The Golden Ratio

So, once you’ve decided on where to paint, and what format to choose, it’s time to start thinking about how to place the elements of the picture on the canvas. And one of the methods artists use to divide up the canvas is geometry. As a former math geek, I have to admit that I really enjoy this topic…

Geometry has been important in art since ancient times; it appeared in ancient Egyptian, Indian, Greek, and Roman art. During the Renaissance, math was seen as the rational key to God’s perfect creation. So geometry was the artist’s way of translating creation into a painting.

So, for example, in della Francesca’s The Baptism of Christ (at right, from Composition, by Sarah Kent) Heaven is represented by a circle, and earth by a rectangle. The spot where they meet – the centre of the circle – is marked by the dove, representing the Holy Spirit. Jesus is placed on the vertical median, and the tree and John are on the verticals that divide the rectangle into thirds.

Because it symbolized the Christian Trinity (God, Jesus, and the Holy Spirit), the triangle (especially the equilateral triangle) was often used to compose religious paintings. It was particularly commonly used in paintings of the Madonna and Child (as in Raphael’s Madonna of the Meadow, at left), and of the Crucifixion.

The most famous mathematical compositional tool, though, is the Golden Ratio, also known as the Golden Mean or the Golden Section. It is based on the number Φ (phi), which is approximately 1.618. Phi is an incredibly cool number (well, to math geeks, at least).

Essentially, the Golden Ratio is the point at which a line can be divided so that the ratio of the long segment to the whole is the same as that of the short segment to the long.  So:

1 : 1.618 = .618 : 1

And did you notice that the short segment’s length is Φ-1?  Even better, Φ² works out to 2.618, or Φ+1.  It has other fascinating properties, too, like being the number toward which the Fibonacci series converges.

A rectangle with sides of length 1 and Φ, called the Golden Rectangle. is shown at right. If you divide off a 1 x 1 square at one end, you’re left with a small rectangle with sides 0.618 and 1 – another Golden Rectangle! And if you divide off a square at the end of that rectangle, you’re left with an even smaller Golden rectangle. And so on. If you then draw quarter circles in all those squares, you get a spiral that duplicates many found in nature:

Totally awesome. No wonder in the Middle Ages it was known as the Divine Proportion, embodying God’s perfect logic!

Ahem. Perhaps I’m not such a former math geek, after all!

In the next post, I’ll look at using the Golden Mean in composition.

Three-Point Perspective

In one-point and two-point perspective, horizontal lines converge toward vanishing points on the horizon, but vertical lines are considered to be parallel. That makes sense, because those lines are usually an equal (or nearly equal) distance from your eyes along their whole length.

But imagine for a moment that you’re in an airplane, looking down at a tall building. The sides of the building are no longer an equal distance from you along their entire length. Instead, they’re receding toward a vanishing point (called the nadir) somewhere down below, as in the diagram of Escher’s Tower of Babel at the left. This viewpoint is sometimes called bird’s eye view.

Vertical lines can also converge at a point above the viewer, called the zenith. Imagine yourself lying on the sidewalk looking straight up at a tall building. From that point of view, the side of the building take on the same position relative to you as, say, a road would if you were standing on it – it recedes away toward a vanishing point at your current eye level. Lying on a sidewalk to study perspective is a bit extreme, but standing and looking up creates a similar effect. This is what happens in the photo of the Canadian mint at the right. This viewpoint is sometimes called worm’s eye view.

Three-point perspective is more common in comics, where it’s often used to an exaggerated degree, than it is in landscape art. However, I think it’s useful to study it anyway because it helps in understanding the concept of multiple vanishing points which are not necessarily on the “normal” horizon. And that understanding will be very useful in the next perspective topic – inclines!

Tom Thomson

Spring
Tom Thomson, 1916
National Gallery of Canada

When I searched for images to illustrate my post on formats, I realized that I don’t know much about art history in general, and landscape history in particular. So I’m starting an ongoing series of posts about famous (or possibly not-so-famous) artists. I’ll start with artists who painted the landscape I know best, and work out from there. And since I am Canadian, that means starting with Tom Thomson and the Group of Seven, because they are pretty much synonymous with Canadian landscape art in most people’s minds.

Canada’s famous Group of Seven almost certainly would have been the Group of Eight if Tom Thomson hadn’t died (under mysterious circumstances) in 1917, before the Group was officially formed. He knew and worked with the other members, and was the first to paint the northern Ontario landscape that they became so well know for. So I’ll start with him.

Tom Thomson was born in 1877 and grew up near the small town of Leith, Ontario, on southern Georgian Bay. He attended two different business colleges (one, in Seattle, owned by his brother), which helped develop his skills as a commercial artist. His early career was rather unsettled, but in 1909 he joined the Toronto design firm Grip, Ltd., working with Franz Johnston under the head designer, J.E.H. MacDonald. In 1911, they were joined by Arthur Lismer & Franklin Carmichael, and in 1912 by Frederick Varley. At a 1911 exhibtion of MacDonald’s sketches, he met Lawren Harris. The story of the Group of Seven was set to unfold.

