The Measure of the Sheet

The idea: proportion and scale

Proportion is the size relationship of the parts of a work to one another and to the whole. Scale is the size of a thing compared with something else, such as the human body or its surroundings.

In a notan: In a notan the whole is the sheet, and every cut is measured against it. A small cut reads as a figure on a field of paper. As a cut grows the balance tips: the ground seen through the hole becomes a field of its own, and the band of paper left around it becomes the figure. The flip adds a second measure, because each piece is exactly the size of its hole and lies outside the sheet. A cut that reaches a third of the way in from one side, folded out over that side, makes the finished design a third wider than the paper it came from. The sheet’s own proportion, square, golden or root two, sets the terms for every cut that follows.

Proportion compares sizes within a work: a part with another part, or a part with the whole. Scale compares a thing with something outside it, usually our own bodies. Both are relationships, not sizes. A cut two inches across is large on a postcard and small on a poster; what the eye reads is the ratio.

Two ratios are especially useful in a notan. The golden ratio, written φ and about 1 to 1.618, gives the rectangle that leaves a smaller rectangle of the same shape when a square is taken off it. The root-two rectangle, about 1 to 1.414, is the one shape that folds in half into two smaller copies of itself, which is why A-series paper uses it. In this activity you set the sheet to these ratios, size cuts by them, and find the point at which a growing cut swaps figure and ground. Then you build both rectangles by hand, with a ruler and a strip of paper, and cut them from a sheet you painted.

Art history

For as long as people have built and drawn, they have argued about which proportions look right. Some systems begin with the body, some with geometry, and some with a good story. These four works show proportion as measurement, as a theory of the body, as a lever of scale and as a working tool.

• Iktinos and Kallikrates, architects The Parthenon, 447–432 BCE Acropolis, Athens

  Its builders worked with whole-number ratios: a ratio of 4 to 9 recurs in the elevation and in the columns’ relation to their spacing. You will often read that the Parthenon was designed with the golden ratio, but measurement does not bear that out; the claim usually rests on a rectangle drawn over a photograph. Checking a proportion before repeating it is part of this lesson.

• Leonardo da Vinci Vitruvian Man, c. 1490 Gallerie dell’Accademia, Venice

  Leonardo tested the Roman architect Vitruvius’s claim that a well-made body fits both a circle and a square. His circle is centered on the navel, his square lower down, and his notes list ratios of part to whole, such as an arm span equal to the height. Proportion here is a web of relationships, as it is between a notan’s cuts and their sheet.

• Utagawa Hiroshige Plum Park in Kameido, 1857

  In this print from One Hundred Famous Views of Edo, one dark plum trunk, cropped by the edges of the sheet, fills the foreground, while visitors behind a fence look tiny. The jump in scale between near and far makes the trunk a figure and the whole garden its ground. Vincent van Gogh copied the print in oil in 1887.

• Le Corbusier Unité d’habitation, 1945–52 Marseille, France

  Le Corbusier sized this apartment block with the Modulor, his scale of measures built from a man 1.83 meters tall whose raised hand reaches 2.26 meters, divided by the golden ratio into two related series. Rooms, windows and fittings share one family of sizes, set to the body.

Design today

Proportion is built into the paper on a designer’s desk. The ISO A series, used in most of the world, gives every sheet sides in the ratio 1 to √2, about 1 to 1.414, so folding a sheet in half makes the next size down in exactly the same shape: A4 (210 × 297 mm) halves into A5 (148 × 210 mm). US Letter (8.5 × 11 inches) lacks the property: halve it and the ratio jumps from about 1.29 to about 1.55.

Typographers borrow the idea as a modular scale: choose a base size and a ratio, such as 1.25, 1.5 or 1.618, and derive every heading and caption size from it, so the sizes relate the way musical intervals do. In architecture and product design, a door, a handrail and a phone are sized to the body, and making something much larger or smaller than we expect changes how we read it at once.

