0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1

This pattern can be seen in the following list of the first 72 Fibonacci numbers:

Lucien Khan arranged these 60 digits of the pattern in a circle, as shown in illustration below:

Here he found other interesting results:

• The zeros align with the 4 cardinal points on a compass.
• The fives align with the 8 other points of the 12 points on a clock.
• Except for the zeros, the number directly opposite each number adds to 10.

Lucien postulates that ancient knowledge of these relationships contributed to the development of our modern use of 60 minutes in an hour, and presentation of numbers on the face of the clock.

I found too that any group of four numbers that are 90 degrees from each other (15 away from each other in the circle) sum to 20, except again for the zeros. As an example, use 1, 7, 9 and 3, which appear one to the right of each of the compass points.

Additionally, every group of five numbers that define the points of the 12 pentagons on the circle also create a pattern. Four of the pentagons have even-numbered last digits of 0, 2, 4, 6, and 8. The remaining eight pentagons have odd-numbered last digits of 1, 3, 5, 7 and 9.

Another interesting pattern yet was observed by Lucien Khan: The 216th number is this sequence is 619220451666590135228675387863297874269396512. The sum of all the digits in that number add up to 216, as well. He notes that it is believed that the secret or hidden name of God contains 216 characters. There are many other fascinating relationships and sacred geometries, which are presented by Lucien Khan in more detail at the links below.

References:

https://docs.google.com/document/d/1mVWd1aLiYZQU8VvYFBnW8kxodeYim3bYDIFfh-w42eU/pub

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• If you square Phi, you get a number exactly 1 greater than itself: 2.618…, or

Φ²  = Φ + 1.

• If you divide Phi into 1 to get its reciprocal, you get a number exactly 1 less than itself: 0.618…, or

1 / Φ = Φ – 1.

These relationships are derived from the dividing a line at its golden section point, the point at which the ratio of the line (A) to the larger section (B) is the same as the ratio of the larger section (B) to the smaller section (C).

This relationship is expressed mathematically as:

A = B + C, and

A / B = B / C.

Solving for A, which on both sides give us this:

B + C = B²/C

Let’s say that C is 1 so we can determine the relative dimensions of the line segments. Now we simply have this:

B + 1  = B²

This can be rearranged as:

B² – B – 1 = 0

Note the various ways that this equation can be rearranged to express the relationship of the line segments, and also Phi’s unique properties:

B² = B + 1

1 / B = B – 1

B² – B¹ – B⁰ = 0

Note:  B^x means n raised to the x power.  Some browsers may not display exponents as superscripts or raised characters.

Now we have a formula that can be solved using the Quadratic formula. This formula allows you to solve a quadratic equation for an unknown x, with a, b, and c as constants. A quadratic equation has this form:

ax² + bx + c = 0

The solution to this is found with the quadratic formula:

So our formula for the golden ratio above (B² – B¹ – B⁰ = 0) can be expressed as this:

1a² – 1b¹ – 1c = 0

The solution to this equation using the quadratic formula is (1 plus or minus the square root of 5) divided by 2:

(  1 +  √5 ) / 2 = 1.6180339… = Φ

(  1 –  √5 ) / 2 = -0.6180339… = -Φ

The reciprocal of Phi (denoted with an upper case P), is known often as by phi (spelled with a lower case p).

Phi, curiously, can also be expressed all in fives as:

5 ^ .5 * .5 + .5 = Φ

This provides a great, simple way to compute phi on a calculator or spreadsheet!

Here’s a little more phi mathemagic, contributed by Abe Ihmeari:

Φ * √5 = 3.6180339… = Φ + 2

Determining the nth number of the Fibonacci series

You can use phi to compute the nth number in the Fibonacci series (f_n): 

f_n =  Φ ⁿ / 5^½

As an example, the 40th number in the Fibonacci series is 102,334,155, which can be computed as:

f₄₀   =   Φ ⁴⁰ / 5^½   =  102,334,155

This method actually provides an estimate which always rounds to the correct Fibonacci number.

You can compute any number of the Fibonacci series (f_n) exactly with a little more work:

f_n = [ Φ ⁿ – (1-Φ)ⁿ ] / √5

Note:  √5 can be expressed as 2Φ-1 to use Φ for all the terms above.

