A Closer Look at Root Rectangles

Pythagoras Theorem

Let's take a closer look at Pythagorean theorem. The surface of the hypotenuse is equal to the sum of opposite surface plus the adjacent surface.

Or another way of putting it:

The area of the tilted square = the sum of the other square areas

To give you an example: Let's say that a = 1 and b = 1

then

c² = √ (1² + 1²)

c = √2

This is the well known Pythagoras Theorem.

You can read more about it here at Wikipedia.

Root Squares

In the example below I'm using diagonals to show the relationship between root numbers and a simple square. The image below explains how simple root number can constructed using a start square with the width and height of 1.

Properties of root squares

Shape	Dimension	Comments
	1 x 1	A simple square
	1 X √2	Din format, 8 pointed star or octagon, European Paper size, A1, A2, A3, A4,…
	1 x √3	Equilateral Triangle, sexagon, tetra ed
	1 x √4	Simply 2 squares
	1 x √5	Related to the "golden mean" and the pentagram or pentagon.

A closer look at 1:√2

1 relates to √2 as (√2 / 2) relates to 1.

The image below shows a more complex way of dividing a square root of 2 rectangle.

The ratio 1 to √2 is used in the A paper format (ISO 216 or DIN 476) because of its properties where this rectangle where the logest side cut in half has the same ratio as the larger rectangle.

A paper format (ISO 216 or DIN 476)

√2 relation to the octagon.

A closer look at 1:√3

A closer look at 1:√5

The interesting thing about this irrational number 1.618 and 0.618 is that the unit 1 relates to 0.618 as 1.618 to 1. In acient greece this ratio was called "phi" or "φ". This ratio was also know as dividing a line in the extreme and mean ratio. In more general terms this ratio is also known as the "golden mean", "golden ratio", "golden section", golden cut", "golden number, "divine proportion", …

We can also find this "golden" number in a pentagram enclosed in a pentagon.

Here is another image showing the irrational number 1.618 or 0.618 relation to √5.

A triangle enclosed in a circle

From the study of phyllotaxis and the related Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...)

What is the root of root numbers?

I believe the truth behind root numbers is quite simpe, the need to have a system to measure the ground when building large structures (i.e. buildings).