The Golden Section Ratio: Phi - Dr Ron Knott
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What is the golden section (or Phi)?
Also called the golden ratio or the golden mean, what is the value of the golden section?A simple definition of Phi
There are just two numbers that remain the same when they are squared namely 0 and 1. Other numbers get bigger and some get smaller when we square them:| Squares that are bigger | Squares that are smaller |
|---|---|
| 22 is 4 | 1/2=0·5 and 0·52 is 0·25=1/4 |
| 32 is 9 | 1/5=0·2 and 0·22 is 0·04=1/25 |
| 102 is 100 | 1/10=0·1 and 0·12 is 0·01=1/100 |
Here is a mathematical derivation (or proof) of the two values. You can skip over this to the answers at the foot of this paragraph if you like.The two values arePhi2 = Phi + 1 or, subtracting Phi + 1 from both sides: Phi2 – Phi – 1 = 0
We can solve this quadratic equation to find two possible values for Phi as follows:
- First note that (Phi – 1/2)2 = Phi2 – Phi + 1/4
- Using this we can write Phi2 – Phi – 1 as (Phi – 1/2)2 – 5/4 and since Phi2 – Phi – 1 = 0 then (Phi – 1/2)2 must equal 5/4
- Taking square-roots gives (Phi – 1/2) = +√5/2 or –√5/2.
- so Phi = 1/2 + √5/2 or 1/2 – √5/2.
| 1 + √5 | = 1·6180339887... |
| 2 | |
| 1 − √5 | = −0·6180339887... |
| 2 |
The large P indicates the larger positive value 1·618... and the small p denotes the smaller positive value 0·618.... Φ = (√5 + 1) / 2, φ = (√5 – 1) / 2
Other names used for these values are the Golden Ratio and the Golden number. We will use the two Greek letters Phi (Φ) and phi (φ) in these pages, although some mathematicians use another Greek letters such as tau (τ) or else alpha (α) and beta (β). Here, Phi (large P) is the larger value, 1.618033.... and phi (small p) is the smaller positive value 0.618033... which is also just Phi – 1.
As a little practice at algebra, use the expressions above to show that φ × Φ = 1.
Here is a summary of what we have found already that we will find useful in what follows:
| Phi phi = 1 Phi - phi = 1 Phi + phi = √5 | |
| Phi = 1.6180339.. | phi = 0.6180339.. |
| Phi = 1 + phi | phi = Phi – 1 |
| Phi = 1/phi | phi = 1/Phi |
| Phi2 = Phi + 1 | (–phi)2 = –phi + 1 or phi2 = 1 – phi |
| Phi = (√5 + 1)/2 | phi = (√5 – 1)/2 |
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A bit of history...
Euclid, the Greek mathematician of about 300BC, wrote the Elements which is a collection of 13 books on Geometry (written in Greek originally). It was the most important mathematical work until this century, when Geometry began to take a lower place on school syllabuses, but it has had a major influence on mathematics.It starts from basic definitions called axioms or "postulates" (self-evident starting points). An example is the fifth axiom that there is only one line parallel to another line through a given point. From these Euclid develops more results (called propositions) about geometry which he proves based purely on the axioms and previously proved propositions using logic alone. The propositions involve constructing geometric figures using a straight edge and compasses only so that we can only draw straight lines and circles. 
For instance, Book 1, Proposition 10 to find the exact centre of any line AB
- Put your compass point on one end of the line at point A.
- Open the compasses to the other end of the line, B, and draw the circle.
- Draw another circle in the same way with centre at the other end of the line.
- This gives two points where the two red circles cross and, if we join these points, we have a (green) straight line at 90 degrees to the original line which goes through its exact centre.
In Book 6, Proposition 30, Euclid shows how to divide a line in mean and extreme ratio which we would call "finding the golden section G point on the line". He describes this geometrically. <-------- 1 ---------> A G B g 1–g Euclid used this phrase to mean the ratio of the smaller part of this line, GB to the larger part AG (ie the ratio GB/AG) is the SAME as the ratio of the larger part, AG, to the whole line AB (i.e. is the same as the ratio AG/AB). We can see that this is indeed the golden section point if we let the line AB have unit length and AG have length g (so that GB is then just 1–g) then the definition means that
| GB | = | AG | i.e. using the lengths of the sections | 1 − g | = | g |
| AG | AB | g | 1 |
| g = | −1 + √5 | or g = | −1 − √5 |
| 2 | 2 |
It seems that this ratio had been of interest to earlier Greek mathematicians, especially Pythagoras (580BC - 500BC) and his "school". There is an interesting article on The Golden ratio at the St Andrew's MacTutor History of Mathematics site.
You do the maths...
