Proportion and scale

Callum's first model for the plane study in lesson 3 was a perfect corner: two card walls meeting at a right angle. It looked lifeless, and he could not say why. Visser asked him to cut one wall shorter, then taller, then longer, photographing each version from eye level. In one version the corner suddenly felt right, like a place you would lean against to read. 'Same element, same angle,' she said. 'All you changed was proportion.' Proportion is the relationship of the sizes of parts to each other and to the whole; scale is the size of something compared with a reference, most often the human body. Together they explain why some rooms feel generous and others cramped, why some facades feel calm and others restless, and why a small cabin can feel more welcoming than a vast hall.

Proportion: the relationship of parts

When we say a room is well proportioned we usually mean that its length, width and height relate to one another in a way that feels balanced. People have tried for thousands of years to find rules that produce good proportion. Some rules come from geometry, some from music, some from the human body. None of them guarantees a good design, but each gives a designer a way to make deliberate choices instead of arbitrary ones, and a way to relate the small parts of a building to the large.

The simplest proportions are whole-number ratios: 1:1 (a square), 1:2 (a double square), 2:3 and 3:4. Renaissance architects, especially Andrea Palladio in his villas of the sixteenth century, used ratios like these for the length, width and height of rooms, partly because the same ratios produce harmonious intervals in music. A room 6 by 9 meters, or 20 by 30 feet, is in the ratio 2:3.

The golden section

The best-known proportion is the golden section, also called the golden ratio or golden mean, usually written with the Greek letter phi (φ). Divide a line into two parts so that the whole is to the longer part as the longer part is to the shorter. The ratio that results is about 1.618. A rectangle with sides in this ratio, a golden rectangle, has a special property: cut a square off one end and the rectangle left over is again a golden rectangle, and this can go on forever, producing a spiral of ever smaller squares.

The golden section is closely related to the Fibonacci sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34 and so on, in which each number is the sum of the two before it. The ratio of each number to the one before it gets closer and closer to φ: 8 to 5 is 1.6, 13 to 8 is 1.625, 21 to 13 is about 1.615. Because these are whole numbers, designers sometimes use Fibonacci sequences to size panels, windows or bays so that they relate to each other in nearly golden proportions.

Be careful with claims about the golden section. It is often said that the Parthenon, the pyramids or famous paintings were designed to it, but many of these claims depend on choosing which lines to measure and fall apart under careful study. What is true is that many designers have used it deliberately, and that it is a convenient ratio for relating parts at different sizes. Use it as a tool, not as a magic formula.

# Fibonacci numbers and how fast their ratios approach the golden ratio.
phi = (1 + 5 ** 0.5) / 2
a, b = 1, 1
print(f"golden ratio phi = {phi:.6f}")
for _ in range(10):
    a, b = b, a + b
    print(f"{b:>4} / {a:<3} = {b / a:.6f}   off by {abs(b / a - phi):.6f}")
# Divide a 6.0 m wall into two parts in golden proportion:
wall = 6.0
longer = wall / phi
print(f"A 6.0 m wall divides into {longer:.2f} m and {wall - longer:.2f} m")

golden ratio phi = 1.618034
   2 / 1   = 2.000000   off by 0.381966
   3 / 2   = 1.500000   off by 0.118034
   5 / 3   = 1.666667   off by 0.048633
   8 / 5   = 1.600000   off by 0.018034
  13 / 8   = 1.625000   off by 0.006966
  21 / 13  = 1.615385   off by 0.002649
  34 / 21  = 1.619048   off by 0.001014
  55 / 34  = 1.617647   off by 0.000387
  89 / 55  = 1.618182   off by 0.000148
 144 / 89  = 1.617978   off by 0.000056
A 6.0 m wall divides into 3.71 m and 2.29 m

Le Corbusier's Modulor

In 1948 Le Corbusier published Le Modulor, a system of proportions that combined the golden section with the dimensions of the human body. He based it on a man about 1.83 meters (6 feet) tall, with his hand raised to 2.26 meters. From these he built two series of measurements, a red series and a blue series, each growing by the golden ratio, so that every dimension in a building could be chosen from numbers related both to each other and to the body. The red series includes values such as 1130, 1829 and 2959 millimeters; the blue series includes 1397, 2260 and 3658. He used the Modulor to size everything in his Unité d'Habitation apartment block in Marseille (1947 to 1952), from ceiling heights to the shape of the concrete panels, and cast the figure of the Modulor man into its concrete walls.

