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=05 and 052 is 025=1/4
32 is 9
1/5=02 and 022 is 004=1/25
102 is 100
1/10=01 and 012 is 001=1/100
One definition of Phi (the golden section number) is that to square it you just add 1
or, in mathematics:
In fact, there are two numbers with this property, one is Phi and another is closely related to it when we write out some of its decimal places.
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.
Multiplying both sides by Phi gives a quadratic equation:
We can solve this quadratic equation to find two possible values for Phi as follows:
First note that (Phi - V2)2 = Phi2 - Phi + V4 Using this we can write Phi2 - Phi - 1 as (Phi - V2)2 - 5/4 and since Phi2 - Phi - 1 = 0 then (Phi - V2)2 must equal 5/4 Taking square-roots gives (Phi - Vz) = W(5/4) or -V(5/4). so Phi=1/2+V(5/4) or Vz-Vp/J.
We can simplify this by noting that V(5/4) = V5/V4 = V5/z The two values of Phi are therefore: !/2 + V5/2 and l/2 - V5/2
Use your calculator to see that the values of these two numbers are 1-6180339887. and -06180339887...
Did you notice that their decimal parts are identical?
We will name the first value Phi and the second - phi using the first letter to tell us if we want the bigger value (Phi) 1618... or the smaller one (phi) 0 618... .
Note that Phi is just 1+phi. As a little practice at algebra, use the expressions above to show that phi times Phi is exactly 1. Here is a summary of what we have found already that we will find very useful in what follows:
Phi phi = 1, Phi - phi = 1, Phi + phi = V5 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 = (V5 + l)/2 phi = (V5 - l)/2
1-61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More..
Euclid, the Greek mathematician who lived from about 365BC to 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 ofany line AB
1. Put your compass point on one end of the line at point A.
2. Open the compasses to the other end of the line, B, and draw the circle.
3. Draw another circle in the same way with centre at the other end of the line.
4. 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".
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 (ie is the same as the ratio AG/AB). 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
Notice that earlier we defined Phi2 as Phi+1 and here we have g2 = 1-g or g2+g=1. We can solve this in the same way as for Phi and we find that
So there are two numbers which when added to their squares give 1. For our geometrical problem, g is a positive number so the first value is the one we want. This is our friend phi also equal to Phi-1 (and the other value is merely -Phi).
It seems that this ratio had been of interest to earlier Greek mathematicians, especially Pythagoras (580BC - 500BC) and his "school".
Things to do
1. Suppose we labelled the parts of our line as follows:
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.
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 16. 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 Egy pt.
^^ 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.
Euclid (365BC - 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 (0) to represent the golden ratio, using the initial letter of the Greek Phidias who used the golden ratio in his sculptures.
Luca Pacioli (also written as Paccioli) 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, mathematicians also use the Greek letter tau (t) , the initial letter of tome which is the Greek work for "cut" as well as phi.
1-61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 . .More. . F^^
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