· First, we must confront [ the reader ] with beautiful specimens …
· Second, some preliminary education related to the selected specimens is needed …
· Third, the neophyte must be encouraged to help himself. The Socratic method is best here … The appreciation of beauty is scarcely to be distinguished from the activity of creation.

H E Huntley, The Divine Proportion: A Study in Mathematical Beauty

Topics

• Here are two images, one from architecture and the other from nature, which seem to suggest a common theme.

   

  

  The Double Spiral Staircase in the Vatican Museums

   

   

  

  Radiograph of the Shell of the Chambered Nautilus Pompilius Shell ¹

   

  An elegant, visual introduction to the basic theme of this lecture can be found at What the Hell is the Fibonacci Series?. Here is a more prosaic alternative. Consider the following infinite sequence

  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

  You will note that each number is the sum of the two immediately preceding ones. These are the Fibonacci numbers. Consider now the following figure, where a large rectangle C appears subdivided into two smaller rectangles A (which is actually a square) and B

   

  

  The Golden Rectangle

   

  If the two rectangles A and B are drawn in such a way that

  x / (x + y) = y / x

  we say that the rectangle A + B is a golden rectangle. In other words, the proportions of rectangle B are the same as those of C. Note that B is also a golden rectangle. A consequence of this last remark is that one can subdivide B in exactly the same way as C was subdivided. And so on. Here is a simple way to construct a golden rectangle:

   

  

  Diagram Showing How to Construct a Golden Rectangle

   

  Using a compass, with center in the midpoint of the bottom side of the square a, draw an arc from the upper right corner to the extension of the bottom side. Mark the intersection and complete the (golden) rectangle a+b. Note again that the rectangle b is also golden.
  x / y is called the golden ratio. It is usually denoted by the Greek letter φ (lowercase 'phi'), and sometimes by the corresponding uppercase Φ, and is equal to

  x / y = φ = ½ (1 + √5) = 1.618033989…

  The Fibonacci numbers and the golden ration are intimately related. As the Fibonacci sequence progresses, the ratio of a Fibonacci number to the preceding one becomes closer and closer to φ. Much more can be said about all this, but, for our purposes, the above is more than sufficient.
• A little historical excursus is now in order. The Fibonacci numbers take their name from the Italian mathematician Leonardo Pisano Fibonacci (1170 - 1250), who "wrote a number of important texts which played an important role in reviving ancient mathematical skills, and made significant contributions of his own." For example, his "Liber Abaci, published in 1202 after Fibonacci's return to Italy [ … ] was based on the arithmetic and algebra that Fibonacci had accumulated during his travels. The book, which went on to be widely copied and imitated, introduced the Hindu-Arabic place-valued decimal system and the use of Arabic numerals into Europe."

  "A problem in the third section of Liber Abaci led to the introduction of the Fibonacci numbers and the Fibonacci sequence for which Fibonacci is best remembered today:

  'A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?'

  The resulting sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … (Fibonacci omitted the first term in Liber Abaci)."
  [ from Leonardo Pisano Fibonacci ]

  For Fibonacci's rabbits and more, read the nicely illustrated   Rabbits, Cows and Bees Family Trees.
  It may now come as a suprise that the golden ratio is, instead, much more ancient. The Golden Ratio seems to have been discovered first by Euclid of Alexandria (ca 325 BC - ca 265 BC). In fact, historian of mathematics now suggest that its origin reaches back to Pythagoras of Samos (ca 569 BC - ca 475 BC), who is credited with the discovery illustrated in the figure below.

   

  

  In a Regular Pentagon, C Divides AB in the Golden Ratio

   

