A diagram comparison - using algebra and numbers of
"the golden mean" and "the rule of thirds"
plus identification of the 'sweet spot' area and how this can be used for the focal point
copyright Katherine Tyrrell
(click on image to download for free for education/non-commercial use only)
Mathematics and art
How do numbers relate to art? Ratios and number intervals are important in rather a lot of famous artwork - just as they are in music.
I've always known that a talent for mathematics and music very much go together - in terms of conceiving patterns and sequences - but have sometimes wondered whether and how much of this translates over into art.
For example, quite a lot of artists seem to be really uncomfortable with numbers while I know a lot of classical art is based on ratios plus I seem to do a lot of visual counting ("too much/too little") and literal counting ( "how many inches?") every time I set up a composition.
This post covers some maths - but I also hope it's accessible to people who don't think they're any good at maths, while also providing detail for those who are interested. Just skip over the bits which you don't relate to and you'll probably still get the gist of what this is all about!
The golden ratio and rule of thirds
A ratio describes how one number relates to another. The most famous ratio used in art has various names and a simplified version for those who don't do maths.
It's been called the divine proportion, the golden ratio, the golden section, the golden spiral. The simple version is the rule of thirds.
It's actually related to a ratio which can be found again and again in the natural world. The golden ratio is approximately to 1·61803 39887 to 1 or 1 to 0.61803 39887. You can read definitions below in the boxes - but skip these if you feel mathematically challenged.
I've included a diagram at the top of the page which compares "the golden ratio" with "the rule of thirds". You can see and download a larger version if you click on it. It shows you the algebra. It also shows you how it works in terms of numbers. You'll note that the rule of thirds is an easy way of remembering the golden ratio - and also that they are not quite the same thing.
Norman Garstin (Newlyn School)
Penlee Gallery, Penzance, Cornwall
Garston's composition of this painting of Penzance promenade on a rainy day was much influenced by Japanese art. It also illustrates how you don't have to place subjects exactly on the sweet spots to help the composition - nearby can be fine.
The title is a quote from Shakespeare who used the phrase in ‘King Lear’ and ‘Twelfth Night’.
Definitions of the golden ration / section / mean
Here are two definitions - the first one from ArtLex
A proportional relation (ratio) obtained by dividing a line so that the shorter part is to the longer part as the longer part is to the whole. Another way to describe this: a proportion between the two dimensions of a plane figure or the two divisions of a line, in which the ratio of the smaller to the larger is the same as the ratio of the larger to the whole: a ratio of approximately 0.618 to 1.0and the second one from Wikipedia.
ArtLex - Golden Mean or Golden Section
In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is approximately 1.6180339887....
At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties.
Wikipedia - The Golden ratio
'La Coiffure' / Combing the Hair (circa 1892 - 1895)
oil on canvas, 114.3 x 146.7 cm
National Gallery London
The sweet spot is where the lines intersect if dividing up both height and width according to the golden ratio or rule of thirds.
On the diagram at the top of this post I've shaded the space inbetween the intersections and circled the area which would be broadly be regarded as the sweet spot.
You don't need to know or understand the maths to know that placing a the focal point on or near the sweet spots (see the diagram above) is very likely to make an image very pleasing to the eye.
The ratio is used extensively in natural forms and lots of items in nature - such as Nautilus shells and the spiral within the seed head of a sunflower work on the principle of the golden ratio - hence the notion that this measure of proportion is divinely inspired. There are lots of examples of images of the golden ratio on Google.
"Combing the hair" , the painting by Degas, is usually studied because of its use of a particularly strong and striking colour palette - but it's also very interesting in terms of how it relates to the sweet spots. What I notice is that the changes in direction in the girl's elbows and the woman's hands and the small yellow vase on the table mark the outer perimeter of the four sweet spots while the girl's face and the hand and brush seem to be plumb on two of them. When you see this painting in person the colour and the length and direction of the main lines seem overpowering - but I find that my focus remains on the action of the hair brushing.
Tip: Use the sweet spot for:
- areas within the drawing/painting which you want to emphasise
- areas where strong contrast is being used to draw attention to the focal point
- areas where there are changes in key elements of the drawing/painting - such as the direction and/or angle of a line, the beginning of a transformation of a shape - particularly if these are being used to guide the eye around the drawing/painting
Tip: the ratio and rule are useful guides - which can also be broken. Your drawing/painting is not a failure because you didn't follow this particulae guides!
You can find more information about use of the rule of thirds in the links below and in the 'rule of thirds' module in Composition and Design - Resources for Artists. The photography links are particularly good at demonstrating how photographers use the rule of thirds to compose photographs.
Thinking in Threes
The other thing I wanted to emphasise around thinking in threes is an old adage which will be well know to gardeners planting out beds - and that's that odd numbers work better than even numbers when it comes to presenting objects. I've no idea why this should be other than that odd numbers tend to be more pleasing to the eye. Maybe because they tend to be easier to fashion into shapes which are also more pleasing or because they are easier to balance within the visual field?
Making a Mark Composition Project:
- Composition - the four most important lines
- Composition and Design - finding and creating a focal point
- Composition and Design - A Digest #1
- Greg Albert - The Simple Secret to Better Painting
- Composition - Principles of Design
- Composition - The Elements of Design
- Composition and Design - An Introduction
- Composition and Design - Resources for Artists