The Golden section ratio: Phi

The golden ratio has a long and ancient history dating back at least to Euclid who called it 'dividing a line into mean and extreme ratio'. This page shows how it relates to the Fibonacci numbers and how to calculate it.
Contents of this page
The Things To Do icon means there is a Things to do section of questions to start your own investigations. The calculator calculator icon indicates that there is a live interactive calculator in that section.

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1 What is the golden section (or Phi)?

Also called the golden ratio or the golden mean, what is the value of the golden section?

1.1 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 biggerSquares 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
One definition of the golden section number is that
to square it you just add 1
or, if we let this value be denoted by the upper-case Greek letter Phi Φ in mathematics:
Phi2 = Phi + 1
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.

Phi2 = 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:

The two values are
1 + √5 = 1·6180339887...
2
1 − √5 = −0·6180339887...
2
Did you notice that their decimal parts are identical?
The larger value 1·618... we denote using the capital Greek letter Phi written as Φ
the smaller value −0·618... is written as −φ using the small Greek letter phi.

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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1.2 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.
bisect a line For instance, Book 1, Proposition 10 to find the exact centre of any 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". 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 = AGi.e. using the lengths of the sections 1 − g = g
AGABg1
which we rearrange to get 1 – g = g2.
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 that we found Phi above:
g = −1 + √5 or g = −1 − √5
22


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". There is an interesting article on The Golden ratio at the St Andrew's MacTutor History of Mathematics site.

1.2.1/ You do the maths... /

  1. 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?

1.2.2 Links on Euclid and his "Elements"

2 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:

3 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".

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4 The Value of Phi

4.1 Phi to many decimal places

Φ has the value  √5 + 1   and φ is  √5 – 1  .
2 2

Both have identical fractional parts after the decimal point. Both are also irrational which means that Later on this page we will show why Φ and φ cannot be written as exact fractions. There is another surprise in store later when we find which fractions are the best approximations to Φ.

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...)

4.1.1 Phi decimal places Calculator

Phi to N dps C A L C U L A T O R
dps in base
R E S U L T S  


 

4.2 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.

4.3 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.

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5 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.

graph of y=Phi x 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.

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5.1 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:
fibratio.gif
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)
But, using the The Fibonacci relationship we can replace F(i+2) by F(i+1) + F(i) and then simplify the resulting fraction, as follows:
F(i+2) = F(i+1) + F(i)
F(i+1)F(i+1)
= F(i+1) + F(i)
F(i+1)F(i+1)
= 1 + F(i)   (B)
F(i+1)
So, replacing F(i+2)/F(i+1) in (A) by the right hand side of (B), we have:
x = F(i+1) = 1 + F(i) = 1 + 1
F(i)F(i+1)x
Multiplying both sides by X gives:
x2 = x + 1
But we have seen this equation before in A simple definition of Phi so know that x is, indeed, exactly Φ!

Remember, this supposed that the ratio of two pairs of neighbours in the Fibonacci series was the same value. This only happens "in the limit" as mathematicians say. So what happens is that, as the series progresses, the ratios get closer and closer to this limiting value, or, in other words, the ratios get closer and closer to Φ the further down the series that we go.

Did you notice that we have not used the two starting values in this proof? No matter what two values we start with, if we apply the Fibonacci relationship to continue the series, the ratio of two terms will (in the limit) always be Phi!

5.2 But there are two values that satisfy X2 = X + 1 aren't there?

Yes, there are. The other value, –phi which is –0·618... is revealed if we extend the Fibonacci series backwards. We still maintain the same Fibonacci relationship but we can find numbers before 0 and still keep this relationship:
i...–10–9–8–7–6–5–4 –3–2–1012345 678910...
Fib(i)...–5534–21 13 –8 5 –3 2 –1 1 0 1 1 235 8 13 21 34 55 ...
When we use this complete Fibonacci series and plot the ratios F(i)/F(i–1) we see that the ratios on the left-hand side of 0 are
1 = -1,  -1 = -0.5,  2 = -0.666..., -3 = -0.6,  5 = -0.625,  ... 
-1 2 -3 5 -8
Plotting these shows both solutions to X2 = X + 1:-
fibPlot

5.3 More on Fibonacci Number ratios

What if we take the ratio not of neighbouring Fibonacci numbers but Fibonacci numbers one apart, i.e. F(n)/F(n-2)?

