is known of Leonardo’s life, except that he was awarded the title of ‘The discreet and wise teacher Leonardo Bigollo’ in recognition of the great progress he made to mathematics. Fibonacci died sometime after 1240, presumably in Pisa.

4

remained in conflict with the abacus – traditionalists who continued to use the abacus in conjunction with Roman numerals.

The second version of Liber Abaci was dedicated to Michael Scot in 1227 CE. No versions of the original 1202 CE book have been found.

Above is both the hindu arabic system and

LIBER ABACI

The first section of the book introduces the Hindu-Arabic numeral system, including different methods for converting between different representation systems.

the numerical system that western culture use to this day

The second section presents examples from Liber Abaci (1202), also spelled as Liber Abbaci, commerce, this includes calculations such as is a historic book on arithmetic by Leonardo of conversions of currency and measurements, Pisa, known later by his nickname Fibonacci. and calculations of profit and interest.

Liber Abaci was among the first Western books to describe Hindu-Arabic numbers traditionally described as ‘Arabic Numerals’. By addressing the applications of both commercial tradesmen and mathematicians, it contributed to convincing the public of the superiority of the HinduArabic numeral system.

The title of Liber Abaci means ‘The Book of Calculation’. Although it has also been translated as ‘The Book of the Abacus’ there is speculation that this has been translated with error. Sigler in 2002 claimed “this is an error: the intent of the book is to describe methods of doing calculations without aid of an abacus”

The third section discusses a number of mathematical problems. For instance, it includes the Chinese remainder theorem, perfect numbers and Mersenne primes as well as formulas for arithmetic series and for square pyramidal numbers. Another example in this chapter, describing the growth of a population of rabbits, was the origin of the Fibonacci sequence for which the author is most famous today.

The fourth section derives approximations, both numerical and geometrical, of irrational numbers such as square roots.

The book also includes proofs in Euclidean geometry. Fibonacci’s method of solving algeOystein Ore confirms for centuries after its braic equations shows the influence of the early publication the algorismists – followers of the 10th-century Egyptian mathematician Abū style of calculation demonstrated in Liber Abaci Kāmil Shujāʿ ibn Aslam.

5

W HAT E X AC T LY I S T H E F I B ONAC C I SE QU E N C E ?

The Fibonacci sequence is a series of numbers in which a given number is the addition of the two numbers before it. So, if you start with 0, the next number will be 1, followed by 1, followed by 2, followed by 3 and so on. As you can see, every number in this series or sequence is obtained by adding the two preceding numbers. This simple series of numbers is referred to as the Fibonacci sequence, or Fibonacci series, and individual numbers in this sequence are often referred to as Fibonacci numbers.

It was found that these Fibonacci numbers have a direct relation to the Greek number phi which is an irrational number and equals 1.6180339885…

If we divide a term greater than 2 by a term preceding it that the ratio always tends to 1.618…

And if we continue this division after the 13th term we will always get a fixed number = 1.618…

When applying the Fibonacci sequence/phi to shapes it results in proportioned shapes, arcs and ratios. This is the most common way we see the sequence/Phi being used in design and this is commonly referred to as the golden ratio, the golden section and the devine proportion.

This entire story starts with a man named Pythagoras, born around 570 BC on the island of Samos

in the Aegean Sea. Now well-known for his famous Theorem, which states that square of the length of the longest side of a right triangle is equal to the sum of the squares of the short sides, Pythagoras was also a respected philosopher and had a considerable following. The Pythagoreans, as they were called, adapted the pentagram as their symbol, as it represented health, and, more interestingly, what is now known for containing golden ratio.

It is possible to establish a rigorous proof that the diagonal and the side of the pentagon are “incommensurable,” meaning that the ratio of their lengths cannot be expressed as a ratio of whole numbers. Therefore, phi is shown to be an irrational number. It has been suggested by several researchers that the Pythagoreans’ discovery of this was the first appearance of incommensurability in history. “The Pythagorean world view was based on an extreme admiration for the arithmos – the intrinsic properties of whole numbers or their ratios and their presumed role in the cosmos. The realisation that there exist numbers, like the Golden ratio, that go on forever without displaying any repetition or pattern caused a true philosophical crisis. In only the fifth century BC, then, our favourite number had already started making waves.”