Warm-up

Fold a sheet of US Letter in half and turn the half so it stands the same way up as the whole. Lay it in the bottom-left corner of an unfolded sheet and draw the whole sheet’s diagonal from that corner. If the half’s far corner lands on the diagonal, the two rectangles share a proportion. It will miss. Now repeat with A4: the half lands on the line, because A4 and A5 are the same shape. You have tested a ratio without measuring a single side.

In the studio

Make it in Studio Notan, where every cut can be moved, refolded and undone.

1. Open the studio. In the Paper section, switch the units to cm and type a width of 14.8 and a height of 21. That is A5, a root-two rectangle. The default sheet, 4 by 5.5 inches, is 1 to 1.375: close, but not the same.
2. Cut a precise parent. Choose the Pen, which snaps to the sheet’s corners, edges and center lines, and click four points: the top-left corner, the point where the top edge meets the vertical center line, the center of the sheet, and the point where the left edge meets the horizontal center line. Click the first point again to close it, and fold it up with ↑. The notch is half the sheet’s width by half its height: an A7 in the corner of an A5.
3. Build the A series inside the sheet. Select the notch with Move and, in Transform, set Scale to 70.7 percent, Rotate to 90 and Copies to 3, then click the bottom-right square of the reference-point grid, which here sits on the center of the sheet. Each copy is the one before turned a quarter turn and scaled by 1 over √2, exactly how A7 relates to A8, A8 to A9 and A9 to A10. Check the preview, then press Apply.
4. Test scale against figure. Save the project from the File menu, then choose Reset, Design: the cuts go, and the sheet and colors stay. Draw a circle near the center with the Ellipse, about a tenth of the sheet’s width across, and fold it down. Select it, set Scale to 150 percent and press Apply. Transform resets after each Apply, so type 150 again and repeat. After each step ask: is the circle a shape on the paper, or has the paper become a frame around a hole? Note the step where it tipped, and the circle’s width as a fraction of the sheet’s.
5. Now the golden rectangle. Reset the design again and set the sheet to 10 by 16.2 cm. Draw a square with the Rectangle (Shift, once dragging) about six-tenths of the sheet’s width, near the top, and fold it up. In Transform set Scale to 61.8, a Move Vertical that carries the first copy clear of its parent (try 5.5 cm) and 3 copies. You get four squares at 100, 61.8, 38.2 and 23.6 percent: each one golden step smaller than the last.
6. Compare proportions without touching the cuts. Change only the sheet’s width: 16.2 for a square, 11.5 for a root-two rectangle, then 10 again for the golden one. The sheet changes about its center; the cuts stay put while every hole and piece is worked out again. Export a PNG of each from File and set the three side by side. Measure the band of paper beside the largest square in each before you decide which sheet holds the cuts best.

With paper and paint: Build the ratio, then cut to it

In the studio you type 10 by 16.2 and trust the arithmetic. By hand you build the ratio. A square, its diagonal and a strip of scrap paper carried from one line to another make a root-two or golden rectangle without a calculator, and folding and measuring the result checks it. Le Corbusier’s Modulor was a scale of measures to build with; a paper strip marked with one diagonal is a small one of your own. Paper also takes away Undo. A hole cannot be scaled back down once it is cut, so you find the tipping point with loose painted squares before the knife touches the sheet. Even the gray of the ground is a proportion, mixed by counted parts, and paint cannot be unmixed.

