15. Algebra and Geometry of Phi

by Tom Gilmore
All graphics by Tom Gilmorr

Part II:  Geometry of Phi
Part III: The Transcendent Equations
Part IV: Number

Part I: Algebra of Phi

The Fibonacci Series

Starting from 1 (and assuming a prior zero), iteratively generating the sum of the result with the previous number converges on a ratio of "phi" between successive numbers (each number in the series is equal to the sum of the previous two numbers).  This iteration of sums is known as the Fibonacci Series.

0,1,1,2,3,5,8,13,21,34,55,89,144,233,377, ...

Each new number has a ratio to the previous number that gets ever closer to the transcendent value of phi, alternating between being less than (<) and greater than (>) phi (to seven decimals 1.6180339).  The ratios are shown below in parentheses, along with the greater or less symbol.  As the fractional tail progresses into established values, the digits are shown in green.  Underlined digits indicate infinitely repeating decimal strings.

1 (1 <) 1 (2 >) 2 (1.5 <) 3 (1.6 >) 5 (1.6 <) 8 (1.625 >) 13 (1.615384 <) 21 (1.619047 >) 34 (1.6176471 <) 55 (1.618 >) 89 (1.6179775 <) 144 (1.6185 >) 233 (1.6180258 <) 377 (1.6180371 >) 610 (1.6180328 <) 987, (1.6180345 >) 1597, (1.6180338 <) 2584, (1.6180340 >) 4181, (1.61803396 <) 6765, (1.618033998 >) 10946 (1.618033985 <) 17711 (1.61803399017  >) 28657 (1.618033988 <) 46368 (1.61803398895 >) 75025 (1.61803398867 <) 121393 (1.61803398878 >) 196418 .....

The violet ratio of 89/55 is utilized in the Great Pyramid (as developed in Part II - The Geometry of Phi).

Each ratio, when inverted, is equal to the decimal portion of the following ratio, for example, for the ratios 8/5, and 13/8,

8/5 inverted is:
5/8 = 0.625
13/8 = 1.625

Notice that (5/8 + 1 = 13/8) and since 8/5 is phi, 5/8 is 1/phi, this means that (1/phi + 1 = phi).
This is the same as saying that the decimal tail of phi is equal to 1/phi.

Another way to express this relationship is that inverting a phi fraction and adding 1 results in the following phi fraction.

Iteration of Adding 1 to the Inverted Fraction

The general form of iteration takes a function (f) and applies it on a number (n), and repeatedly applies the function on the result.

The iteration f(n) = 1/n + 1 eventually converges on the value phi for any positive integer (n).  For the purpose of this treatise, we use (n)=1.

1/n+1  (n=1)
1/1+1=2/1;
1/(2/1)+1=2/1+1=3/2;
1/(3/2)+1=2/3+1=5/3;
1/(5/3)+1= 3/5+1=8/5;
1/(8/5)+1=5/8+1=13/8,
1/(13/8)+1=8/13+1=21/13 …

Note that this produces the identical ratio series (bold fractions) resulting from the Fibonacci numbers.

This iterative inversion, followed by adding 1, is the same functionality as Euclid's “continued fraction of 1”.

The Continued Fraction of 1

In fact there is only one number and it is the number one (refer to the article on Number).

Euclid is credited with developing a "continued fraction expansion" utilizing the number 1 that is “equal” to Phi.

1 + 1 / (1 + 1 / (1 + 1 / (1 + ... …))) = Phi.

(This continued fraction is simply a transposition of (f(n) = 1/n + 1) to (f(n) = 1 + 1/n), for n=1).

In reducing this equation from a finite portion of the continued fraction (as detailed below), the final fractions of the 2-step reduction (shown in bold) are the Fibonacci ratios.  In the 2-step reduction, ratios are inverted by the fractional pattern (1/n), and in the next step are added to 1, resulting in the next Fibonacci ratio in the series.  In order to exactly follow the Fibonacci ratios, although the first reduction results in the number 2, it is replaced by (2 = 2/1).

1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1)))))))) =
1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (2/1)))))))) =
1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1/2))))))) =
1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (3/2))))))) =
1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 2/3)))))) =
1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (5/3)))))) =
1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 3/5))))) =
1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (8/5))))) =
1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 5/8)))) =
1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (13/8)))) =
1 + 1 / (1 + 1 / (1 + 1 / (1 + 8 /13))) =
1 + 1 / (1 + 1 / (1 + 1 / (21/13))) =
1 + 1 / (1 + 1 / (1 + 13 /21))
1 + 1 / (1 + 1 / (34/21)) =
1 + 1 / (1 + 21/34) =
1 + 1 / (55/34) =
1 + 34/55 =
89/55 = 1.618 = Phi

The ratio 89/55 is the Egyptian Royal Phi ratio used in constructing the Great Pyramid of Giza.

