** **

Algebra and Geometry of Phi

by Tom Gilmore**
**Copyright 2018

See also Phi in the Great Pyramid

**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.__61____5384__
<) 21 (1.__61____9047__
>) 34 (1.6176471 <) **55 (1.6 18 >) 89** (1.6179775 <) 144 (1.618

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=

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 = Phi^{2},
then 1^{2} + sq-rt(Phi)^{2 }= 1+Phi = Phi^{2}, 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+sqrt****5)/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+sqrt****5)/2
**

**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 Phi^{2}.

This means that, using the series numbers (**21,
34, 55**), Phi = 34/21, and Phi^{2} = 55/21.

Substituting Phi^{2} = 55/21 in the formula **(X = Phi ^{ 2}
x 6/5),
**X = (55/21 x 6/5) = 330/105 =

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 Sqrt****5 in terms of Phi**

**Demonstrating that sqrt****5 =**** ****Phi + 1/Phi**

**Phi = (1 + sqrt5) /2
**And thus:

2Phi = 1+ sqrt

sqrt

sqrt

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 ****Sqrt****5**

**Pi = Phi ^{2} x 6/5,
and Phi^{2} = Phi +1, **so

**Pi = (Phi + 1) x 6/5 **and** ****Phi**** = (1+sqrt****5)/2**** , so
Pi = ((1+sqrt**

= (1 + sqrt

= (1.5 + (sqrt5)/2) x 6/5

= 9/5 + (sqrt

**Pi = 9/5 +
.6 sqrt5 **

**Confirmation usimg the
following approximations:
sqrt**

**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.

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

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.

**The Regular Polygons and Transcendence**

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 180^{0}.

Sides Angle Total /180

3 60 180 1

4 90 360 2

5 108 540 3

6 120 720 4

7 1284/7 900 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 180^{0}, which is the tangent to the
circle.

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 180^{0}, 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 180^{0}. 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 10,000,000,000^{3} =
1,000,000,000,000,000,000,000,000,000,000.