Experiment with a spreadsheet (or calculator) and you will soon find that F(n)/F(n-2) gets closer and closer to 2.618... as n gets larger. This is Φ+1 and also Φ2.

What if the numbers are 3 apart i.e. F(n)/F(n-3)?
This time the ratio gets closer to 4.236... or Φ3
Is there a general rule for the limit of F(n)/F(n-k)?
Yes! This ratio gets closer and closer to Φk.

If is easy to verify this using a small variation on the method above (where we showed F(i)/F(i+1) gets closer to Φ as i gets larger), or using Binet's Formula that we met earlier on the page called A Formula For Fib(n).

6 Another definition of Phi

We defined Φ to be (one of the two values given by)
Phi2 = Phi+1

Suppose we divide both sides of this equation by Phi:
Phi = 1 + 1/Phi

Here is another definition of Φ - that number which is 1 more than its reciprocal
(the reciprocal of a number is 1 over it so that, for example, the reciprocal of 2 is 1/2 and the reciprocal of 9 is 1/9).

7 Phi as a continued fraction

Look again at the last equation:
Phi = 1 + 1/Phi
This means that wherever we see Phi (Φ) we can substitute (1 + 1/Phi).
Since we see Phi on the right hand side, lets substitute it in there!
Phi = 1 + 1/(1 + 1/Phi)
In fact, we can do this again and again and get:
Phi = 1 + 1
1 + 1
1 + 1
1 + ..
This unusual expression is called a continued fraction since we continue to form fractions underneath fractions underneath fractions.

This continued fraction has a big surprise in store for us....

7.1 Phi is not a fraction

But Phi is a fraction .. it is (√5 + 1)/2.
Here, by a fraction we mean a number fraction such as 2/3 or 17/24 or 12/7. The first is a proper fraction since it are less than 1. Also 5.61 is a fraction, a decimal fraction since it is 561/100, the ratio of a whole number and a power of ten.
Strictly, all whole numbers can be written as fractions if we make the denominator (the part below the line) equal to 1! However, we commonly use the word fraction when there really is a fraction in the value.

Mathematicians call all these fractional (and whole) numbers rational numbers because they are the ratio of two whole numbers and it is these number fractions that we will mean by fraction in this section.

It may seem as if all number can be written as fractions - but this is, in fact, false. There are numbers which are not the ratio of any two whole numbers, e.g. √2 = 1.41421356... , π = 3.14159..., e = 2.71828... . Such values are called ir-ratio-nal since they cannot be represented as a ratio of two whole numbers (i.e. a fraction). A simple consequence of this is that their decimal fraction expansions go on for ever and never repeat at any stage!
Any and every fraction has a decimal fraction expansion that either

Can we write Phi as a fraction?
The answer is "No!" and there is a surprisingly simple proof of this. Here it is. [with thanks to Prof Shigeki Matsumoto of Konan University, Japan]
First we suppose that Φ can be written as a fraction and then show this leads to a contradiction, so we are forced to the conclusion Φ cannot be written as a fraction:
Suppose Φ = a/b and that this fraction is in its lowest terms which means that:
But we know from the alternative definitions of Φ and φ that
Φ = 1/φ and φ = Φ – 1 which we put in the fraction for Φ:
Φ = 1/(Φ –1).
Now we substitute a/b for Φ:
a/b = 1/(a/b –1) = b/(ab)
But here we have another fraction for Φ that has a smaller numerator, b since a > b
which is a contradiction because we said we had chosen a and b were chosen to be the smallest whole numbers.
So we have a logical impossibility if we assume Φ can be written as a proper fraction
and the only possibility that logic allows is that Φ cannot be written as a proper fraction - Φ is irrational.