 Determining Phi with Trigonometry and Limits

Phi can be related to e, the base of natural logs,
through the inverse hyperbolic sine function:

Φ = e ^ asinh(.5)

It can be expressed as a limit:

or

Other unusual phi relationships

There are many unusual relationships in the Fibonacci series.  For example, for any three numbers in the series Φ(n-1), Φ(n) and Φ(n+1), the following relationship exists:

 Φ(n-1) * Φ(n+1) = Φ(n)² – (-1)ⁿ

(  e.g.,   3*8 = 5²-1   or   5*13=8²+1 )

Here’s another:

 Every nth Fibonacci number is a multiple of Phi(n),
where Phi(n) is the nth number of the Fibonacci sequence.

Given 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946

(Every 3rd number., e.g., 2, 8, 34, 144, is a multiple of Phi(3), which is 2)

(Every 4th number, e.g., 3, 21, 144 and 987, is a multiple of Phi(4), which is 3)

(Every 5th number, e.g., 5, 55, 610, and 6765, is a multiple of Phi(5), which is 5)

(Every 6th number, e.g., 8, 144, 2584, is a multiple of Phi(6), which is 8)

(Every 7th number, e.g., 13, 377, 10946, is a multiple of Phi(7), which is 13)

And, as contributed by Abe Ihmeari, for any Fibonacci sequence number f(n), we find that f(n)-f(n-5)-f(n-10) = 10 ∙ f(n-5). This is easiest to see when the Fibonacci sequence numbers are grouped in fives. As an example, 4181-377-34 = 3770, which is 10 ∙ 377, and 28657-2584-233 = 25840, which is 10 ∙ 2584!

And another:

The first perfect square in the Fibonacci series, 144,

is number 12 in the series and its square root is 12!

0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144

or, if not starting with 0:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144

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The powers of phi have unusual properties in that they are related not only exponentially, but are additive as well.  We know that:

Phi ² = Phi + 1

Which is the same as:

Phi ² = Phi ¹ + Phi ⁰

And this leads to the fact that for any n:

Phi ⁿ⁺² = Phi ⁿ⁺¹ + Phi ⁿ

Thus each two successive powers of phi add to the next one!

Powers of Phi and its reciprocal:

Another little curiosity involves taking phi to a power and then adding or subtracting its reciprocal:

For any even integer n:

Phi ⁿ  +  1 / Phi ⁿ = a whole number

For any odd integer n:

Phi ⁿ  –  1 / Phi ⁿ = a whole number

Examples are shown in the tables below:

for n = even integers

for n = odd integers

The whole numbers generated by this have a relationship among themselves, creating an additive series, similar in structure to the Fibonacci series, and which also converges on phi:

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Phi (Φ) and pi (Π) and Fibonacci numbers can be related in several ways:

 The Pi-Phi Product and its derivation through limits

The product of phi and pi,

1.618033988…  X  3.141592654…,  or  5.083203692,

is found in golden geometries:

Golden Circle	Golden Ellipse
Circumference = p * Φ	Area = p * Φ

Ed Oberg and Jay A. Johnson have developed a unique expression for the pi-phi product (pΦ) as a function of the number 2 and an expression they call “The Biwabik Sum,”a function of  phi, the set of all odd numbers and the set of all Fibonacci numbers, as follows:

This relationship was derived after Oberg noticed an interesting relationship between pi and phi while contemplating geometric questions related to the location of the King and Queen’s burial chambers in the Great Pyramid, Cheops, of Giza, Egypt, the design of which is based on phi.You can access the complete paper published by Ed Oberg and Jay A. Johnson, The Pi-Phi Product, in Word, or the Pi-Phi Product in Excel to see their formulation illustrated numerically.

Trigonometric functions relating phi (Φ) and pi (Π)

Divide a 360° circle into 5 sections of 72° each and you get the five points of a pentagon, whose dimensions are all based on phi relationships.

Accordingly, it shouldn’t be too surprising that phi, pi and 5 (a Fibonacci number) can be related through trigonometry:

Or, a much simpler way involving, contributed by Dale Lohr:

Pi = 5 arccos (.5 Phi)

Note:  The angle of .5 Phi is 36 degrees, of which there are 10 in a circle or 5 of in pi radians.

