The Proportion of Paper: the hidden mathematics of catalog and editorial designThe Proportion of Paper: the hidden mathematics of catalog and editorial design

In 1786, in Göttingen, the physicist Georg Christoph Lichtenberg described a small observation in a letter to a colleague: there was a rectangle that, folded in half across its middle, kept its proportions. This rectangle, with sides in the ratio 1 to √2, was something mathematicians knew about but nobody used. In 1922 the German engineer Walter Porstmann wrote the ratio into the DIN 476 standard and added a rule: the largest sheet, A0, would be exactly one square meter. That is how A4's 210 by 297 millimeters came to be; today it is the one paper language used throughout the world except in the United States and Canada. A catalog's first design decision was made two hundred and forty years ago, before the designer draws a single line.

The value of Lichtenberg's rectangle is economic. Fold A0 in half and you get A1; fold again and you get A2; every size is exactly half the one before it, and no size produces waste. The same design scales from an A5 brochure to an A2 poster without its proportions breaking; the press sheet divides without trimming; envelopes, folders, and shelves are made to the same measure. American Letter paper (8.5 by 11 inches, a ratio of 1.29) does not have this property, and when folded in half it drops to a different ratio. The A4 PDF that a Turkish manufacturer preparing an export catalog sends to the United States is either cropped at the edges or shrunk on the printer there; two markets call for two page layouts, and this small detail asks for the grid to be built on two ratios from the start.

In 1946 the Dutch typographer J. A. van de Graaf measured the page layouts of medieval manuscripts and early printed books and found the same geometry in all of them: a page ratio of 2 to 3, a text block also of 2 to 3; the height of the text block equal to the width of the page; inner, top, outer, and bottom margins in the proportion 2 to 3 to 4 to 6. Jan Tschichold called this structure the "secret canon" and showed that it could be drawn without a ruler, using the line-dividing method in the sketchbook of the thirteenth-century architect Villard de Honnecourt. In 1947 the Argentine Raúl Rosarivo reached the same result by dividing the page into nine: the layout of Gutenberg's 42-line Bible had been built on ninths.

What the canon tells design is that the margin is not empty space but part of the proportion. An outer margin twice the inner one makes room for the thumb holding the book; the bottom margin being the widest keeps the text block from "falling" down the page, because the eye takes a point slightly above the geometric center as the center. This five-hundred-year-old knowledge applies unchanged today to the page margins of a product catalog.

In 1928, in Die neue Typographie, Tschichold argued for asymmetric layout and functional typography; the Swiss school turned it into a system. Josef Müller-Brockmann's Grid Systems in Graphic Design, published in 1981, became the foundational text of design education for the grid that divides the page into modules of columns and rows. The grid provides two things: consistency and speed. In a catalog of a hundred products, when every page's photograph, code, dimension table, and description sit in the same modules, the buyer learns where to look after the second page and scans the rest instead of reading it.

The grid's invisible second layer is the baseline: every line of text advances down the page at the same interval, so that when two columns sit side by side their lines align, and the type showing through from the back of the sheet falls on the lines of the front. The thinner the catalog's paper, the more this alignment stops being a matter of aesthetics and becomes one of legibility.

There is a scientific case for the catalog being printed. In 2018, in Educational Research Review, Pablo Delgado, Cristina Vargas, Rakefet Ackerman, and Ladislao Salmerón published a meta-analysis covering 54 studies from 2000–2017 and more than 170,000 participants: for informational texts, readers on paper comprehended significantly better than readers on screen, and the gap, rather than closing as the screen generation grew up, was widening. The researchers named this "screen inferiority"; a product catalog containing technical data is exactly this kind of text.

Visual memory, on the other hand, recognizes no distinction between paper and screen, and its scale is surprising. In 1973 Lionel Standing showed participants ten thousand photographs over five days; in the recognition test that followed, the success rate was 83 percent. Roger Shepard had measured 98 percent recognition with 612 photographs in 1967. This is the "picture superiority effect" that Allan Paivio theorized in 1971: an image is remembered far better than a word. The reason a product photograph and a technical table sit on the same page of a catalog is not aesthetics; it is that the table is remembered by hanging on the photograph.

In the 1960s the newspaper designer Edmund Arnold proposed the "Gutenberg diagram": on an evenly distributed page the eye starts at the top left and traces a Z toward the bottom right; the bottom left corner is the "fallow" area. In 2006 Jakob Nielsen found the F-pattern with eye tracking on text-heavy web pages: the top line in full, the second line in part, then down the left edge. The two patterns belong to two different kinds of page; the catalog designer must know which one applies on each spread. A promotional page with a single large image is read in a Z, a dense product list in an F; the heading and the code are placed on these paths, not outside them.