7.2 Rational Approximations to Phi

If no fraction can be the exact value of Φ, what fractions are good approximations to Φ?

The answer lies in the continued fraction for Φ that we saw earlier on this page.
If we stop the continued fraction for Φ at various points, we get values which approximate to Φ:

Φ ≈ 1
 
Φ ≈ 1 + 1 = 2
1
 
Φ ≈ 1 + 1 = 3
1 + 12
1
 
Φ ≈ 1 + 1
1 + 1
1 + 1
1
= 5
3

The next approximation is always 1 + 1/(the previous approximation) shown in green.
Did you notice that this series of fractions is just the ratios of successive Fibonacci numbers - surprise!

The proper mathematical term for these fractions which are formed from stopping a continued fraction for Φ at various points is the convergents to Φ. The series of convergents is

1, 2, 3, 5, 8, ...
11235

7.3 Why do Fibonacci numbers occur in the convergents?

This is an optional section where we show exactly why the Fibonacci numbers appear in the successive approximations (the convergents) above. Skip to the next section if you like

The convergents start with 1/1 = F(1)/F(0)
where F(n) represents the n-th Fibonacci number.

To get from one fraction to the next, we saw that we just take the reciprocal of the fraction and add 1:
so the next one after F(1)/F(0) is

1 +
1
F(1) / F(0)
= 1 +
F(0)
F(1)
=
F(1) + F(0)
F(1)

But the Fibonacci numbers have the property that two successive numbers add to give the next, so F(1) + F(0) = F(2) and our next fraction can be written as

F(1) + F(0)
F(1)
=
F(2)
F(1)

So starting with the ratio of the first two Fibonacci numbers the next convergent to Φ is the ratio of the next two Fibonacci numbers.

This always happens:
if we have F(n)/F(n – 1) as a convergent to Φ, then the next convergent is F(n+1)/F(n).

We will get all the ratios of successive Fibonacci numbers as values which get closer and closer to Φ.

You can find out more about continued fractions and how they relate to splitting a rectangle into squares and also to Euclid's algorithm on the Introduction to Continued Fractions page at this site.

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8 Other ways to find Phi using your calculator

Here are two more interesting ways to find it.

8.1 Calculator Method 1: Invert and Add 1

Keep repeating these two operations (take the reciprocal, add 1) and you will find that soon the display does not alter and settles down ("converging" as mathematicians call it) to a particular value, namely 1.61803... .

In fact, you can start with many values but not all (for instance 0 or -1 will cause problems) and it will still converge to the same value: Phi.

8.1.1 Why?

The formula Φ = 1 + 1/Φ shows us where the two instructions come from.
To start, we note that the simplest approximation to the continued fraction above is just 1.

8.1.2/ You do the maths... /

  1. In Calculator method 1, 0 causes a problem because we cannot take its reciprocal.
    So if x is -1, when we take its reciprocal (1/-1 = -1 ) and add 1 we get 0. So 0 and -1 are bad choices since they don't lead to Phi.
    What value of x will give -1? And what value of x would give that value?
    Can you find a whole series of numbers which, in fact, do not lead to Phi with Calculator method 1?
    [Thanks to Warren Criswell for this problem.]

8.1.3 Method 1 Calculator: 1/x and +1

+ and / C A L C U L A T O R
x=

R E S U L T S

calculator: 1/x and +1 + and √ +n and 1/x

8.2 Calculator Method 2: Add 1 and take the square-root

Here is another way to get Φ on your calculator. Keep repeating these two instructions and you will find it too converges to Φ.

8.2.1 Why?

This time we have used the other definition of Phi, namely
Φ2 = Φ + 1
or, taking the square root of both sides:
Φ = √Φ + 1
Can you see why we must start with a number which is not smaller (i.e. is not more negative) than –1?