1, 1, 2, 3, 5, 8, 13, 21, 34 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765 Above is an example of the first twenty numbers of the Fibonacci sequence

89/55 = 1.618, 144/89 = 1.618, 233/144 = 1.618’ Above is an example of Fibonacci numbers greater than 2 divided by the preceding Fibonacci number

Above is an example of the ratio when applied to shapes and an arc

7

T H E G R E AT DE BAT E

Images of subjects such as the Parthenon with the golden ratio on top becoming more common than ever there are supporting arguments to the overlay

ANCIENT GREECE Greek mathematician Eulicid published 12 books in 300 BC collating Greek mathematical knowledge of the time, including the first known theory of Phi. Though the Parthenon was finished in 432 BC, the books also contained mathematical theories that date back to 570 BC so it is argued that the Greeks may have used Phi before the Parthenon was created. There is however disagreements with this viewpoint “this is wishful thinking put into a diagram this is rigging the diagram and faking things to support a belief.”

A fellow mathematician, Robb Enzmann, also stated this was just a “camera placed to fit”. Even amongst sceptics there are differing opinions “this isn’t to say that there are no civilisations that used the golden ratio, we know for a fact that the ancient Mayans used it.” Contrary to this Devlin believes that the Mayans may have used the ratio, however, there is no evidence that they understand the ratio or made practical use of it.

8

LEONARDO DA VINCI Another heavily debated pioneer of the ratio is Leonardo Da Vinci. Da Vinci was friends and illustrated for Luca Pacioli on his book ‘The Divine Proportion’, which contained the mathematics of the golden ratio and its application in art and architecture. Having collaborated with Pacoili, it is possible that he would have known about the golden ratio and used it in his work. Da Vinci’s ‘Vitruvian Man’ potentially shows application of the golden ratio, but it is argued by Keith Devlin that the measurements are…

“Close, but in maths close is not enough”

9

T H E F I B ONAC C I SE

Though there is debate over who has used t the most famous design works that are s

2600 B.C.

1500 B.C.

447 B.C.

PYRAMIDS OF GIZA

MAYA RUINS OF COPAN

THE PARTHENON

The Great Pyramid of Giza is the oldest and largest of the pyramids in the Giza pyramid complex bordering present-day Giza in Greater Cairo, Egypt. It is the oldest of the Seven Wonders of the Ancient World, and the only one to remain largely intact. The pyramids of Giza were royal tombs built for three different pharaohs. The Great Pyramid of Giza has a base of 230.4 meters (755.9 feet) and an estimated original height of 146.5 meters (480.6 feet). This creates a height to base ratio of 0.636, which indicates it is indeed a Golden Triangles, at least to within three significant decimal places of accuracy. If the base is indeed exactly 230.4 meters then a perfect golden ratio would have a height of 146.5367.

This varies from the estimated actual dimensions of the Great Pyramid by only 0.0367 meters (1.4 inches) or 0.025%, which could be just a measurement or rounding difference.

Copán is an archaeological site of the Maya civilization in the Copán Department of western Honduras. Copán was occupied for more than two thousand years. The city developed a distinctive sculptural style within the tradition of the lowland Maya. The fundamental shape of Maya geometry is the golden section, and that the Maya composed such sections using a procedure that is brilliant in its simplicity. Using a cord, it is easy to construct a square. If the cord is doubled back on itself it obviously becomes half the length, and that halved cord can be used to find the midpoint of one of the sides of the square. Next, if the cord is placed on the midpoint and extended to one of the opposite corners, it can be swung like a compass in an arc that will define the length of a golden section, from which the final rectangle can be constructed. Markings on Maya doors have been found supporting this theory of design.

The Parthenon is a temple on the Athenian Acropolis, Greece, dedicated to the goddess Athena, whom the people of Athens considered their patroness. It was completed in 438 BC and is the most important surviving building of Classical Greece. The Parthanon beam on top of the columns is in a golden ratio proportion to the height of the columns. The grid lines is a golden ratio proportion of the one below it, so the third golden ratio grid line from the bottom to the top at the base of the support beam represents a length that is phi cubed, 0.236, from the top of the beam to the base of the column. The beam on top of the columns has a horizontal dividing line that is in golden ratio proportion to the height of the support beam and the width of the columns is in a golden ratio proportion formed by the distance from the centre line of the columns to the outside of the columns.