  Notice also that "the term golden ratio [ … ] was never used by the [ ancient ] mathematicians [ … ] The common term used by early writers was simply 'division in extreme and mean ratio.' Pacioli certainly introduced the term 'divine proportion' [ … ] The names now used are golden ratio, golden number or golden section." This observation is taken from The Golden Ratio, where you will find much more historical information.
• We are now ready to explore the significance of Pythagoras', Euclid's, Fibonacci's findings, and why these findings, seemingly of interest only to mathematicians, are greatly relevant in a whole range of phenomena, many of which are definitely of the 'everyday' kind.
  There are many excellent resources on the web and in the library. Here, among the books, I will only suggest, in addition to Huntley's work, the famous collection edited by Gyorgy Kepes, Module, Proportion, Symmetry, Rhythm (George Braziller, New York 1966). Among the most useful websites, I will recommend here Ron Knott's Fibonacci Numbers and the Golden Section at Surrey University; Phi: That Golden Number Φ; Jim Loy's Fibonacci Numbers and The Golden Rectangle and the Golden Ratio; ThinkQuest's The Fibonacci Series. Finally visit A Golden Proposal for the World Trade Center, which features numerous, very beautiful illustrations. You can also watch an animation of another method of construction at The Golden Rectangle Demonstration. Other useful sites are mentioned under Readings, Resources and Questions below.
• Perhaps a good starting point is Christopher Green's All That Glitters: A Review of Psychological Research on the Aesthetics of the Golden Section, originally published in the journal Perception, 24, 937-968 (1995). It includes a very useful list of references. [ You may also want to read the companion paper Mathematical and Historical Background to "All That Glitters..." ] This is the abstract:

  "Since at least the time of the Ancient Greeks, scholars have argued about whether the golden section—a number approximately equal to 0.618—holds the key to the secret of beauty. Empirical investigations of the aesthetic properties of the golden section date back to the very origins of scientific psychology itself, the first studies being conducted by Fechner in the 1860s. This paper reviews both historical and contemporary issues with regard to the alleged aesthetic properties of the golden section. An introductory section (1) describes the most important mathematical occurrences of the golden section. As well, brief reference is made to research on natural occurrences of the golden section, and to Ancient and Medieval knowledge and application of the golden section, primarily in art and architecture. Major sections following the introduction discuss and critically examine (2) empirical studies of the putative aesthetic properties of the golden section dating from the mid-19th century up to the 1950s, and (3) the empirical work of the last three decades. It is concluded that there seems to be, in fact, real psychological effects associated with the golden section, but that they are relatively sensitive to careless methodological practices."

  The basic idea is, to quote Green again, that "it has often been claimed, since the time of the Ancient Greeks, and perhaps much earlier, that the golden section is the most aesthetically pleasing point at which to divide a line." [ my emphasis ] One of the most famous ancient monuments to illustrate this claim is the Parthenon, where the base and several modules seem to have been built according to the golden section. Notice that the Greeks were very aware of the visual impact of their buildings. For example, the base of the Parthenon is intentionally not perfectly horizontal, in order that it may appear horizontal!
  The 'canon' introduced by Polyclitus (middle of the 5th century BC) seems to have been based, in part, on the golden section. This claim is supported by a careful examination of Roman copies of his famous Doryphorus. See, e.g.,   Polyclitus's Canon and the Idea of Symmetria, or E Bresnan's Canon of Polyclitus.

   

  

  Polyclitus' Doryphoros (Roman Copy in Marble of Bronze Original, ca 450 BC)

   

  The number of art works embodying, one way or another, the golden section, is immense, ranging from Polyclitus to Leonardo da Vinci, to the architect Le Corbusier and The Modulor. Read a good, short review of Le Corbusier's Modulor and Modulor 2.

   

  

  Le Corbusier's Modulor

   

  Notice that here I use 'art' to include also, for example, music:

  "The golden mean ratio can be found in many compositions mainly because it is a 'natural' way of dealing with divisions of time. One can find it in a lot of works by Mozart, Beethoven, Chopin, etc, etc. It is a question if it was used in a deliberate way or just intuitively (probably intuitively). On the other hand, composers like Debussy and Bartok have made a conscious attempt to use this ratio and the Fibonacci series of numbers which produces a similar effect (adjacent members of the series give ratios getting closer and closer to the golden mean ratio). Bartok intentionally writes melodies which contain only intervals whose sizes can be expressed in Fibonacci numbers of semitones. He also divides the formal sections of some of his pieces in ratios corresponding to the golden mean. Without going into much detail, Debussy also does this in some of his music and so does Xenakis (a composer who writes exclusively by using stochastic distributions, set theory, game theory, random walks, etc.) in his first major work, Metastasis. The idea is not new, already in the Renaissance composers used it and built melodic lines around the Fibonacci sequence—just like Bartok's Music for Strings, Percussion and Celesta." [ from Golden Ratio and Fibonacci Numbers ]