Even though few architects use the Modulor exactly today, its idea endures: a building feels right when its dimensions relate to the body that uses it. The height of a counter, a stair step, a door, a window sill and a ceiling should come from how people sit, stand, reach and move.

Scale: size compared with the body

Scale describes how big something feels compared with a reference. Human scale means that a space or element relates comfortably to the size of a person. A small café with a low ceiling and a window seat has intimate human scale. A cathedral nave or a hotel atrium has monumental scale; its size is meant to impress and make individuals feel small. Neither is wrong: a courthouse may want to feel monumental, a reading nook should feel intimate. The problems come when the scale does not suit the purpose, like a children's ward with vast blank walls or a public square so large that it feels empty.

Designers manage scale with elements of known size. A door, a stair, a window or a brick tells the eye how big a building is because we know roughly how big those things are. A building with oversized doors and windows can look smaller than it is; a tall building broken into floors with ordinary windows can feel friendlier than its height suggests. This is why the street front of a city building is so important: the lowest floors, the ones people walk past, should have doors, windows, canopies and details at human scale.

Drawing scale

Architects also use the word scale in a technical sense: the ratio between a drawing and the real thing. A plan drawn so that 1 unit on paper equals 100 units in reality is at a scale of 1:100. Common scales are chosen so that the drawing fits the paper while showing the detail needed: small scales for site plans and large ones for details.

In the United States, architectural drawings usually use imperial scales that are written as a length on paper equal to one foot in reality. Common ones are 1/16 inch = 1 foot for a site, 1/8 inch = 1 foot or 1/4 inch = 1 foot for building plans, 1/2 inch = 1 foot or 1 inch = 1 foot for parts of a building, and 3 inches = 1 foot for details. At 1/4 inch = 1 foot, a wall 16 feet long is drawn 4 inches long. The architect's scale ruler has these scales printed along its edges, and engineers use a separate scale with decimal divisions.

# Converting real sizes to drawing sizes at common scales.
scales = {'1/16" = 1\'-0"': 1/16, '1/8" = 1\'-0"': 1/8, '1/4" = 1\'-0"': 1/4, '1/2" = 1\'-0"': 1/2}
items = {"pavilion length": 20.0, "door height": 7.0, "bench length": 6.0, "stair tread": 11/12}  # feet
for s, inch_per_ft in scales.items():
    sizes = ", ".join(f"{k} {v * inch_per_ft:.2f} in" for k, v in items.items())
    print(f"{s:>12}: {sizes}")
print("1/4 in = 1 ft is a ratio of 1:48, close to the metric 1:50.")

1/16" = 1'-0": pavilion length 1.25 in, door height 0.44 in, bench length 0.38 in, stair tread 0.06 in
1/8" = 1'-0": pavilion length 2.50 in, door height 0.88 in, bench length 0.75 in, stair tread 0.11 in
1/4" = 1'-0": pavilion length 5.00 in, door height 1.75 in, bench length 1.50 in, stair tread 0.23 in
1/2" = 1'-0": pavilion length 10.00 in, door height 3.50 in, bench length 3.00 in, stair tread 0.46 in
1/4 in = 1 ft is a ratio of 1:48, close to the metric 1:50.

Precedent: Palladio's villas

Andrea Palladio, working near Vicenza in northern Italy in the sixteenth century, designed villas whose rooms follow simple ratios such as 1:1, 2:3, 3:4 and 3:5, and published them with their dimensions in The Four Books of Architecture in 1570. The Villa Rotonda, begun in the 1560s, is a square plan with four identical porticoes and a central domed hall. His work spread across Europe and later to North America; Thomas Jefferson's Monticello in Virginia and many public buildings in Canada and the United States owe their calm proportions to Palladio's example.

Exercise 4: proportion studies and a measured self

This exercise takes about five hours.

• Part A: measure yourself. Record your standing and seated eye heights, your reach upward and sideways, your stride length, and the height of your elbow when standing. Draw yourself to scale beside a wall with these dimensions marked.
• Part B: draw the same simple facade (a wall with one door and two windows) four times, using a different proportional system each time: whole-number ratios, the golden section, the Fibonacci sequence and your own body measurements. Write a short note on which feels best and why.
• Part C: draw one room you know at a proper drawing scale, 1/4 inch = 1 foot, in plan and section, with yourself drawn in at the same scale.
• Deliverable: three sheets, each labeled with the scale used, and a half-page reflection.

Self-critique checklist

• Is every drawing clearly labeled with its scale?
• Have I drawn a person in each drawing to show human scale?
• Can I explain why one proportional system worked better than another for this facade?
• Do my measured dimensions match the typical ranges, and if not, have I noted why?