- Suppose we labelled the parts of our line as follows: A G B x 1 so that AB is now has length 1+x. If Euclid's "division of AB into mean and extreme ratio" still applies to point G, what quadratic equation do you now get for x? What is the value of x?
Links on Euclid and his "Elements"
- From Clarke University comes D Joyce's exciting project making Euclid's Elements interactive using Java applets. But why not look at the books themselves: they are classics in every sense!
This 3-volume set is cheap and has been the standard book for many years: - The Thirteen Books of Euclid's Elements, Books 1 and 2
- The Thirteen Books of Euclid's Elements, Books 3 to 9
- The Thirteen Books of Euclid's Elements, Books 10 to 13 all are by Thomas L Heath who translated and annotated Euclid's work, each is about 464 pages, published by Dover in paperback, second edition 1956.
- MacTutor's History of Maths Archives has an interesting page on the golden ratio and Euclid's Elements. The ratio itself is mentioned in several places and, since Euclid seems to have collected already known methods and facts then the golden ratio would have been known well before his time too. Of course is was not called the "golden ratio" then (a term originating in the 1820's probably), but Euclid's term (translated into English) is dividing a line in mean and extreme ratio.
- Euclid Book 6 Proposition 30: To cut a given finite straight line in extreme and mean ratio. This proposition refers back to an earlier one:
Euclid Book 2, Proposition 11 To cut a given straight line so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment
Phi and the Egyptian Pyramids?
The Rhind Papyrus of about 1650 BC is one of the oldest mathematical works in existence, giving methods and problems used by the ancient Babylonians and Egyptians. It includes the solution to some problems about pyramids but it does not mention anything about the golden ratio Phi.The ratio of the length of a face of the Great Pyramid (from centre of the bottom of a face to the apex of the pyramid) to the distance from the same point to the exact centre of the pyramid's base square is about 1·6. It is a matter of debate whether this was "intended" to be the golden section number or not. According to Elmer Robinson (see the reference below), using the average of eight sets of data, says that "the theory that the perimeter of the pyramid divided by twice its vertical height is the value of pi" fits the data much better than the theory above about Phi. The following references will explain circumstantial evidence for and against:
- The golden section in The Kings Tomb in Egypt.
- How to Find the "Golden Number" without really trying Roger Fischler, Fibonacci Quarterly, 1981, Vol 19, pp 406 - 410 Case studies include the Great Pyramid of Cheops and the various theories propounded to explain its dimensions, the golden section in architecture, its use by Le Corbusier and Seurat and in the visual arts. He concludes that several of the works that purport to show Phi was used are, in fact, fallacious and "without any foundation whatever".
- The Fibonacci Drawing Board Design of the Great Pyramid of Gizeh Col. R S Beard in Fibonacci Quarterly vol 6, 1968, pages 85 - 87; has three separate theories (only one of which involves the golden section) which agree quite well with the dimensions as measured in 1880.
- A Note on the Geometry of the Great Pyramid Elmer D Robinson in The Fibonacci Quarterly vol 20 (1982) page 343 shows that the theory involving pi fits much better than the one regarding Phi.
- George Markowsky's Misconceptions about the Golden ratio in The College Mathematics Journal Vol 23, January 1992, pages 2-19. This is readable and well presented. You may or may not agree with all that Markowsky says, but this is a very good article that tries to debunk a simplistic and unscientific "cult" status being attached to Phi, seeing it where it really is not! He has some convincing arguments that Phi does not occur in the measurements of the Egyptian pyramids.
Other names for Phi
Euclid (about 300BC) in his "Elements" calls dividing a line at the 0.6180399.. point dividing a line in the extreme and mean ratio. This later gave rise to the name golden mean.There are no extant records of the Greek architects' plans for their most famous temples and buildings (such as the Parthenon). So we do not know if they deliberately used the golden section in their architectural plans. The American mathematician Mark Barr used the Greek letter phi (φ) to represent the golden ratio, using the initial letter of the Greek Phidias who used the golden ratio in his sculptures.
Luca Pacioli (sometimes written as Paccioli), 1445-1517, wrote a book called De Divina Proportione (The Divine Proportion) in 1509. It contains drawings made by Leonardo da Vinci of the 5 Platonic solids. It was probably Leonardo (da Vinci) who first called it the sectio aurea (Latin for the golden section).
Today, some mathematicians use phi (φ) for the golden ratio as on these web pages and others use the Greek letters alpha (α) or tau (τ), the initial letter of tome which is the Greek work for "cut".
- A Mathematical History of the Golden Number R Herz-Fischler, Dover (1998) paperback. This is an informative book, densely packed with historical references to the golden mean and its other names.
