mathcad prime - the relationship between the cubit, meter, pi and the golden ratio

MathCAD Prime - The relationship between the Cubit, meter, pi and the Golden Ratio.mcdx

The relationship of the Cubit to the meter, and the golden ratio, π φ

by Julio C. Banks, PE

This paper is a summary of the relationships of the egyptian unit of measurement, the Cubit, to the meter, pi and the golden ratio, f. The information presented here is based on published work found in the public domain via the Internet. The work begins with the derivation of the golden ratio, . It has been speculated the the Egypcians knew about the φ

pi, f, the meter and the speed of light as "coded" in the dimensions of the greate pyramics. This work is not an endorsement of such beliefs but simply describes such relationshps for the elegance and mystery of such parametric relationships.

1.0 The Golden ratioThe golden ratio is defined as the the ratio of the largest number, a, to that of the smallest, b, as the ratio of the sum of the numbers. a + b, to the largest number a. Therefore,

＝＝φ ―a

b――+a b

a((1))

That is to say,

＝φ ―a

b((2))

Express equation 1 in terms of the Golden ratio, φ

＝＝φ ―a

b+1 ――

1⎛⎜⎝―a

b

⎞⎟⎠

＝φ +1 ―1

φ

＝−−φ2

φ 1 0 ((4))

Equation 4 is a quadratic equation which is most practically solved using the following procedure. Transform equation 4 into the form given in equation 5

＝++φ2

⋅⋅2 B1 φ B0 0 ((5))

Julio C. Banks, PE of 3


Comparing equations 4 and 5, we obtain the values of the constant parameters, and B0 B1

≡B0 −1 and ≡B1 −―1

2

The solution of the general quadratic equation, 6, is

＝φ ⋅−B1

+1 ‾‾‾‾−1 Λ

−1 ‾‾‾‾−1 Λ

⎡⎢⎣

⎤⎥⎦

≔Λ =――B0

B1

2−4

Since and the solution sought, , is a positive number, then the golden ratio, , is>−B1 0 φ φ

＝φ ⋅B1⎛⎝ +1 ‾‾‾‾−1 Λ

⎞⎠

＝φ ⋅―1

2⎛⎝ +1 ‾‾‾‾‾‾‾−1 ((−4))⎞⎠

≔φ =⋅―1

2⎛⎝ +1 ‾‾5⎞⎠ 1.618 ((6))

2.0 The CubitThe cubit is the unit of length that was used to build the Great Pyramid of Giza. It seems to have some amazing properties. For instance. If you draw a circle with a diameter of 1 meter, one sixth of the circumference will be equal to 1 cubit. Keep in mind that we weren't using the meter as a unit of measurement until some time after 1789.

≔Cubit ⋅―1

6⎛⎝ ⋅ππππ DM⎞⎠ ((7))

Where ≡DM ⋅1 mmmm

=Cubit 0.5236 mmmm =cubitcubitcubitcubit 0.4572 mmmm Is this MathCAD defintion correct?

3.0 The Cubit relationship to and π φ

A Cubit can be defined in terms of and as followsπ φ

=−ππππ φ2

0.5236



Now, we can symbolically represent a relationship between , , and the Cubit to π φ

subsecquently the meter represented by the unit-meter diameter, DM

＝−π φ2

Cubit

≔Cubit ⋅―1

6⎛⎝ ⋅ππππ DM⎞⎠ ((7))

((8))

Substitute Eq. 7 into Eq. 8

＝−π φ2

⋅―1

6⎛⎝ ⋅π DM⎞⎠ ((8))

＝DM ⋅6⎛⎜⎝

−1 ――φ

2

π

⎞⎟⎠

((9))

Numerically, the r.h.s. of equation 9 is unity corresponding to a one-meter diameter

=⋅6⎛⎜⎝

−1 ――φ

2

ππππ

⎞⎟⎠

1

An intriguing combination of irrational numbers combining to produce a finite number, unity.


BanksJ

Rectangle

BanksJ

Rectangle

mathcad prime - the relationship between the cubit, meter, pi and the golden ratio

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