## 48 Golden Ratio Spirals

How to construct a perfectly harmonized Cube and Sphere:

First, draw 8 golden rectangles beginning with this self-explanatory geometric configuration:

A simple 2 x 2 square, where A-B (φ 0.618...) + B-C (φ 0.618...^{2} + 1) = 2

The line A-C consists of an infinite number of evenly spaced prime numbers

as explained in a Cubical Universe, and expressed there as 1 to infinity + 1 = 2

Keep in mind that in actuality straight lines do not exist, arcs and circles do.

8 golden rectangles and their converging points form a circle, and it represents one

face of a cube. Therefore, 48 golden rectangles (logarithmic spirals) signify a sphere within

the cube, and it shows unique and fixed ratios between the Cube and Sphere.

To simplify the idea:

__Volume Calculations:
__

Cube 2³ = 8

Sphere 4/3πr³ (r = φ 0.618...) = 0.98883923...Volume 1 (V1)

8 / 0.98883923...(V1) = 8.09029389... x φ 0.618... = 5.00007660...

What if:

8 / 0.98885438...(V2) = 8.09016994... x φ 0.618... = 5 (has a slightly altered fraction 4/3)

1.6 / ϕ** =** 0.98885438...(V2) / π / r³ (r = φ 0.618...) **= **1.33335376...** **the adjusted "4/3"

A sphere volume of 1.33335376... x π x r³ is more acurate than the usual 4/3πr³ formula.

__In this example, Volumes for a Cube and Sphere of equal diameters:
__

Cube (if side = φ 0.618... x 2) ³ = 1.8885438...

Sphere (1.6 / ϕ) = 0.98885438...(V2)

0.98885438... x 10 = 9.8885438... **-** 1.8885438... = 8

__A Sphere in a Cylinder:
__

Imagine a cylinder volume πr**²**h ( r = φ 0.618... and h = 2 x r ) which contains

a sphere with the common formula 4/3πr³ (r = φ .618...) = 0.98883923...(V1)

Cylinder V 1.48325884... / Sphere 0.98883923...(V1) = 1.5 Here the fraction 4/3

reduced the sphere size, so as to accommodate and keep the cylinder volume!

__Explanation for the two different Sphere volumes:__

Sphere one 0.98883923...(V1) is restricted and the second 0.98885438...(V2) is not.

Sphere one is of slightly lesser volume because it is surrounded and "compressed"

within a "Cube", whereas the second sphere in a free state is not. Imagine a sphere

within a cube of equal diameters, where one or the other has to give up space.

So, V2 (1.6 / ϕ) represents a sphere that exists in a harmonious, free and balanced state.

(Imagine a filled balloon and that same shape without the balloon material)