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The Value of Phi
Phi to many decimal places
| Φ has the value | √5 + 1 | and φ is | √5 – 1 | . |
| 2 | 2 |
- They cannot be written as M/N for any whole numbers M and N;
- their decimal fraction parts have no pattern in their digits, that is, they never end up repeating a fixed cycle of digits
Here is the decimal value of Φ to 2000 places grouped in blocks of 5 decimal digits. The value of φ is identical but begins with 0·6.. instead of 1·6.. . Read this as ordinary text, in lines across, so Φ is 1·61803398874...)
Phi decimal places Calculator
Phi to N dps C A L C U L A T O R| dps in base |
  |
- Here is a text file of the first 10,000 decimal places (opens in a new window, 12K).
Phi to 1,500,000,000 places!
Simon Plouffe of Simon Fraser University notes that Greg J Fee programmed a method of his to compute the golden ratio (Phi) to ten million places in December 1996. He used Maple and it took 29 minutes and 16 seconds on an SGI R10000 194MHz computer. He lists the first 15,000 places. The latest record is 1,000,000,000,000 places in 2010 by A.J. Yee.Phi in Binary
Using base 2 (binary), is there a pattern in the digits of Phi? Let's have a look at the first 500 binary-digits (or bits):1·10011 11000 11011 10111 10011 01110 01011 11111 01001 01001 11110 00001 01011 11100 11100 11100 11000 00001 10000 00101 100 11001 11011 01110 01000 00110 10000 01000 01000 00100 01001 11011 01011 11110 01110 10001 00111 00100 10100 01111 11000 200 01101 10001 10101 00001 00011 10100 00110 00001 10001 11010 01010 10010 01110 11001 11111 10000 10110 00101 01001 11101 300 00100 11110 11011 11111 00000 01101 00011 10000 01000 10110 11010 11011 11110 00110 00001 00111 11110 00000 01100 01000 400 01101 11100 00100 10010 10000 10000 00001 10000 00000 01011 00000 11101 01100 10010 11101 00100 00001 11100 11001 10101 500 Here is a text file of 25,000 binary places (opens in a new window, 28K).
Neither the decimal form of Phi, nor the binary one nor any other base have any ultimate repeating pattern in their digits. This is because Phi is not a fration, an ir-rational number, and so its "decimal fraction" never gets into a repeating pattern in any base. Fractions always stop or else end up repeating the same pattern in their "decimal fraction". For more on this, the patterns and why any fraction either stops or repeats and much more see Fractions and Decimals.
1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More..
Phi and the Fibonacci numbers
On the Fibonacci and Nature page we saw a graph which showed that the ratio of successive Fibonacci numbers gets closer and closer to Φ.Here is the connection the other way round, where we can discover the Fibonacci numbers arising from the number Φ.
The graph on the right shows a line whose gradient is Φ, that is the line y = Φ x = 1·6180339.. x Since Φ is not the ratio of any two integers, the graph will never go through any points of the form (i,j) where i and j are whole numbers - apart from one trivial exception - can you spot it? So we can ask What are the nearest integer-coordinate points to the Φ line? Let's start at the origin and work up the line. The first is (0,0) of course, so these are the two integer coordinates of the only whole-number point exactly on the line! In fact ANY line y = k x will go through the origin, so that is why we will ignore this point as a "trivial exception" (as mathematicians like to put it). The next point close to the line looks like (0,1) although (1,2) is nearer still. The next nearest seems even closer: (2,3) and (3,5) is even closer again. So far our sequence of "integer coordinate points close to the Phi line" is as follows: (0,1), (1,2), (2,3), (3,5) What is the next closest point? and the next? Surprised? The coordinates are successive Fibonacci numbers!
Let's call these the Fibonacci points. Notice that the ratio y/x for each Fibonacci point (x,y) gets closer and closer to Phi = 1·618... but the interesting point that we see on this graph is that the Fibonacci points are the closest points to the Φ line.
1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More.. >
The Ratio of neighbouring Fibonacci Numbers tends to Phi
On the Fibonacci Numbers and Nature page we saw that the ratio of two neighbouring Fibonacci numbers soon settled down to a particular value near 1·6:
In fact, the exact value is Φ. As we take larger and larger Fibonacci numbers, so their ratio gets closer and closer to Phi. Why? Here we show how this happens. The basic Fibonacci relationship is
| F(i+2) = F(i+1) + F(i) The Fibonacci relationship |
The graph shows that the ratio F(i+1)/F(i) seems to get closer and closer to a particular value, which for now we will call x. If we take three neighbouring Fibonacci numbers, F(i), F(i+1) and F(i+2) then, for very large values of i, the ratio of F(i) and F(i+1) will be almost the same as the ratio of F(i+1) and F(i+2), so let's see what happens if both of these are the same value: x.
| F(i+1) | = | F(i+2) | = x (A) |
| F(i) | F(i+1) |