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General: 2* COS 144 (REVELATION 21:17)=1.618033=GOLDEN RATIO=LAST SUPPER
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"Harmonic 360 of The Circle"
The ancients selected the division of 360 degrees for a specific reason.
(Observe the outer ring of this complex Ying-Yang Chart having specifically 360 divisions).
The factors of the number 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36
indicating that it has an unusually large range of divisors and therefore more friends with other numbers.…
So irrational! Euler's identity = e ^ (ix) = cos x + i sin x
As cos (pi) = -1 and sin (pi) = 0,
e^(i.pi) = -1 + i.0 = -1
Hence, e^(i.pi) + 1 = 0
Unfortunately, it had to be said, this is very incorrect.
With Circle's diameter, also equal to (A * G) = Real_Pi = 3.144605511...
A = Base Cathetus = Diameter / G = 1.94347308702659
Nowhere close to Phi....
And B = Opposite Cathetus = A * SQRT (G) = 2.47213595499959
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"Internal consistency (and sanity) check"With r = 1 and Real_Pi = 3.144605511After the conversion from deg to rad as follows:"Real_Radians" = Theta(deg) * (Real_Pi/180)ALL sin(Theta_rad)^2 + cos(Theta_rad)^2 = 1
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The mathematical Golden Spiral or Phi Spiral should mimic nature.
And this is what a Golden Spiral or Phi Spiral really should look like.
These are the formulas used in the calculation of the plotting:
Values for Theta in degrees:
(-2400, -2390, -2380... Zero, 10,20...1430, 1440)
r = Phi ^ (theta / 360)
Sine = SIN(Theta*Real_Pi/180) <== this is the true way
Cosine = COS(Theta*Real_Pi/180)
Corroborated the above values, all the way through with: '=sin^2(theta)+cos^2(theta) = 1
Plot points:
x = r * Cosine(Theta) as calculated above
y = r * Sine(Theta) as as calculated above
Extrapolated in the minus(deg) for Theta, that is, below or beyond zero, so you can see the
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Contribution (in agreement) by Liddz:
To get the correct measure for a circle’s diameter and to prove that Golden Pi = 4/√φ = 3.144605511029693144 is the true value of Pi by
applying the Pythagorean theorem to all the edges of a Kepler right triangle when using the second longest edge length of a Kepler right triangle as the diameter of a circle then the shortest edge length of a Kepler right triangle is equal in measure to 1 quarter of a circle’s
circumference. Also if the radius of a circle is used as the second longest edge length of a Kepler right triangle then the shortest edge length of a Kepler right triangle is equal to one 8th of a circle’s circumference:
Example 1:
The circumference of the circle is 12 but the measure for the diameter of the circle is not yet known. To discover the measure for the diameter of the circle apply the Pythagorean theorem to both 1 quarter
of the circle’s circumference and also the result of multiplying 1 quarter of the circle’s circumference by the Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895. Divide the diameter of the
circle by the square root of the Golden ratio = 1.272019649514069 to confirm that the edge of the square that has a perimeter that is equal to the numerical value for the circumference of the circle is equal to 1 quarter of the circle’s circumference.
Multiply the edge of the square by 4 to also confirm that the perimeter of the square has the same numerical value as the circumference of the circle.
Divide the measure for the circumference of the circle by the measure for the diameter of the circle to discover the true value of Pi. Multiply Pi by the diameter of the circle to also confirm that the circumference of the circle has the same numerical value as the perimeter of the square.
The second longest edge length of a Kepler right triangle is used as the diameter of a circle in this example. 12 divided by 4 is 3 so the shortest edge length of the Kepler right triangle is 3. The hypotenuse
of a Kepler right triangle divided by the shortest edge length of a Kepler right triangle produces the Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895.
According to the Pythagorean theorem the hypotenuse of any right triangle contains the sum of both the squares on the 2 other edges of the right triangle.
The shortest edge length of the Kepler right triangle is 3 and since the ratio gained from dividing the hypotenuse of a Kepler right triangle by the measure for the shortest edge of the Kepler right triangle is the Golden ratio of Cosine (36) multiplied by 2 =
1.61803398874989 then the measure for the hypotenuse of a Kepler right triangle that has its shortest edge length as 3 is 4.854101966249685.
4.854101966249685 divided by 3 is the Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895.
The square root of the Golden ratio = 1.272019649514069
4.854101966249685 squared is 23.562305898749058.
3 squared is 9.
23.562305898749058 subtract 9 = 14.562305898749058
The square root of 14.562305898749058 is 3.816058948542208.
Remember that the second longest edge length of the Kepler right triangle is used as the diameter of a circle. The measure for both the second longest edge length of this Kepler right triangle and the diameter of the circle is 3.816058948542208.
Remember that the shortest edge length of this Kepler right triangle is 3 and is equal to 1 quarter of a circle’s circumference that has a measure of 12 equal units.
Circumference of circle is 12
Diameter of circle is 3.816058948542208.
Diameter of circle is 3.816058948542208 divided by the square root of the Golden ratio = 1.272019649514069 = 3 the edge of the square.
3 multiplied by 4 = 12.
The perimeter of the square = 12.
12 divided by 3.816058948542208 = Golden Pi = 3.144605511029693144.
4/√φ = Pi = 3.144605511029693144 multiplied by the diameter of the circle = 3.816058948542208 = 12.
The circumference of the circle is the same measure as the perimeter of the square.
4/√φ = 3.144605511029693144 is the true value of Pi.
PYTHAGOREAN THEOREM: https://en.wikipedia.org/wiki/Pythagorean_theorem
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