1. Paint first, so the paint can dry while you work in the studio. Brush black gouache over one side of the cardstock in two thin coats, the second across the first, and leave the back white. Then mix two grays by counting equal dabs of paint: one of black to four of white, and one of black to twelve. Paint a sheet of watercolor paper and a scrap with each, and write the recipe on the back.
2. Build a root-two sheet. When the black is dry, press the sheet flat for a few minutes under a heavy book, between clean sheets of paper, since cardstock painted on one side curls. Then rule two strips 4 inches wide on its white back and cut them free with the knife against the metal ruler. On one strip, rule a line across 4 inches from one end, making a square, and rule the square’s diagonal. Mark the diagonal’s length on the paper strip, lay the strip along the long edge from the end corner where the diagonal starts, and mark again. Cut across at that mark. The sheet is 4 by about 5.66 inches, or 1 to 1.414.
3. Build a golden sheet. On the second strip, rule the same 4-inch square at one end. Mark the midpoint of the square’s side that runs along the long edge, 2 inches from the end. With the paper strip, measure from that midpoint to either corner on the far side of the square, then lay the strip along the long edge from the midpoint, away from the end, and mark. Cut across there. The sheet is 4 by about 6.47 inches, or 1 to 1.618.
4. Check both by folding, then by arithmetic. Fold the root-two sheet in half, short edge to short edge: the half, 4 by 2.83 inches, is the same shape turned on its side, as A5 is to A4. Fold the golden sheet along its square’s line: what lies beyond the fold, about 2.47 by 4 inches, is a smaller golden rectangle. Divide each long side by its short side to confirm. Pinch each fold only at the edges, since a full crease can crack the paint and will show in the finished notan; the pinch marks can guide your cuts, as the sheet’s center lines guide the studio’s Pen.
5. Test before you cut, because a hole cannot be scaled back down. From the painted scraps, cut loose gray squares a tenth, a fifth, a third and a half of the 4-inch short side: 0.4, 0.8, 1.33 and 2 inches. Lay each on the black front, where it reads as a hole. At which size does the black stop being a sheet and become a frame? Try the other gray. Does the tipping point move with the value you mixed?
6. Choose one sheet and cut by its ratio. On the golden sheet’s white back, rule a parent square 0.618 of the short side, about 2.47 inches, a child 0.618 of that, 1.53 inches, and a grandchild, 0.94 inches. On the root-two sheet, start at half the short side, 2 inches, and scale by 0.707: 1.41 inches, then 1. Set each square against an edge or close to one. Cut each free, turn the sheet black side up, and turn each square over its nearest edge like a page, so it lands white side up, as far outside the edge as its hole is inside.
7. Glue the sheet to the ground whose gray passed your test, with a glue stick on its white back, then glue each piece in place, black side down. A glue stick is drier than liquid glue, so it is less likely to buckle the paper or wet the gouache again.
8. Compare. Lay your notan beside a partner’s built on the other ratio, and measure the band of black beside each parent cut. Then set yours beside the same design on screen: the ratios match and the scale does not. Which sheet holds its cuts best, and at which size does the design feel like an object for the hand?

Take care: Cut on a self-healing mat with an adult supervising, the blade drawn away from your body and the fingers of your holding hand behind the metal ruler, and change the blade when it starts to drag, since a dull blade needs more force and slips more easily. Use paints that carry the ACMI AP seal, cover the table, let the black coat dry fully before you measure or cut, and rinse brushes before the paint dries in them.

Take it further

Take the design off the screen at true scale. Set the sheet to A4 (21 × 29.7 cm), export the Cut lines SVG, which is drawn at the sheet’s real size, and print it at 100 percent. Cut it from black paper with a craft knife and fold each piece out by hand. Then print the same file at 50 percent, which makes it A6 (10.5 × 14.8 cm), and cut that too. The proportions are identical and the scale is not: which version feels like an object for the hand, and which like a picture on a wall?

Reflect

• At what fraction of the sheet’s width did your circle, or your loose gray square, stop being a figure, and did the gray you mixed move that point?
• Which of your three sheets held the cuts best, and what is your evidence?
• Why is it worth checking a claim such as “the Parthenon follows the golden ratio” instead of repeating it?
• Once the pieces fold out beyond the sheet, which proportion matters more: the sheet’s, or the whole design’s?

What to look for

• The sheet is set to a stated ratio, and its size is recorded.
• Every cut’s size follows from one ratio, and the learner can show the scale values used.
• At least one cut is large enough that the paper around it reads as the figure.
• The learner states the tipping point as a measured fraction of the sheet, not a guess.
• The hand-built sheet’s long side divided by its short side is within 0.02 of 1.414 or 1.618, and each paper cut is scaled from its parent by that ratio.