10

QU E N C E T I M E L I N E

the Fibonacci sequence, here are some of speculated to have used the sequence 670 A.D.

1160 A.D.

1804 A.D.

MOSQUE OF KAIROUAN

THE NOTRE-DAME OF PARIS

SYMPHONY NO.5 IN C MINOR

Established by the Arab general Uqba ibn Nafi in the year 670 AD at the founding of the city of Kairouan, the mosque occupies an area of over 9,000 square metres (97,000 sq ft). It is one of the oldest places of worship in the Islamic world, and is a model for all later mosques in the Maghreb.

The golden section appears repeatedly in some part of the building measurements. It is found in the overall proportion of the plan, in the prayer space, the court and the minaret. The existence of the golden section in some parts of Kairouan mosque indicates that the elements designed and generated with this principle may have been realised at the same period. This suggests and opens the possibility for further inquiry into the transformations that took place in this mosque. Theories about the mosque demonstrate the geometric constructions for the golden section and applications of constructions to the mosque

Notre-Dame is a medieval Catholic cathedral on the Île de la Cité. The cathedral is considered one of the finest examples of French Gothic architecture. Several of its attributes set it apart from the earlier Romanesque style, particularly its pioneering use of the rib vault and flying buttress, its enormous and colourful rose windows, and the naturalism and abundance of its sculptural decoration. Notre-Dame appears to have golden ratio proportions in a number of its key proportions of design. Although it is rather asymmetrical in its design and difficult to measure photographically because of parallax distortions, the golden ratio appears vertically between the tiers of the building. As well as this the distance between the top two tiers also follow the golden section vertically. One of the tiers conforms to the golden ratio when applying the proportions to the other top tier and space between them.

As well as visual design the Fibonacci sequence has claimed to have been used in multiple different fields. One of the most historic claims for the use of the sequence is in Beethoven’s iconic Symphony number five in C Minor. The piece is one of the best-known compositions in classical music and one of the most frequently played symphonies. It is widely considered one of the cornerstones of western music. It was first performed in Vienna’s Theater an der Wien in 1808. In the piece, the opening motto occurs exactly at the golden mean point of 0.618, namely in bar 372 of 601. What’s more, the coda is 129 bars long, and if you divide it using the golden section, you get 49:80. After the first 49 bars of the coda, Beethoven actually introduces a completely new melody that has not appeared in the movement so far, a real first in the history of classical music composition.

11

www.

3 IN 4

Engaged with art with the last 12 months

28%

Digitally engaged with art in the last year

MOST EXPENSIVE ART SALES IN HISTORY Salvator Mundi, £347.63 Million, Leonardo da Vinci, Rome, Italy, c. 1500 Oil on walnut panel Les Femmes d’Alger, £143.18 Million, Pablo Picasso, Málaga, Spain, 1955 Oil on canvas Nu Couché, £136.08 Million, Amedeo Modigliani, Paris, France, 1917 Oil on canvas Portrait of Dr. Gachet, £119.48 Million, Vincent van Gogh, Zundert, Netherlands, 1890 Oil on canvas Nu Couché (sur le côté gauche), £118.46 Million, Amedeo Modigliani, Paris, France, 1917 Oil on canvas Three Studies of Lucian Freud £115.68 Million, Francis Bacon, Dublin, Ireland, 1969 Oil on canvas Bal du moulin de la Galette, £113.12 Million, Pierre-Auguste Renoir, Limoges, France, 1876 Oil on canvas Twelve Landscape Screens, £108.73 Million, Qi Baishi, Xiangtan, China, 1925 Ink on paper Garçon à la pipe, £104.35 Million, Pablo Picasso, Málaga, Spain, 1905 Oil on canvas The Scream, £98.86 Million, Edvard Munch, Oslo, Norway, c. 1893 Mixed media on cardboard

FINACIAL EXPORT OF ART RANKING 12

52,000 EMPLOYED

-42.8%

Art sales dropped -42.8% from 2019 to 2020

artists in the UK – enough to fill hampden stadium

TOP 5 EXPORTERS The 5 biggest financial exporters of paintings and drawings are the United States, United Kingdom, Hong Kong, Switzerland and France. Collectively, this leading quintet of art suppliers generated 83.6% of overall exported paintings and drawings during 2020. That percentage indicates a relatively concentrated set of art exporting nations.

