## Pi Phi and e Link

__The rotating phenomenon in the Pi and Phi numbers led me to the following ideas:__

Pi, is the length of the outside of a circle (circumference), divided by the length across its center (diameter).

Circle = πd = 2πr

Area = πr²

Volume = 1/3πr²h

Phi is the Golden Ratio that results when a line (diameter) is divided in a very special way, and this proportion appears throughout creation.

**ϕ** = Phi 1.618033989… (1 + √5) / 2 **or** √1.25 + .5

**φ** = phi .618033989…

**ϕ + ****φ = ****√5**

__An Illustrative relationship between pi and phi:__

AC represents a diameter of 1 or √1, and a circle circumference of 1 x **π** 3.14159...

HL represents a diameter of **ϕ** + **φ** = √5, and a circle circumference of √5 x π

HL diameter **/** AC diameter = √5

HL circle circumference √5 x π **/** AC circle circumference 1 x π = √5

As cones, DHL and ACD divided = 10

__Two triangles DFE and EIH, and their six segments from which the other measurements can be obtained:__

EF = 0.5

DF = 0.809016994…

DE = 0.951056516…

HI = 0.618033989…

EI = 1

EH = 1.175570505…

This fish shape contains many interesting ratios, such as:

DP / HL = ϕ, BD / AB = ϕ, AD / AB = √(ϕ + 2), AB / BD = φ, EI / HI = ϕ, HL / AC = √5, AC / HL = √.2, DL / AD = √5,

AG / DL = √.8, DL / AG = √1.25, AL / AG = ϕ, AL / HK = √ (ϕ + 2), DP / BD = √20, DJ / HJ = ϕ, etc.

*John 21:11 Simon Peter went up, and drew the net to land full of great fishes, an hundred and fifty and three: Matthew 17:27 and cast an hook, and take up the fish that first cometh up; and when thou hast opened his mouth, thou shalt find a piece of money: …* i.e. Circular!

*153 + 1 = 154*

*154 / 49 = 3.1412859 (22/7)*

*><>*<><*

The tube idea was covered in pages The silver lining and DNA and Primes.

__It seems that e is linked to this phenomenon as well!__

(**e**) 4 / 3.14507891392169474652 ... = 1.2718281828459045235 ...² = 1.61754692668111554934 ...

Pg 4 / 3.14465408805031446540 ... = 1.272 ² = 1.617984

(**ϕ**) 4 / 3.14460551102969314427... = 1.27201964951406896425 ...² = 1.61803398874989484820 ... (√1.25 + .5)

(**π**) 4 / 3.14159265358979323846 ... = 1.27323954473516268615 ...² = 1.62113893827740434310 ...

Pg 4 / 3.13990744037023851904 ... = 1.27392290249433106575 ...² = 1.62287956149958093592 ...

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** e*** = 2.71828182845904523536028 ...*

**(e** / 10) + 1 = 1.27182818284590452353602 ...

**√** **ϕ**** **= 1.27201964951406896425242 ...

4 /**π**** **= 1.27323954473516268615107 ...

Introducing segment WZ, which represents **e**.

To obtain the external rectangle, extend lines AL, CH and DI, DK as shown. From the horizontal center line BP,

up and down respectively, a distance of **e** / 2. Install the horizontal and vertical margins to complete. **∞**

__Again, so many interesting and exact ratios linking pi, phi and e. Here are just a few:__

QW / WT(e) = ϕ, QZ / ZW = √(ϕ + 2), QT / QU = ϕ + 1, XP / Xc = ϕ, WT(e) / WX = 2, DX / XW = ϕ,

RH / HS = ϕ, RH / RV = ϕ, DJ / AD = √(ϕ + 2), DJ / BD = √5, DX / BD = **e**, BX / BD = **e + 1**, AD x **e** = DZ, etc.

Imagine e as the outside diameter of a box suggesting a physical visible reality, and Phi + phi as its inner diameter, which is a reflection of the invisible true reality as represented by Pi, which occupies the space within.

The 3 constants Pi, Phi and e, illustrated as perfectly interlinked horizontal, vertical and diagonal ratios.

Diameter AC / (ϕ + 2) = rC

rC x √5 = Ht x (ϕ + 2) = Diameter HL (√5)

rC x e = Wd x (ϕ + 2) = Diameter WZ (e)

__I see an analogy between this illustration and man’s threefold nature, his triune being; spirit, soul, and body.__

The body (**e**) as the third and outermost segment of man’s being comes in contact with matter, affecting it and being affected by it (aging, decay, etc.). The soul (**ϕ + φ**) is the second segment and is the medium which stands between, connecting the spirit and body. The soul is eternal and has an invisible bodily form. The spirit (**π**) is the ﬁrst and innermost segment of man, that which was formed by God: “The breath of life” is the noblest part of man, that which joins him to God. Why not, mathematicians have no problem using the word transcendental.