The golden ratio
Two lengths a (longer) and b (shorter) are in the golden ratio if a ÷ b = (a + b) ÷ a. This number is called φ ("phi") and equals (1 + √5) ÷ 2 ≈ 1.618. It is irrational, so its decimal never ends. Also φ² = φ + 1.
A golden rectangle has long side ÷ short side = φ. Cut off a square of the short side and what is left is again a golden rectangle, smaller but the same shape. Repeating this gives a chain of squares, and quarter-circles drawn in them make the golden spiral. Ratios of neighbouring Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55...) get closer and closer to φ: 55 ÷ 34 = 1.6176.
Be careful: some claims that old buildings or famous paintings follow φ exactly are exaggerated. Artists use it as one helpful guide, not a rule. See the full lesson The golden ratio.
Perspective
Perspective is the maths that lets a flat paper show depth. Two facts do the work.
- The horizon is the line at your eye level. Lines that are parallel in real life and go away from you (rails, road edges, tops of a fence) meet at a vanishing point on that horizon. This is one-point perspective. With two or three vanishing points you can draw corners of buildings.
- Things look smaller when they are farther. By similar triangles, size on the picture = real size × (distance from eye to picture) ÷ (distance from eye to object). Double the distance and the picture size is half.
If your eye level is high, the horizon is high and you see tops of things; if it is low, you look up at things. Learn more in Perspective drawing.
Fractals
A fractal is a shape in which a small part looks like the whole. This is self-similarity. You make one by repeating a simple rule many times.
- Fractal tree: draw a trunk, then at its top draw two shorter branches at an angle, then at each of those two shorter branches again. A tree of depth d has 2d+1 − 1 branches.
- Koch curve: replace the middle third of a line with a bump. Each step makes the line 4/3 times longer: after n steps its length is L × (4/3)n.
- Sierpinski triangle: cut out the middle of a triangle again and again.
Ferns, broccoli, river networks and coastlines look like fractals. Computer artists and film makers use fractal rules to make mountains, clouds and trees. See Fractals.
Try it: draw all three
On paper: (1) Draw a rectangle 8 cm by 13 cm, divide off an 8 cm square, then keep going with 5, 3, 2, 1 cm squares. Draw the spiral. (2) Draw a horizon, one point on it, and a road to that point. (3) Draw a Y shape, then put a smaller Y on each branch, three times. Count the branches.
Key formulas and definitions
- φ = (1 + √5) / 2 ≈ 1.618; φ² = φ + 1
- Golden rectangle: long ÷ short = φ
- Size on picture = real size × (eye-to-picture distance) ÷ (eye-to-object distance)
- Fractal tree of depth d has 2^(d+1) − 1 branches
- Koch curve: length after n steps = L × (4/3)^n
Worked examples
1. A golden rectangle has a short side of 10 cm. Find the long side.
Long = 10 × 1.618 = 16.18 cm.
2. A golden rectangle has a long side of 30 cm. Find the short side.
Short = 30 ÷ 1.618 ≈ 18.54 cm.
3. Show that 55 / 34 is close to φ.
55 ÷ 34 = 1.6176, and φ = 1.6180. They differ by less than 0.001.
4. A 6 m tall tree stands 30 m from the eye. The picture plane is 40 cm in front of the eye. How tall is the tree on the picture?
Picture height = 6 m × 0.40 m ÷ 30 m = 0.08 m = 8 cm. At 60 m it would be 4 cm.
5. A Koch line starts 9 cm long. How long is it after 3 steps?
Length = 9 × (4/3)^3 = 9 × 64/27 = 21.33 cm.
6. How many branches does a fractal tree of depth 5 have?
2^(5+1) − 1 = 64 − 1 = 63 branches.
Common mistakes
- Writing φ as exactly 1.6. It is 1.6180339... and never ends.
- Placing the vanishing point anywhere. For lines on the ground it must lie on the horizon (at eye level).
- Thinking bigger on the page always means closer. Size on the page depends on both real size and distance.
- Believing a fractal has to be drawn forever. We draw a few steps; the rule is what makes it fractal.