Part II: Geometry of Phi

The Golden Section (Phi as a Rectangle)

Geometrically, the number Phi is defined as a ratio represented by a point on a line positioned such that the two segments have the property that the short segment has the same proportion to the long segment as the long segment has to the entire line.

A----------B----------------C

The formula is Phi = AC/BC = BC/AB ~ 1.618

Da Vinci named the rectangle with sides the proportion of 1 to 1.618 (AB to BC) the Golden Section.
The phi-rectangle is shown below.

Phi in the Pentagram

Connecting the 5 points that are seperated by 72 degrees with all possible lines, creates a pentagram (5-pointed star) within a pentagon, as shown below.

At the center of the pentagram is an inverted pentagon, and connecting the vertices of that pentagon produces an inverted pentagram.  As the symbol of magic, the outer pentagram represents “white magic”, meaning majic used to help others (outer), and the inner pentagram represents “black magic”, meaning magic used for selfish purposes (inner).

The pentagram illustrates two equations involving Phi.  The phi ratios are integral to the proportions of the line segments.
In the diagram above, two equations are shown on the upper right, and are illustrated (color coded) by lined-up spanned lengths between intersections of a pentagram line, such that the sums match the equations.

Powers of Phi

The two equations above are specific examples of the general equation that
Phi n = Phi n-1 + Phi n-2, or that each power of Phi is the sum of the previous 2 powers of Phi.

Using n=1,
Phi 1 = Phi 0 + Phi -1
means that Phi = 1 + 1/Phi, the first equation in the graphic above.
Using n=2,

Phi 2 = Phi 1 + Phi 0
means that Phi 2 = Phi + 1, the second equation in the graphic above.

The following demonstration of the general equation uses (for an example) the series numbers and ratios of
(5, 8/5, 8, 13/8, 13, 21/13, 21).

Phi 1 = 8/5
Phi 2 = Phi x Phi = (8/5 x 13/8) = 13/5, and
Phi 3 = Phi x Phi x Phi = (8/5 x 13/8 x 21/13) = (13/5 x 21/13) = 21/5.

The product of any number of consecutive phi ratios always reduces to a ratio with the denominator of the first ratio's denominator and the numerator of the last ratio's numerator.

Logical proof of the exponential equation: Since all consecutive powers of Phi have the same denominator, the general exponential equation (Phi n = Phi n-1 + Phi n-2), reduces to the Fibonacci series numbers (see example below).  In other words, the powers of the ratio Phi reduce to the Fibonacci series numbers, which by definition sum the previous 2 numbers.

Example: Using Phi n = Phi n-1 + Phi n-2, where Phi n =21/5, Phi n-1 =13/5, and Phi n-2 =8/5,
(21/5 = 13/5 + 8/5), or (21 = 13 + 8).

The Phi Right-Triangle

By the Pythagorean Theorem the square of the hypotenuse of a right triangle equals the sum of the squares of the other 2 sides.  Thus if the to the 2 sides are 1 and the square-root of Phi (see illustration below), since 1+Phi = Phi2, then 12 + sq-rt(Phi)2 = 1+Phi = Phi2, proving that the hypotenuse equals Phi.

This triangle relates to Phi in the Great Pyramid

Phi by Compass and Ruler

The geometric derivation of Phi is shown below.
It is half the length of a unit square plus the diagonal of the half-square rectangle.

The diagonal is the hypotenuse of a right triangle with sides of 0.5 and 1.
Thus, the diagonal is sqrt(1x1+.5x.5) = sqrt(1.25) = sqrt(5/4).

So (Phi = .5 + sqrt (5/4)) or
Phi = (1+sqrt5)/2

Part III: The Transcendent Equations

The 3 transcendent numbers of Phi, Pi, and sqrt5 have a common transcendent root.  This is established by the equations between them.

Phi = (1+sqrt5)/2
Pi = Phi 2 x 6/5

The (Approximate) Transcendent Equation Proof

Pi = Phi 2 x 6/5

Proof using Pi ~ 22/7 and the Fibonacci series numbers (21, 34, 55).