The Grip artists often organized weekend painting trips near Toronto. Thomson’s work at that point was quite conventional, like the watercolour landscape to the right which he painted in 1910. His turning point came in 1912 when he made two trips into northern Ontario, first to Algonquin Park, then canoeing in the Mississagi Forest Reserve west of Sudbury.

His sketches from those trips, like Pine Stump and Rocks at left, were still conventional and dark, showing none of the strong colour and brushwork he would  later develop, but his friends felt that they caught the character of the northern landscape and encouraged him to continue. In 1913, his first major painting A Northern Lake was purchased by the Ontario government. Also that year, Dr. James MacCallum offered to support him for a year to allow him to paint, and introduced him to A. Y. Jackson.

Unable to enlist during WW I for health reasons, Thomson spent a great deal of time in Algonquin Park from 1914 until his mysterious death in 1917. He sometimes worked as a guide and fire ranger as well as painting small sketches. He was often joined by his friends, whose more formal art training and knowledge helped him develop his style. During the winters he returned to Toronto and worked the sketches up into finished paintings. From 1915, he lived in “Thomson’s shack”, seen at right in its current setting at the McMichael Gallery.

Thomson’s work over these years developed quickly. He applied paint thickly, in an impressionistic style. At times he experimented with pointilism and abstraction. He became fascinated with recording the changes in the landscape over the course of the year; in 1917 he painted 62 daily sketches recording the progress of spring. Some of his work is shown below, in roughly chronological order.

Sunset, Algonquin Park 1914 (oil study on board):

The 1915 painting Northern River is below on the right; the gouache study for it is on the left. Looking through trees across water to the far shore is a frequent motif for Thomson.

Thaw in the Woods (study on board) 1916

Thomson’s two most famous works were both painted in the winter of 1916-17. The West Wind is now in the Art Gallery of Ontario in Toronto, and The Jack Pine is in the National Gallery in Ottawa. The sketch for The Jack Pine is shown beside it. It’s interesting to compare the two; the energy of the sketch gives way to a stylized calmness in the painting.

Finally, this sketch, Early Spring, was done in 1917, only a few months before Thomson died:

More about Tom Thomson:

• Thomson works at the National Gallery
• Thomson works at the Art Gallery of Ontario (the AGO’s collection isn’t searchable, but some of his works are included in this page of highlights of the Canadian collection.)
• Tom Thomson Art Gallery
• The McMichael Collection
• Wikipedia

Format

As I said in an earlier post, I’m working my way through Mitchell Albala’s book Landscape Painting. I chose the photo above as an example of a scene with good spatial cues, based on the Site Selection chapter. Next, in the chapter on Composition, he spends a good deal of time talking about finding the best crop of format for a scene.

“If a good composition can’t be found by imposing a picture window around the scene, then the scene itself is probably not a viable starting point.” -Mitchell Albala

Below I take a look at the various format options. For each format, I’ve included an example by a historical artist, as well as a crop (or two) of the photo. It’s quite interesting how the choice of format changes the effect of the scene.

Horizontal

A horizontal rectangle is the most common format for a landscape painting. In fact, a horizontal rectangle is often called landscape orientation – just ask your word processing software! Even the word “horizontal” is derived from “horizon”. So it seems a natural choice for a landscape. The horizontal format seems to suggest an open space.

Standard sized canvases seem to use a ratio of 4:3 (1.33 : 1), 5:4 (1.25 : 1), or 6:4 (1.5 : 1), but any ratio can be used to suit the scene. Constable’s The Hay Wain is 185.4 x 130.2 cm, or 1.424 : 1. The crops below are all 1.25:1.

Vertical

A vertical format is less common in landscapes, but can create space by leading the eye into an up/down movement on the canvas. Emily Carr often used a vertical canvas for her energetic paintings of trees, like Dead Tree in the Forest, and totem poles.

The first of the two vertical crops below is an attempt to create depth by leading the eye back through the overlapping forms. The second tried to lead the eye up the tree, although that would work better on a taller, thinner tree! Compare it to the similar square format below.

Square

The square format is unusual in landscape, at least until quite recently. In fact, I didn’t find any truly square historical examples; Tom Thomson’s Decorative Landscape: Birches is actually  77.1 x 72.1 cm, so using it here is a bit of a cheat!

Square formats can create a sense of balance and calmness, allowing the eye to follow the elements of the composition rather than the edges of the format. They can also act as a sort of “bulls-eye”, by keeping the eye centred.

Panoramic

The panoramic format, with its 2 : 1 ratio, is an exaggerated case of the horizontal format. It can create a strong sense of openness, but keeping the picture integrated can be difficult. In Henry A. Duessel’s Winding River Landscape, the vertical trees help to break up the space and keep the painting integrated.

In the cropped photo, I tried to create a wide, restful view of the scene while using the vertical trees near each edge to keep the scene tied together.

