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## 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)

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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.