Panagiotis Stefanides |

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All triangles derive their origin from two triangles [ Pl.Ti. 53 D].
By
*Panagiotis Chr. Stefanides**

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....all triangles derive their origin from two triangles, each having one angle
right and the others acute....These we lay down as the principles of fire and
all the other bodies...Pl.Ti. 53 D.
One of these two triangles is of course the 45/45 deg [tan 45 equals
1] orthogonal triangle and my interpretation of the other scalene
orthogonal is that whose angle between hypotenuse and the
horizontal smaller has as tangent the sqrt[Φ],
contrary to the currently accepted interpretation of the 30/60 deg
scalene orthogonal one.
We consider the following:**

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sqrt[Φ]=
sqrt[[sqrt(5) + 1 ]/2] = 1.27201965…= T , with corresponding angle
θ,
so that:
θ
= Arctan(T) = 51.82729238… deg.
An implementation of the relationships involving the two triangles [angles
tan(45) deg. and tan(θ)],
is:
Sqrt[ [tan(45)+sqrt[5*tan(45)]*cos(60)] = tan(51.82729238…)=1.27201965..=T =sqrt[Φ],
or**

**
[1+sqrt(5)] / 2] =
Φ,
where,
Φ
= 1.618033989… , with corresponding angle
φ,
so that:
φ
= Arctan(Φ)=
Arctan(T^2) = 58.28252559…deg.
**

**
θ
= Arctan(T)
θ
= 51.82729238… deg., and
Tan [θ]
= T ,
Tan [φ]
= T^2= 1.618033989..=Φ
[Tan [θ]]^2
= Tan [φ],
or
Tan [θ
]= sqrt[Tan [φ]]
θ
= Arctan[sqrt[Tan[φ]]]
Angle 60 deg. [the constituent equilateral triangle
angle[Pl.Ti.54 E],
formed by 6 scalene orthogonal triangles of the 30/60 deg. form]
is
related to: 2 =[sqrt(5) +1]/
Φ,
as 60 deg. =Arcos[1/2].**

**
Since:
θ
= Arctan[sqrt[Tan [φ]]],
i.e angle
θ
is related to angle
φ,
and
Since:
Sqrt[ [tan(45)+sqrt[5*tan(45)]*cos(60)] =tan(51.82729238…)=1.27201965..=T =sqrt[Φ]
= tan[
θ],
then
angles
θ,
and 45, are related also to angle 60 deg.
**

**
Logarithmic Relationships:**

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According to my manipulation of basic logarithms [Using the known
relationship for finding logarithm to base 2 of a NUMBER as the ratio :
Ln[NUMBER] / Ln[2] , Ln being the Natural Logarithm [base e], all [positive
Real] numbers [ or angles ] are related to angles Arctan[T] =51.82729237..deg.,
and 45 deg:**

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[tan(45)+[sqrt(5)+1] / e^{[ln[ln[NUMBER]] ]- ln[ln[sqrt[NUMBER]]]}]^0.25 = =
1.27201965..=T =sqrt[Φ].
Examples:
[tan(45)+[sqrt(5)+1] / e^{[ln[ln[2]] ]- ln[ln[sqrt[2]]]}]^0.25 = 1.27201965..=T
=sqrt[Φ]
[tan(45)+[sqrt(5)+1] / e^{[ln[ln[0.1]] ]- ln[ln[sqrt[0.1]]]}]^0.25= 1.27201965..
**

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Number: 2=2*tan(45)=1/[cos(60)], **
**
Logarithmic Relationship****:
The Logarithmic form involving number 2, [giving the relationship for the 30/60
deg. orthogonal triangle, from which the equilateral triangle is
produced], is as follows:
Ln[Ln[+positive real NUMBER or angle]] – Ln[Ln[sqrt(+positive real NUMBER or
angle )]] = =0.693147181..
Ln(2) = 0.693147181..=Ln[1/[cos(60)]], = Ln [[ sqrt(5*tan(45) +tan(45)]/T^2].
e^[0.693147181.]=2
Example:
Ln[Ln[16]] – Ln[Ln[sqrt(16)]] = Ln(2)= Ln [[ (5*tan(45) +tan(45)]/T^2] =
0.693147181...
**

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A Great Pyramid Model [G.P.] relationship with the 5 solids.
A Great Pyramid Model [G.P.], its structure being based upon this **

By cutting in half one of this face [by a vertical from its vertex perpendicular to its base] we get two orthogonal triangles, bearing angle φ [and 90- φ].

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Adding them along their hypotenuse we get a parallelogramme plane.
Placing three such planes each perpendicular to each other, automatically we get
20 equilateral triangles[60 deg.] defined by the vertices of the such structured
planes.
So we constructed the**

In a similar way we do so for the structure of

The 45 deg. [ Arcsin[1/sqrt(2) = Arcos [1/sqrt(2)] structures the square and the

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Theory of Lines Building the World
Solids are formed by triangle surfaces originating from the two triangles the
special scalene orthogonal and the isosceles orthogonal.
Triangles sides are formed from lines based on kinds of 4 lines in **

The special scalene orthogonal is formed from the lines T^3, T^2, T^1, the isosceles orthogonal from 1, forming the square with sides 1.

A second triangle similar to the special scalene is formed with sides T^2, T^1,

and 1. This is constituent orthogonal triangle of the first.

Two pairs of such triangles joined together build[

© Copyright 1986 - 2011 Panagiotis Chr. Stefanides

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*Eur
Ing Panagiotis Stefanides BSc(Eng) Lon(Hons)
MSc(Eng)NTUA TEE
CEng MIET**

Emeritus Honoured Member of the Technical Chamber of Greece [

Ex Hellenic Aerospace Ind S.A., R and D Lead Engineer and Directorate Aircraft Engineer

Manufacturing Engineering Methods’ Superintendent. (c.v.)

“Golden Root Symmetries of Geometric Forms”