38.9% 30% 6.5% 4.1% 4%

TOP 25

26–50

51–75

76–100

100 + 13

W H Y D OE S DE SIG N I N G W I T H T H E F I B ONAC C I SE QU E N C E WOR K ?

USE IN NATURE Observing the geometry of plants, flowers or fruit, it is easy to recognise the presence of recurrent structures and forms. For example he Fibonacci sequence plays a vital role in phyllotaxis, which studies the arrangement of leaves, branches, flowers or seeds in plants, with the main aim of highlighting the existence of regular patterns. The various arrangements of natural elements follow surprising mathematical regularities. D’arcy Thompson observed that the plant kingdom has a curious preference for particular numbers and for certain spiral geometries, and that these numbers and geometries are closely related. We can easily find the numbers of the Fibonacci sequence in the spirals formed by individual flowers in the composite inflorescences of daisies, sunflowers, cauliflowers and broccoli.

Even the ratio’s biggest critics support the evidence of the ratio within nature. An example of this use in nature is sunflower seed patterns which contain many spirals that if counted will always result in a Fibonacci number.

14

We know for a fact that the ancient Maya civilisation used it and this isn’t because we matched diagrams digitally on top of their doors. It’s because we went and we looked at the doorways they had made which were made of stone and there were nick marks and actual carvings in the door that indicated the use of a geometric construction of the golden ratio. This is much like how you can build the golden section with the squares, but obviously the ancient Maya civilsation did not have the level of mathematics as we have today. They didn’t have the same rigger, but they did what was important they looked for patterns around things specifically patterns in things like the images above. These images from left to right are: a succulent, sunflower and pinecone. You can see the spirals coming from the centre and going clockwise and counterclockwise. It turns out these are the golden spirals that we are looking for and it doesn’t seem very intuitive at first but the mayans saw these patterns. They looked at the petals and stem formations and saw that they were growing outwards and were usually a fibonacci number of spirals.

T H E P SYC HOL O G Y

One of the most common experiments to test the golden ratio was first carried out by psychologist Gustav Fechner THE EXPERIMENT Fechner arranged ten rectangles of different ratios, including one rectangle having the golden ratio dimensions, and had subjects select which they found most appealing. From his experiment he found that out of the ten rectangles the majority (30.3%) of the subjects selected the rectangle using the golden ratio.

SPECULATION The experiment has been replicated multiple times with adaptations and although the majority of experiments support Fechner’s findings there are experiments that have found no preference to the golden ratio rectangle. One of the theories behind why this golden ratio rectangle is selected is that Western culture has become familiar with this ratio through art. When the same Fechner experiment was carried out in Japan the results showed that the golden ratio rectangle was not the most preferred. Another theory behind why the golden ratio rectangle is selected is due to the order, format and how the golden ratio rectangle fits in context to the other shapes presented in the experiment.

Devlin claimed he is part of an ongoing, unpublished exercise at Stanford. He has worked with the university’s psychology department to ask hundreds of students over the years what their favourite rectangle is. He shows the students collections of rectangles, then asks them to pick out their favorite one. Devlin says “If there were any truth behind the idea that the golden ratio is key to beautiful aesthetics, the students would pick out the rectangle closest to a golden rectangle. But they don’t. They pick seemingly at random. And if you ask them to repeat the exercise, they pick different rectangles. It’s a very useful way to show new psychology students the complexity of human perception.”

This is the main argument that the golden ratio isn’t more aesthetically pleasing to people at all.

Devlin’s experiments aren’t the only ones to show people don’t prefer the golden ratio. A study from the Haas School of Business at Berkeley found that, on average, consumers prefer rectangles that are in the range of 1.414 and 1.732. The range contains the golden rectangle, but its exact dimensions are not the clear favorite.