Note:  Above formulas expressed in radians, not degrees

Alex Williams, MD, points out that you can use the Phi and Fives relationship to express pi as follows:

5arccos((((5^(0.5))*0.5)+0.5)*0.5) = pi

Robert Everest discovered that you can express Phi as a function of Pi and the numbers 1, 2, 3 and 5 of the Fibonacci series:

Phi = 1 – 2 cos ( 3 Pi / 5)

Pi and Phi in the Great Pyramid of Egypt

Another interesting relationship between Pi and Phi is related to the geometry of the Great Pyramid of Giza.  This relationship connects dimensions of the Great Pyramid to both Pi and Phi, but it is not known with certainty whether this was an intentional aspect of its design, whether its design was based on Pi or Phi but not both, or whether it is a simple coincidence. It relates to the fact that 4 divided by square root of phi is almost exactly equal to Pi:

The square root of Phi (1.6180339887…) = 1.2720196495…

4 divided by 1.2720196495… = 3.14460551103…

Pi = 3.14159265359…

The difference of these two numbers is less than a 10th of a percent.

See the Phi, Pi and the Great Pyramid page for more details.

Pi squared (Π²) and 987

Pi squared (Π²) is 9.8696…, which, if you round to 9.87 and ignore the decimals, is 987, the 17th number of the Fibonacci series. (Contributed by William Erman.)

More on the relationship of Phi squared and Pi

If you’re looking for other interesting ways to relate pi and phi, 6/5 * Phi^2 = 3.1416, which approximates pi. (Contributed by Steve Lautizar.)

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It’s been noted by some who say they can “demystify phi” that phi is just one of an infinite series of numbers that can be constructed from the following expression using the square root (√) of integer numbers:

(1+√n) / 2

It just so happens that you get phi when you let n equal 5.  Let n be other integers and you get a series of numbers whose squares (see Phi2 in table in green) each exceed their root by a difference (see Δ in table in blue) that increases by 0.25 for each number in the series, as shown below.

Phi, being the 5th one in the series, just happens to be the one that produces a difference of 1 with its square, leading to the unique property that it shares with no other number:

Phi + 1 = Phi ²

So does this demystify phi, making it just one of a series of phi-like numbers?  Not necessarily, as this is only one aspect of phi’s unique properties.  Phi is also the only number that produces a difference of 1 with its reciprocal:

Phi – 1 = 1 / Phi

This is the key to its relationship to the golden section, which is based on sectioning a line in a way that fulfills two requirements:

A = B + C

and

A/B = B/C

A is to B as B is to C, where
A is 161.8% of B and B is 161.8% of C, and
B is 61.8% of A and C is 61.8% of B

Let n be any integer other than 5 and you won’t find the same pattern of consistent differences as shown above or the unique reciprocal and additive properties of phi.

Insights on phi’s formula in the table above contributed by Joseph Conklin.

The post The Phi Formula appeared first on The Golden Ratio: Phi, 1.618.

Pascal’s Triangle, developed by the French Mathematician Blaise Pascal, is formed by starting with an apex of 1.  Every number below in the triangle is the sum of the two numbers diagonally above it to the left and the right, with positions outside the triangle counting as zero.

The numbers on diagonals of the triangle add to the Fibonacci series, as shown below.

Pascal’s triangle has many unusual properties and a variety of uses:

• Horizontal rows add to powers of 2 (i.e., 1, 2, 4, 8, 16, etc.)

• The horizontal rows represent powers of 11 (1, 11, 121, 1331, 14641) for the first 5 rows, in which the numbers have only a single digit.

• Adding any two successive numbers in the diagonal 1-3-6-10-15-21-28… results in a perfect square (1, 4, 9, 16, etc.)

• It can be used to find combinations in probability problems (if, for instance, you pick any two of five items, the number of possible combinations is 10, found by looking in the second place of the fifth row.  Do not count the 1’s.)

• When the first number to the right of the 1 in any row is a prime number, all numbers in that row are divisible by that prime number

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Numeric reduction is a technique used in analysis of numbers in which all the digits of a number are added together until only one digit remains.  As an example, the numeric reduction of 256 is 4 because 2+5+6=13 and 1+3=4.

Applying numeric reduction to the Fibonacci series produces an infinite series of 24 repeating digits:

1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9

If you take the first 12 digits and add them to the second twelve digits and apply numeric reduction to the result, you find that they all have a value of 9.

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This is a little curiousity involving the number 89, one of the Fibonacci series numbers.

1/89 is a repeating decimal fraction with 44 characters:

.01123595505617977528089887640449438202247191

You can see the beginning of the Fibonacci sequence in the first 6 digits of the decimal equivalent of 1/89. (i.e., 0,1,1,2,3,5 appears as 0.011235..)

If you take each Fibonacci number, divide it by 10 raised to the power of its position in the Fibonacci sequence and add them all together, you get 0.011235955…, the same number as the reciprocal of 89.

