# 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:√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).

### Other interesting readings

Jay Hambidge, Dynamic Symmetry, ISBN 0-7661-7679-7

Ernst Mössel,

Architecture and mathematics in ancient Egypt, Corinna Rossi, web

### The Web

HEAMedia, The Giza Pyramid and Root numbers (my own site)

HEAMedia, The Flagellation of Christ (my own site)

HEAMedia, A Closer Look at Root Rectangles (my own site)

Wikipedia, Pythagorean theorem

Wikipedia, Dynamic rectangle

Wikipedia, Golden ratio (1:1.618)

Wikipedia, Trigonometric_functions

Wikipedia, Silver ratio (1:√2)

Golden Section

The Golden Rectangle and the Golden Ratio

Proportions: Golden Section or Golden Mean, Modulor, Square Root of Two, Theorie and Construction