  The golden section appears also in humbler incarnations, such as the shape of many windows and doors (check it out!), and even in items such as paper and photographic film sizes ( see Proportions: Theorie and Construction: Golden Section or Golden Mean, Modulor, Square Root of Two ). Even stock market trends seem to patterned to some extent on Fibonacci 'Golden Ratio' of 1.6180339887498949 (FIBO).
• Even more impressive is the ubiquity of the golden ratio in nature. Before looking at some examples, I should mention the 'golden or logarithmic spiral.'

   

  

  The Geometrical Construction of the Golden Spiral

   

  ACIF and BJKF are golden rectangles. Using a compass, with center in B, draw the arc AD; then, with center in J, draw the arc DK, etc. The resulting curve is a golden or logarithmic spiral. See an elegant, animated presentation at Golden.Spiral. Visit also A K Erbas' Spira Mirabilis. Please note: 1) Not all spirals are golden! 2) Phyllotaxis (see below) includes many more patterns which are not based on the golden ratio!

   

  

  The Arrangement of Seeds in a Sunflower

   

  Notice "that the seeds seem to form spirals curving both to the left and to the right. If you count those spiralling to the right at the edge of the picture, there are 34. How many are spiralling the other way? You will see that these two numbers are neighbours in the Fibonacci series." [ from Where Equiangular Spirals Are ]

   

  

  Tendrils Exhibiting Logarithmic Spirals

   

  The two examples illustrated above, are instances of a more general phenomenon, phyllotaxis. "Leaves, petals and other organs in plants are not arranged randomly, rather they seem to follow well-defined rules. This geometrical arrangement of organs is one of the main questions in plant morphogenesis, and the study of patterns thus produced is named phyllotaxis, from the Greek phyllo (leaf) and taxis (order)." [ from Phyllotaxis ] Visit Smith College's Phyllotaxis: An Interactive Site for the Mathematical Study of Plant Pattern Formation.
  Recent work by Patrick Shipman and Alan Newell has shown that, at least in cacti, "spiral patterns, and the Fibonacci relationships among the spirals, arise out of simple mechanical forces acting on a growing plant."
  Read   Cactus Patterns Buckle Up.

   

  

  Computer Simulation (bottom) of Spiral Patterns in a Cactus (top)

   

• Once again we have barely scratched the surface. Use the references given here, directly and indirectly, to explore this topic further, and look around, near and far, for more examples.

 
Readings, Resources and Questions

• ¹ This image was kindly scanned by Daniel J Denis from H E Huntley, The Divine Proportion: A Study in Mathematical Beauty (Dover Publications, New York 1970). This book is a classical survey of the golden section and its recurring appearances in nature, the arts and the sciences.
• A very interesting website is the Russian Museum of Harmony and Golden Section, where you can even read about Fibonacci computers. See also Laura Smaller's nice introduction to The Fibonacci's Sequence; and The Life and Numbers of Fibonacci Fibonacci Sequence Around the World; Der Goldene Schnitt: Theoretische Überlegungen und Beispiele in Wissenschaft und Kunst (if you can read German). Finally a useful, interactive site is Ask Dr Math.
• In the second Topics bullet, it is noted that "Fibonacci omitted the first term" in the sequence. Why would he have done that?
• A beautiful photograph of a Chambered Nautilus-Halved by Edward Weston can be found at the Amon Carter Museum.
• A good list of links can be found at DMoz, under Fibonacci Numbers, and a very useful bibliography of applications of Fibonacci's numbers in various areas, such as biology, physics, archeaeology, architecture and the fine arts, is at Fibonacci Facts: Information Sheet on Fibonacci Numbers.
• A very good recent book on these topics is The Golden Ratio. The Story of Phi, the World's Most Astonishing Number, by Mario Livio (Broadway Books, NY, 2002, 2003).
• Although I can not enter into any detail here, it is interesting to note that the golden ratio plays an important role in chaotic systems, which goes to show that even 'chaos' is not so chaotic after all.