The reciprocal of 109 is also based on the Fibonacci series, forwards and backwards

Here’s another curiousity involving the number 109, discovered and contributed (10/20/2003) by Rick Toews.

1/109 is a repeating decimal fraction with 108 characters:

.009174311926605504587155963302752293577981651376146788
990825688073394495412844036697247706422018348623853211

You can see the beginning of the Fibonacci sequence in the LAST 6 digits of the decimal equivalent of 1/109, appearing in REVERSE order starting from the END of the decimal. (i.e., 0,1,1,2,3,5, 8 appears as …853211)

If you take each Fibonacci number, divide it by 10 raised to the power of 109 MINUS its position in the Fibonacci sequence (starting with 0) and add them all together, you get the reciprocal of 109.

Lastly, here’s one more curiosity involving the number 109.

If you take each Fibonacci number, divide it by 10 raised to the power of its position in the Fibonacci sequence and add and subtract each alternate term together, you get .00917431… again, the reciprocal of 109.

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Phi can be derived from several formulas based on the number 5.  The most traditional, based on the geometric construction of phi is this:

This formula for phi can also be expressed all in fives as:

Φ = 5 ^ .5 * .5 + .5

Another formula for phi based entirely on 5’s, an original insight contributed by Erol Karazincir (pcerol@yahoo.com), is as follows:

And, as pointed out by W. Nathan Saunders, the terms in above representation of phi can be expressed in yet another way that involves four 5’s:

(5 + √5) x (5 – √5) = 5 + 5 + 5 + 5

Phi appears in the geometry of the 5-sided pentagon

Take a pentagon with 5 equal sides and connect all the points to form a 5-pointed star.  The ratios of the lengths of the resulting line segments are all based on phi.

Phi appears in the natural logs and trigonmetric functions

Phi can be related to e, the base of natural logs,
through the inverse hyperbolic sine function:

Phi = e ^ asinh(.5)

Determining the nth number of the Fibonacci series

You can compute the nth number in the Fibonacci series (f_n) using phi and root 5:

f_n =  Phi ⁿ / 5^½

 5 is the 5th Fibonacci number

5 is also the 5th of the Fibonacci numbers, including 0, 1, 2, 3, and 5.

5 appears in the human body, which has proportions based on phi

Another interesting aspect of phi and five is in relation to the design of the human body, which in addition to being based on phi relationships in its proportions, has:

• 5 appendages from the torso, in the two arms, two legs and a head,
• 5 appendages on each of legs and arms in the five fingers and five toes,
• 5 openings on the face, and
• 5 senses in sight, sound, touch, taste and smell.

5 deserves a “high 5” for its role in phi, don’t you think!

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Math isn’t tough, but it can be mean.  The term “mean” in mathematics simply reflects a specific relationship of one number as the middle point of two extremes.

Arithmetic means

The arithmetic mean of 2 and 6 is 4, as 4 is equally distant between the two in addition:

2 + 2 = 4
and
4 + 2 = 6

For the arithmetic mean (b) of two numbers (a) and (c):

b = ( a + c ) / 2

4 = ( 2 + 6 ) / 2

The arithmetic mean is thus the simple average between two numbers.

Geometric means

The geometric mean is similar, but based on a common multiplier that relates the mean to the other two numbers. As an example, the geometric mean of 2 and 8 is 4, as 4 is equally distant between the two in multiplication:

2 * 2 = 4
and
4 * 2 = 8

So 2 is to 4 as 4 is to 8.

b =  √( a * c )

4 = √( 2 * 8 )

The Golden Mean

The Golden Mean is a very specific geometric mean.  In the geometric mean above, we see the following lengths of line segments on the number line:

Yellow line = 2
Blue line = 4
White line = 8

Here, 2 x 2 = 4 and 4 x 2 = 8, but 2 + 4 = 6, not 8.  The Golden Mean imposes the additional requirement that the two segments that define the mean also add to the length of the entire line segment:

This occurs only at one point, which as you can see above is just a little less than 5/8ths, or 0.625.  The actual point of the Golden Mean is at 0.6180339887…, where:

A is to B as B is to C
AND
B + C = A

B = √ ( A * C )  AND  B + C = A

1 = √ ( Phi * 1/Phi ) AND 1 + 1/Phi = Phi

1 =  √ ( 1.618 … * 1/1.618 … )

AND

1 + 1 / 1.618 … = 1.618 …

Note also that:

1 / 1.618 …    =    0.618…    =    1.618 … – 1

1 / Phi    =    0.618…    =    Phi  – 1

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