Ellipses

If you look straight down at the top of a cylinder, it’s a circle. But if you look at it at an angle, it becomes an ellipse, which is essentially a squashed circle.

You can draw a basic ellipse using string and push pins, as shown in the photo. (By the way, notice how smoothly the ends are rounded – it’s not a football shape.) You’ll often see a basic, symmetrical ellipse like that used to represent a circle seen in perspective. And sometimes that’s accurate enough. But circles are not immune to the laws of perspective, so let’s look a bit deeper.

A circle can be inscribed in a square. Let’s start there. Notice that the square and the circle have the same centre point, and that the circle touches the square at the midpoint of each side:

Now here it is viewed in perspective:

The perspective centre of the square is still the perspective centre of the circle also. And the circle still only touches the square at the (perspective) middle of each side – but that’s no longer the (measured) halfway point. One other thing to notice: the top and bottom halves of the circle are not mirror images. Although the circle only touches the side of the square at the midpoint, it actually continues widening slightly on the near side of the tangent point (past the green vertical line).

Similarly, the easiest way to draw a cylinder in correct perspective is to draw it inside a box, as shown in the video below. Also, the roundness of the ellipses in a cylinder increases as they move farther from eye level. So the top of the cylinder on the left in the video  is a flatter ellipse than the bottom.

Perspective Centre

One of the first things you learn when you start studying perspective is that identical objects – like fence posts, telephone poles, or the trees in Van Gogh’s Avenue of Poplars - appear smaller and closer together as they get farther away. But how much smaller, and how much closer together? Sometimes an “eyeballed” estimate is close enough, but if you want accurate spacing you need to understand perspective centre.

You probably remember from your high school geometry that the centre of any rectangle is the point where its diagonals cross. Well, the same thing applies to a rectangle drawn in perspective; the point where its diagonals cross is its perspective centre:

Let’s change that rectangle-in-perspective into the side of a house, and add a door in the middle of it. (Again, eyeballing it might be fine most of the time, but I’m going to be super-accurate here just for the practice.)

1. Here’s our little house, with the orthogonals, or perspective lines, converging on a vanishing point on the horizon. I’ve added a blue line, receding to the same vanishing point, to show where the top of the door will be.
2. Next, I’ve drawn the diagonals of the side of the house (in red), to find its perspective centre. (Notice that the half of the rectangle that’s closer to us appears a bit bigger than the half that’s farther away.) Drop a vertical line (green) through the perspective centre to mark the centre of the door.
3. Here I’ve added a vertical line to indicate one side of the door. There’s nothing special about its placement – it just depends on how wide you want the door to be. I’ve also added a dot on the green centre line to indicate the middle of the door’s height (measured with a ruler). That point will be the perspective centre of the door. Then I drew a diagonal (orange) through that point to find the nearer top corner of the door.
4. Now drop a vertical line from that top corner, and there it is – a door in perfect perspective centre!

We can use a similar method to determine the spacing of fence posts, etc.

1. Draw the first post. Extend the orthogonals to the vanishing point, and add the second post between the orthogonal lines. The distance between the first two posts is arbitrary – just put it wherever it looks right!
2. Find the perspective centre of the rectangle created by the first two posts, by drawing the diagonals. Connect the perspective centre to the vanishing point. This line (blue) will cross the middle of each post we will draw.
3. Now draw a line from the top of post #1 through the middle of post #2. The point at which this line intersects the bottom orthogonal is where the third post will go. Draw the vertical line for post #3 between the two orthogonal lines. (Notice that the the space between posts #2 & #3 appears less than between #1 and #2.)
4. Repeat step 3 as many times as necessary to complete the row of posts:

You can use this method for any equal-sized rectangles in perpective, such as a tiled floor.

Site Selection

I’ve been reading Mitchell Albala’s book Landscape Painting. One of the first things that struck me about it is that he divides the usual chapter on “composition” into two chapters: Site Selection and Composition. This is the first time I’ve seen site selection discussed so explicitly, and it seems to me to be an excellent idea because as he explains, a lot of attractive scenes don’t make good paintings.

Albala sums up his thoughts on choosing a good site by saying:

“A landscape painter’s subject can be almost anything, but the choice is never arbirtrary. A potential site must contain the types of visual cues painters rely on to differentiate forms and suggest depth.”

He goes on to explain those cues in more detail: patterns of light/shade that create a sense of volume, overlap and scale, distinct foreground, middle ground and background, and linear perspective. No new concepts, just applied a bit earlier in the painting process (i.e, before it starts!).

He also discusses types of scenes that are problematic, including “walls” of trees. He says, “They appear to be dimensional in real life, but to create an illusion of space in a painting, they must have clear patterns of lights and darks to define structure.” So the photo on the left below wouldn’t be a good choice of scene to paint, although in real life it was quite attractive, while the scene of the boys fishing has much better depth cues. Next time I go plein air painting, I’ll be thinking of this chapter before I set up my easel!