Since each number in the series increases by Phi (because Phi is the ratio between the numbers), 2 numbers in the series increase by Phi2.
This means that, using the series numbers (21, 34, 55), Phi = 34/21, and Phi2 = 55/21.

Substituting Phi2 = 55/21 in the formula (X = Phi 2 x 6/5),
X = (55/21 x 6/5) = 330/105 = 22/7 ~ Pi.

Using higher numbers in the series increases the accuracy of both Phi and Pi in the equation, but because Pi and Phi are transcendent there is no exact proof.

The 6/5 in the equation is also the disparate measure of approximation of Pi~22/7 using Phi~89/55, because
Pi=(Phi x Phi x 6/5) =
(89/55 x 89/55  x 6/5) =
(7921/3025 x 6/5) = 47526/15125,
and 22/7 x 2160 = 47520/15120
(leaving a residual 00006/00005 (the 6/5 in the formula).

Solving for Sqrt5 in terms of Phi

Demonstrating that sqrt5 = Phi + 1/Phi

Phi = (1 + sqrt5) /2
And thus:
2Phi = 1+
sqrt
5
sqrt
5 = 2Phi –1
sqrt
5 = Phi+(Phi –1)

And since
Phi = 1 + 1/Phi
(Phi – 1) = 1/Phi), and substituting 1/Phi for (Phi – 1) above,
sqrt
5 = Phi + 1/Phi

Solving for Pi in terms of Sqrt5

Pi = Phi2 x 6/5, and Phi2 = Phi +1, so

Pi = (Phi + 1) x 6/5 and Phi = (1+sqrt5)/2 , so
Pi = ((1+
sqrt
5)/2) +1) x 6/5
= (1 +
sqrt
5 +2)/2) x 6/5
= (1.5 + (sqrt5)/2) x 6/5
= 9/5 + (
sqrt
5 x 6/10), so

Pi = 9/5 + .6 sqrt5

Confirmation usimg the following approximations:

sqrt
5 ~2.236, Pi ~3.1416)

Pi = 9/5 + .6 x 2.236 = 1.8 + 1.3416 = 3.1416

Part IV: Number

There is only one actual number and it is the number one.  All the integers and fractions can be expressed using only the number one.  For example,
(2 = 1+1), and (2/3 = 1+1 / (1+1+1)).  Phi is all about the number 1.

The Unit Measure

Zero does not exist.  It is used in number schemes as a positional placeholder, but mathematical operations with zero are imaginary.  Infinity is also an imaginary construct.  For every number there can be stated a larger number, but every larger number is nevertheless finite.  In theoretical mathematics there are "magnitudes of infinity".  For example, between the numbers 1 and 2 there are an infinity of fractions, while at the same time there are an infinity of integer numbers, with an infinity of fractions between each of the integers.  From this it can be seen that there exist powers of infinity, which violates logic, because if an infinity is smaller than another infinity then the smaller infinity must be finite.

Einstein developed a laborious proof of the unit measure, using the Cartesian co-ordinate system.  Einstein developed the "relativity" principle to deal with the issue of separate personal viewpoints, or the relativity of perception.  His proof of the unit measure established that all personal viewpoints (termed "special relativity") were commensurate, and could be exactly related through a common "general relativity".

The proof of the unit measure disproved the concept of "space symmetry", or the infinite continuum of space.  Of course this had been proved many times before by various different arguments.  One ancient Greek proof was that in a continuum, motion could not begin, and this was because before moving any distance a smaller distance must be moved first, and there would be no end to smaller distances in a continuum, so it would take infinite time and still go nowhere.

The mathematician Leibniz approached the proof by establishing that number and line length are equivalent.  Leibniz asserted that any quantity of zero length "points" lined up are still of zero combined length, so a line cannot be made up of zero length points, and any such imagined points contained in the line contribute nothing to the length.  His argument is that since the point cannot be of zero length, the elemental point must have a given length.  It follows that every length of line is precisely defined by the number of elemental points it contains, and thus number is equivalent to line length.  This "elemental point of a given length" is in fact the same as Einstein's "unit measure".

The Greek mathematician Zeno wrote various proofs of the discontinuity of motion using the half-way concept, such as the race between a turtle and a hare, where giving the turtle a head-start means that the rabbit would be closing by half the distance, even as the turtle crossed the finish line.

A similar argument is that if a finite line is considered to contain an infinite number of points, then movement along the line requires an infinite amount of time no matter how little time it takes to pass through each point because infinity times any positive number equals infinity.  And this applies no matter how short the finite line is.  Thus, a continuum results in infinite transit times, and reality contradicts this.

