Drawing and Painting the Landscape – Root Rectangles

 

St Ives Harbour - Root-4 Rectangle

Lesson 31 of  Drawing and Painting the Landscape by Philip Tyler is another lesson about composition with a foundation in mathematics. 

Wikipedia defines a root rectangle as 

A rectangle in which the ratio of the longer side to the shorter is the square root of an integer, such as √2, √3, etc..

Phillip explains how to draw a root-2 rectangle (√2 rectangle) by starting with a square, measuring the diagonal and extending two opposite sides to be the length of the diagonal. 

Root-2 Rectangle

You can then draw a root-3 rectangle by measuring the diagonal of the root-2 rectangle and extending the long sides to be the length of this diagonal.

Root-3 Rectangle

You draw a root-4 rectangle by extending the longer sides to be the length of the root-3 rectangle’s diagonal, and so on.

One of the interesting properties of root rectangles is:

If you divide a root-N rectangle along its long side into N equal subdivisions, all the little rectangles are also root-N rectangles, for example, if you divide a root-5 rectangle into 5 equal slices, all the little rectangles are also root-5 rectangles

Root-5 Rectangle with Subdivisions

Some people believe this symmetry is a thing of beauty and provides a good underlying structure for the composition of a picture (if you are interested, check out the Wikipedia article - Dynamic rectangle). 

Philp points out the paper sizes in the A series (A0, A1, A2, A3, A4, etc) are all root-2 rectangles. If you divide a root-2 rectangle in half, both halves are also root-2 rectangles. Cut a piece of A0 in half and you have 2 pieces of A1, cut a piece of A1 in half and you have 2 pieces of A2, and so on.

A Paper

Phillip suggests a sketchbook exercise creating a series of compositions in squares and different root rectangles. I did a slightly different exercise. I took a holiday photograph which isn't an obvious candidate to paint.

St Ives Harbour - Source Photo

I searched this picture for squares and root rectangles that have the most promise as paintings. My favourites are the image at the top of the post and this one.

St Ives Harbour - Square

I will refine and improve them in the next lessons.