16

SELECT WHAT

SHAPE YOU FIND MOST APPEALING

KEY

The ten recrangles in Gustav Fechner’s golden ratio experiment

The rectangle with the orange border is the golden ratio rectangle – the shape the majority of people chose in Gustav Fechner’s experiments

The rectangle with the bold blue outline is the rectangle that the majority people in Japan preferred and has a ratio of 1.414

The rectangles above the orange bracket are rectangles made with the ratio between 1.414 and 1.732 – the rectangles which Haas school claim are equally as preferred 17

T H E F I B ONAC C I SE QU E N C E I N MODE R N DE SIG N

SOCIAL MEDIA A quick scour of social media platforms Instagram and Youtube shows just how common the use of the sequence is. From initially opening the Instagram app, after just 5 posts a logo design showing how the design used the golden ratio had appeared. After scrolling past 12 more posts a similar post showing the use of the sequence was found. As social media is tailored to individual

18

interests, it is understandable that this would not be every users experience. On searching ‘golden ratio logo tutorial’ on Youtube, there are hundreds of videos with the highest reaching over 2.9 million views. From these two platforms it is evident that people are aware of, and that they are using the golden ratio in modern graphic design.

“It’s become more of a meme or a bit of a laugh in the industry – always makes me laugh when I see it popping up in a project rational” WHAT DO MODERN DESIGNERS THINK? To try and get a deeper understanding of how regularly the Fibonacci sequence is applied to modern design, research was conducted on the experience of graphic designers with the Fibonacci sequence.

When asking the question, “Are you aware of the Fibonacci sequence/golden ratio in design?”, 42.8% of those asked stated that they were not aware of the sequence/ ratio. When asked, “How often do you use the sequence or golden ratio when designing?” there was a wide variety of answers.

These ranged from: “It’s not something I have ever looked at or thought about when working on a project” to: “Not very often at all. It’s mostly been for the purposes of learning/experimentation”, and, “all the

time, whenever I am designing a logo I always use the ratio”. Another question asked was: “What are your thoughts on the sequence/ratio, do you feel it works or is it too limiting?”. This question generally gathered similar viewpoints in that the sequence did work, but that there is an uncertainty of why “there is obviously science behind the reason it produces visually appealing results that I am unaware of ” it was also stated “you can create visually appealing work without it, so it isn’t a necessity by any means”. Though the response “In my opinion it’s become more of a meme or a bit of a laugh in the industry. Always makes me laugh when I see it popping up in a project ration-

ale” shows the impact the internet and social media has had on graphic design within the industry. Contrary to this, one of the subjects of the questionnaire was a design lecturer and curriculum manager of graphic design and illustration at Glasgow Clyde College. He talked about the use of the Fibonacci sequence during his classes and explained “we go deeper into the sequence and see how it can be applied to a lot of design” so although the sequence may be known for having a “meme” presence and being popular online the Fibonacci sequence is also being taught in academic environments which in itself shows there is a still an authetic use in design for it.

19

20

R OB B JA N OF F – I N T E RV I EW

A sit down interview with the Robb Janoff – the designer behind the iconic Apple logo When did you design the original Apple logo with the colorful stripes?

It was early 1977. The agency got the account (Apple) sometime January. The logo was introduced with the new product Apple II in April of that year.

Did you get a brief from them?

Really there was no brief. But the really funny thing was the only direction we got from Steve Jobs is: “don’t make it cute”. There were briefs on subsequent jobs but it was pretty loose at that time. There was a logo previous to my logo. It was a logo done by Ron Wayne who was a very brief partner of the two Steves early on. Ron did a pen and ink drawing of Sir Isaac Newton sitting under an Apple tree with a poem all around the border. And, I think when Steve Jobs started to get serious about the Apple II and getting a prototype for the design of the shell he realized that logo would not do. So he needed a new logo.

How many versions did you do for the presentation?

We presented two versions of the logo. One with and one without the bite. Just in case he thought the bite was too cute. Fortuanately he went with the one that gave it the most personality with the bite.

Frankly it was a no brainer and you would miss the mark if you don’t show some kind of an apple. When I presented I showed him several variations. Striped version, solid color version, metallic version. All those with the same shape.

So even then you knew you needed a solid color version and a metallic version?