Cubic Matrix Numerology

The Geocubic Model is based on the unit measure in 3-dimensions, which is a unit-cube.

All cubic forms in space are the size of an integer cubed (see above).  There is a hidden progressive numerological pattern of the integers cubed.  The list below shows integers cubed.  The “^” column is (n3 – (n-1)3), and the “*” column is (^ - prior ^).

n         n3         ^          *
1         1          1          0  (there is no prior ^)
2         8          7          6
3         27        19        12
4         64        37        18
5         125      61        24
6         216      91        30
7         343      127      36
8         512      169      42
9         729      217      48
10        1000    271      54
11        1331    331      60
12        1728    397      66
13        2197    469      72
14        2744    547      78
15        3375    631      84
16        4096    721      90
17        4913    817      96
18        5832    919      102
19        6859    1027    108
20        8000    1141    114

In the n3 column the final digit repeats (1,8,7,4,5,6,3,2,9,0) which is all 10 digits.
In the ^ column the final digits repeats (1,7,9,7,1).
In the * column the final digit repeats (0,6,2,8,4) which is all the even numbers, and advances uniformly by the number 6.

The Regular Polygons and Transcendence

Transcendent numbers are imaginary, because any values assigned to transcendent numbers must be finite, and therefore not transcendent.

An analysis of regular polygons reveals that, in a quantumized Cosmos, transcendence is merely unimaginable immensity.

All regular (equal-sided) polygons fit inside a circle, the vertices touching the circumference.  As charted below, the interior angles of the regular polygons follow a linear pattern of unit increase in the total sum of interior angles divided by 1800.

Sides   Angle  Total   /180
3          60        180      1
4          90        360      2
5          108      540      3
6          120      720      4
7          128
4/7  900      5
8          135      1080    6
9          140      1260    7
10        144      1440    8

From the above chart it is evident that the interior angle equals 180 times the number of sides minus 2 divided by the number of sides.

(interior angle = (Sides – 2) / Sides x 180)

The formula reveals the progression of the interior angle is asymptotic to 1800, which is the tangent to the circle.

For the triangle ((3-2)/3 x 180 = 1/3 x 180 = 60)
For the square ((4-2)/4 x 180 =1/2 x 180 = 90)
For the pentagon ((5-2)/5 x 180 = 3/5 x 180 = 108)
For 8 sides ((8-2)/8  x 180 = 6/8 x 180 = 135)
For 100 sides (98/100 x 180 = 176.4)
Notice the progression asymtotic to 180 of the following (ending in the digits 64 with 9’s inserted)
For 1,000 sides, the angle is (998/1000 x 180) = 179.64
For 10,000 sides, the angle is (9998/10000 x 180) = 179.964
For 100,000, the angle is (99,998/ 100,000 x 180) = 179.9964
For 1,000,000 (999,998/ 1,000,000 x180) = 179.99964

The interior angle never reaches 1800, no matter how large the number of polygon sides.  Even if the polygon could have an infinite number of sides, the angle would only be infinitely marginally less than 1800.  This asymptotic separation of the arc from the polygon shows that space is a quantum discontinuity.

The quantum discontinuity implies that the circle is actually a regular polygon.  It follows that for any circle, the size determines the number of polygon sides that it consists of (the quantum is 10 –10 meters).  The ratio of the circumference of a circle to its diameter is a transcendent number (Pi), and this is because conceptually for every circle there exists a larger circle, however this concept of transcendence is based on the illusion that space is infinite.  If space is actually finite (and it must be because infinity is proved mathematically invalid) then Pi is not actually transcendent.  This same principle applies to the number Phi, which is derived from progressive number ratios, and is considered transcendent because number is considered to be infinite, but if number is restricted to representing the quanta of space, then number is also finite, and no number is actually transcendent.

Leibniz used his "method of fluxions" to prove that

Pi/4 = 1/1 –1/3 +1/5 –1/7 +1/9 ….

This equation reveals an underlying regularity in the nature of Pi, however it converges very slowly on the value of Pi and thus does not lend itself to manual calculation of the numerical value of Pi.

To appreciate how staggeringly immense the transcendence is, consider the miniscule size of the quantum (size of the atom) multiplied by the enormous size of the known universe.

Size of unit-cube: 10-10 meters.  Unit-cubes in one cubic meter is 10,000,000,0003 = 1,000,000,000,000,000,000,000,000,000,000.

Index of all Articles by Tom Gilmore