At some point in every catalog brief the words "golden ratio" come up. In 1876 Gustav Fechner wrote that he had measured people preferring the golden rectangle to other rectangles; the replications of the following hundred years either failed to find this preference or found it very weak. In 1992, in the College Mathematics Journal, George Markowsky examined the golden ratio claims one by one, from the Parthenon to the Mona Lisa, and showed that most were measurement errors or selectively drawn rectangles. The proportions of editorial design are not mystical but practical: √2 preserves itself when folded, 2 to 3 has worked in books for five hundred years, 3 to 4 is the photo sensor. A ratio is chosen by use, not by story.

A page that looks right on screen prints differently on paper, and the difference has a name, dot gain: as ink is absorbed into the paper it spreads, halftone dots grow, and midtones darken. On coated paper this gain is around ten percent; on matte and uncoated wood-free paper it can exceed twenty percent; the same file yields two different catalogs on two papers. The paper's weight and opacity determine how visible the back of the sheet is; on paper under 90 grams, a full-page photograph shadows the table behind it. This is why the press proof is made on the chosen paper, not on screen.

The file itself is a standard too: PDF/X, from the ISO 15930 family, began in 2001 with PDF/X-1a and gained transparency in 2008 with PDF/X-4; this is the file the printer expects, not a "high-quality PDF." The bleed is 3 millimeters, and total ink coverage is limited to around 300 percent depending on the paper. In a catalog with hundreds of products, text and tables are merged automatically from a single data source; the same data flows to the website and the technical sheet, so that three documents do not one day state three different dimensions. Every print run carries the version and date in its colophon: for the buyer to know which year the document in hand dates from is the cheapest insurance against a wrong order.

The IKEA catalog began in 1951; in 2016 it was printed in 32 languages and roughly 200 million copies, and in 2021, seventy years on, it came to an end; the company announced that its readers had moved to digital. In the same years Jonathan Zhang of Colorado State University described a field experiment in Harvard Business Review: when an online retailer that had communicated only by email began sending a printed catalog as well to a portion of its customers, sales and product inquiries rose significantly. Where the catalog has ended is the consumer mass market; where it remains is the document left on the table after a sales meeting. What an exporting manufacturer leaves with a buyer after a trade fair is still paper, and Delgado's meta-analysis says that paper will be understood better than a screen.

This is why our studio's catalog protocol is built as follows: a grid in two ratios (A and Letter), Tschichold's margin proportions, a baseline, merging from a single data source, a dot gain proof on the chosen paper, PDF/X-4 delivery, and a version colophon. Lichtenberg's letter of 1786 is still on the table today.

Lichtenberg, G. C. (1786). Letter to Johann Beckmann, October 25, 1786; Porstmann, W. (1922). DIN 476; ISO 216 (1975).

Van de Graaf, J. A. (1946). Nieuwe berekening voor de vormgeving. Tschichold, J. (1975). Ausgewählte Aufsätze über Fragen der Gestalt des Buches und der Typographie (Eng. The Form of the Book, 1991). Rosarivo, R. (1947). Divina proportio typographica.

Tschichold, J. (1928). Die neue Typographie. Berlin.

Müller-Brockmann, J. (1981). Grid Systems in Graphic Design. Niggli.

Delgado, P., Vargas, C., Ackerman, R. & Salmerón, L. (2018). Don't throw away your printed books: A meta-analysis on the effects of reading media on reading comprehension. Educational Research Review, 25, 23–38.

Standing, L. (1973). Learning 10,000 pictures. Quarterly Journal of Experimental Psychology, 25(2), 207–222. Shepard, R. N. (1967). Recognition memory for words, sentences, and pictures. Journal of Verbal Learning and Verbal Behavior, 6(1), 156–163. Paivio, A. (1971). Imagery and Verbal Processes.

Arnold, E. C. (1969). Modern Newspaper Design. Nielsen, J. (2006). F-shaped pattern for reading web content.

Fechner, G. T. (1876). Vorschule der Ästhetik. Markowsky, G. (1992). Misconceptions about the golden ratio. College Mathematics Journal, 23(1), 2–19.

ISO 15930 (PDF/X-1a: 2001; PDF/X-4: 2008/2010).

IKEA (2020). Announcement of the discontinuation of the catalog; Zhang, J. Z. (2020). Why catalogs are making a comeback. Harvard Business Review, February 2020.