A Closer Look at Root Rectangles

Disclaimer: Please note that my perspective is aesthetic inspiration, practicality and real life theoretical implications rather than mathematical.

What is the root of root numbers?

The original root of root numbers is probably geometry, which literally means measuring earth
Try to think of root numbers more like an area instead of the length of a line
Root numbers could be viewed as a mathematical way of describing how a surface (a two dimensional area) relates to a line (one dimension)

In the examples below I am using simple shapes and diagonals to show the relationships between different primitive shapes and root numbers.

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 (c²) = the sum of the other square areas (a² + b²)

Example: Let's say that the area of a = 1 and b = 1

then the area of c will be

c² = 1² + 1²

c = √2

This is the well known Pythagoras Theorem.

View an interactive demonstration of the Pythagoras Theorem.

You can read more about it here at Wikipedia.

Root Rectangles

In the example below I'm using diagonals to show the relationship between root numbers and a simple square. It explains how root numbers is achieved using a start square with the unit height and width of 1.

Properties of root rectangles

Shape	Apspect	Comment
	1 : 1	A simple square
	1 : √2	Din format, 8 pointed star or octagon, European Paper size, A1, A2, A3, A4, ...
	1 : √3	Equilateral Triangle, sexagon, tetraed
	1 : √4	Simply 2 squares
	1 : √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, the longest side cut in half, has the same ratio as the larger rectangle.

European paper format (ISO 216 or DIN 476)

√2 rectangle in relation to the octagon

A closer look at 1:√3

A square root of 3 rectangle is simply half an equilateral triangle

A square root of 3 rectangle inscribed in a hexagon

A more complex division of a square root of 3 rectangle inscribed in a unit circle with a radius of 1

Vesica piscis, two unit circles with the radius of 1, intersecting in such a way that the center of each circle lies on the circumference of the other

A more complex division of a equilateral triangle inscribed in a circle

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 ancient 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", etc.

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, etc)

What is the root of root numbers

I believe the truth behind root numbers is quite simple, the need to have a system to measure the ground when building large structures (i.e. buildings). The only tools needed to construct root rectangles is to drive a stick in the ground with a string attached to it. From these simple tools, a person can draw circles, right angles, squares and diagonals. Based on number one (can be any scale), diagonals in those squares are what we call root numbers. Think of what you leaned in school, "What's the square root of...?". That's really how simple it is.

All the figures I have done above can be done with very simple tools, a stick and a string. From that, deduct the simplicity and beauty of root rectangles. Why is it beautiful? Personally I think it's because it is organized dynamic symmetry, not unlike nature itself.