8.2.2 Method 2: Calculator: +1 and √

+1 and √ C A L C U L A T O R
  x=

R E S U L T S

calculator: 1/x and +1 +1 and √ +n and 1/x
If one value tends towards Phi, what about the other value (the result after pressing the "x+1" button)?

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9 Similar numbers

Robert Kerr Baxter wrote to me about other numbers that have the Φ property that when you square them their decimal parts remain the same:
Phi = 1.618033.. and Phi2 is 2.618033..
Φ has the value  √5 + 1
2

Rob had noticed that this happens if we replace the sqrt5 with sqrt13 or sqrt17 or sqrt21 and so on. The series of number here is 5, (9), 13, 17, 21, (25), 29, ... which are the numbers that are 1 more than the multiples of 4. The numbers 9 and 25 are in brackets because they are perfect squares, so taking their square roots gives a whole number - in fact, an odd number - so when we add 1 and divide the result by two we just get a whole number with .00000... as the decimal part.

9.1 Why does this happen?


Algebra can come to our help here and it is a nice application of "Solving Quadratics" that we have already seen in the first section on this page.

We want to find a formula for the numbers (x, say) "that have the same decimal part as their squares". So, if we subtract x from x2, the result will be a whole number because the decimal parts were identical.
Let's call this difference N, remembering that it is a whole number.
So

the difference between x2 and x is N, a whole number
is a description of these numbers in words. We can write this in the language of mathematics as follows:
x2 – x = N spacer or, adding x to both sides: spacer x2 = N + x
and we can "solve" it in exactly the same way as we did for Φ's quadratic: x2 = 1 + x. The formula for x this time is
x = 1 ± √1 + 4 N
2
You can see that, under the square-root sign, we have 1 plus a multiple of 4 which gives the series:
N:12345...
1+4N:59131721...
just as Rob had found.

For example: if we choose N = 5, then the number x (that increases by exactly 5 when squared) is

x = 1 ± √1 + 4×5 = 1 ± sqrt21 = 2.791287847.. and x2 = 7.791287847... = 5 + x
22
Checking we see that the square of this x is exactly N = 5 more than the original number x.

Another example: take Φ, which is (1 + √5)/2 or (1 + √(1+4×1))/2 so that N = 1.
Thus we can "predict" that Φ squared will be >1 more than Φ itself and, indeed, Phi = 1.618033.. and Phi2 = 2.618033.. .
We can do the same for other whole number values for N.

More generally: There is nothing in the maths of this section that prevents N from being any number, for instance 0·5 or π. Suppose N is π = 3.1415926535... . We can find the number x that, when squared, increases by exactly π! It is

x = 1 ± 1 + 4 π = 1 ± 12.566370614... = 2·3416277185...
22

and x2 = 5·483220372... = 2·3416277185... + 3.1415926535... = x + π as requested!

9.2 Add and Invert Calculator

+n and 1/x; C A L C U L A T O R
      x=

R E S U L T S

calculator: 1/x and +1 +1 and √ +n and 1/x

9.2.1/ You do the maths... /

  1. Make a table of the first few numbers similar to Phi in this way, starting with Phi and its square.
  2. We have only used the + sign in the formula for x above, giving positive values of x.
    What negative values of x are there, that is negative numbers which, when squared (becoming positive) have exactly the same decimal fraction part?
  3. What is the number that can be squared by just adding 0·5?
  4. Is there an upper limit to the size of N?
    Can you use the formula to find two numbers that increase by one million (1,000,000) when squared?
  5. Can N be negative?
  6. For instance, can we use the formula to find a number (as we have seen, there are two of them) that is 0·5 smaller when it is squared?
  7. What about a number that decreases by 1 when it is squared?
  8. Is there a lower limit for the value of N?
We look at some other numbers similar to Phi but in a different way on the (optional) Continued Fractions page. This time we find numbers which are like the Golden Mean, Phi, in that their decimal fraction parts are the same when we take their reciprocals, i.e. find 1/x. They are called the Silver Means.

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Created: 1996, Updated: 30 August 2023