## 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 AB= φ .618... and BC= φ .618...² + 1

Here 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 a unique and fixed relationship between this Cube and Sphere.

To simplify the idea:

**Volume Calculations:**

**Cube** 2³ = **8**

**Sphere** 4/3πr³ (r = φ .618...) = **.98883923...**Volume 1 (V1)

8 / .98883923...(V1) = 8.09029389... x φ .618... = 5.00007660...

What if:

8 / .98885438...(V2) = 8.09016994... x φ .618... = **5** (has a slightly altered fraction 4/3)

**1.6 / ϕ =**

**0.98885438...**(V2)

**/ π / r³**(r = φ .618...)

**= 1.33335376...(**the new "4/3")

**This Volume formula represents an unrestricted Sphere:**

**V = (√1.25 + 1.5) x 1.6 x r³**

**V**** = [** (√1.25 + 1.5) x 1.6 **]** __or__ **[** π x 1.33335376... **]** __or__ **[** (1.6 / ϕ) + 3.2 **]** __x r³__

**In this example Volumes for a Cube and Sphere of equal diameters:**

Cube (side = .618... x 2) ³ = 1.8885438...

Sphere (1.6 / ϕ) = 0.98885438...

1.8885438... / 0.98885438... = 1.90983005... [ 5 / (ϕ + 1) ]

And
0.98885438... **-** (1.8885438... / 10) = 0.8

8 / 1.90983005... = 4.18885438... [ (1.6 / ϕ) + 3.2 ]

**A Sphere in a Cylinder:**

Imagine a cylinder volume πr**²**h ( r = φ .618... and h = 2 x r ) which contains

a sphere with the common formula 4/3πr³ (r = φ .618...) V = **.98883923...**

Cylinder V 1.48325884... / Sphere V .98883923... = 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, the formula V = (√1.25 + 1.5) x 1.6 x r³ consisting of true numbers pi and phi,

represents spheres that exist in a perfectly harmonious, free and balanced state.

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