Article 072 - A Further explanation of the Golden ratio to a Pythagorean Cube

A Further explanation of the Golden ratio to a Pythagorean Cube

The golden ratio 1: 1.618 is a ratio with an infinite number sequence in it and

so cannot be accurately established and fixed for use in actual work without

degrading the 'beauty' of the ratio.

The pre-Euclid subdivision of a side of a 1:1:1:1 cube tries to divide the face of

the 1:1:1:1 cube into 4 or 5 equal subdivisions using the diagonal of the 1:1:1:1

cube as a radius.

This allows the side of the 1:1:1:1 cube to be related to the diagonal of the

square of the 1:1:1:1 cube with a unit of 1.414 the square root of 2.

This does however have an error within it since the face of the 1:1:1:1 cube

cannot be completely divided into 4 or 5 equal subdivisions.

1.414 / 4 = 0.3535 creates the progression 0.3535 , 0.7070, 1.0605 which

exceeds the side length of the cube of length 1    

1.414 / 5 = 0.2828 creates the progression 0.2828 , 0.5656, 0.8484, 1.1312 

which exceeds the side length of the cube of length 1      

The Euclid subdivision of a side of a 1:1:1:1 cube tries to divide the face

of the 1:1:1:1 cube into not 4, not 5, but the next number in sequence,  

6 equal subdivisions.

This allows the side of the 1:1:1:1 cube to be divided into 6 equal subdivisions.

1 / 6 = 0.166 creates the progression 0.166, 0.333, 0.5, 0.666, 0.833, 1.00 which

matches the side length of the cube of length 1

This then allows the diagonal 1.414 to be equally subdivided by the 6 divisions.

This matches the side of the cube to the diagonal, the square root of 2.

The additional 0.618 is then made up of one or more divisions of the same harmonic

to allow a match to the golden ratio 1 for the side of the cube to 0.618 and to the whole ratio length of 1: 1.618         

The six subdivisions also allow the 1:1:1:1 cube to be divided into 6 and be related by geometry and proportion to the hexagon and the other ratios from Pythagorean Geometry and Solids.           

Ian K Whittaker

Websites:

https://sites.google.com/site/architecturearticles

Email: iankwhittaker@gmail.com

06/01/2014

14/10/2020

383 words over 1 page.