Yes, you kind of had to. When you’re doing printing of either one or two color you need to have some way to go and I realized that the stripes would not always get it. The stripes really didn’t work as a halftone greyscale.

Do you use Macs today? Do you still work?

I’d really like to retire, but in this economy I really can’t. I do work on a Mac, it’s all I ever worked on. I would not know what to do with a left click and a right click. Been brand loyal all the way, even though the products cost a little bit more. I wouldn’t think of using anything else. Plus, for graphics and design Apple has it all over Microsoft.

You obviously didn’t design the logo on a computer?

No, and it’s a revelation to a lot of young designers. I get emails about the logo all the time asking me questions about the logo from all over the world and it’s really kind of very satisfying because it’s not something every designer gets a chance to talk to everybody because of some work you did. And, people ask me “did you design it on a computer?” and of course at the time computers couldn’t really do that for me. It was only years later till the Mac was designed, developed and refined that I even started working on a computer. At the time it was all pencil and paper, glue and cut paper, pens and all that stuff.

Do you like the changes Apple made to your original design over the years?

I do like them. The stripes served their purpose and they are definitely dated. I think it’s very important that a company like Apple keep very up-to-date and Steve Jobs is obviously very conscious of that and he has fabulous designers working for him in industrial design and graphic design. I feel great that it’s still the same basic silhouette even though it went through lots and lots of changes. The apple shape changed slightly from my original design in the early 80’s. The design firm Landor & Associates made the changes. They brightened the colors, they made the shapes much more symmetrical, much more geometric. When I designed it I pretty much did it freehand. I often think to myself, why didn’t I do that? It’s because it wasn’t where I was coming from at the time. I think they did a great job and it will be fascinating to see the next iteration and how it works out.

“they made the shapes much more symmetrical, much more geometric” Final question, what is your suggestion to our younger readers, what should they focus on to become great designers?

This is something I tell my kids: I could do this even if I’m not getting paid for it, because I like it so much, because now more than ever before there are so many people trying to become designers and work for agencies just because the tools are available. So, it’s harder and harder to get work.

And, the way some people have to get work is by apprenticing and working for nothing for somebody until they get that job, because there is so much competition.

21

T H E A P P L E LO G O A N D G OL DE N R AT IO

With great debate around the apple logo and if it conformed to the golden ratio we put the it to the test

The circles above are proportioned to the golden ratio and are overlayed on the logo 22

Inspecting the overlap on the logo shows that it does not conform to the golden ratio

Inspecting the overlap on the logo shows that it does not conform to the golden ratio

L E C OR BU SI E R The first designer to openly admit using the sequence?

It is evident that designers use of the Fibonacci sequence in both ancient and current design is heavily debated. It is evident that the sequence appears in nature, however, there is no clear evidence of when it was first used in design. Though there is still speculation in some modern design whether the sequence has been used, it is becoming more common that designers are openly showing their use of the Fibonacci sequence within their design.

Swiss architect/painter Le Corbusier designed the UN building in New York in 1952 and confirmed that the golden ratio was used as measurement for his design.

Corbusier also created art within his architecture called ‘Modulor Man’, which was a piece used solely with the Fibonacci sequence.

23

LE CORBUSIER’S ‘MODULAR’ SEGMENT Le Corbusier developed ‘Modulor’ in the long tradition of Vitruvius, Leonardo da Vinci’s Vitruvian Man, the work of Leon Battista Alberti, and other attempts to discover mathematical proportions in the human body and then to use that knowledge to improve both the appearance and function of architecture. The system is based on human measurements, the double unit, the Fibonacci numbers, and the golden ratio. Le Corbusier described it as a “range of harmonious measurements to suit the human scale, universally applicable to architecture and to mechanical things.”

With the Modulor, Le Corbusier sought to introduce a scale of visual measures that would unite two virtually incompatible systems: the Anglo-Saxon foot and inch and the international metric system. The numbers used and marked by Le Corbusier below are not Fibonacci numbers however they do follow the ratio with the two previous numbers adding up to following which was the purpose of the design. For example the green numbers on the right 76 + 126 =204, 126 + 204 =330 and so on until 2260.

432

2260

698

1829

1397

1130

863

1130

698

534 432 330 267 165 102

204 126 76