Sunday, March 27, 2011


Paper and Computer Screen Sizes

Once upon a time, television screens were all about the same shape, and there was a good reason for that: they were all intended to display standard shaped video images in a 4:3 ratio of width:height. That was the proportion standard in the motion picture industry at the time television began, so I suppose that made it an obvious standard to adopt for television as well. However, motion picture aspects evolved to a wider, flatter aspect ratio during the 1950s in particular, to 1.85:1 for the U.S. and 1.66:1 for some European countries. The now-popular for television and digital video 16:9 ratio is approximately 1.78:1, not quite as broad as the American motion pictures but nearly so.

The 4:3 ratio was also a standard for computer screens, at least since the IBM PC with VGA shipped in 1987; more and more computer screens are now 16:9, as are television screens (for HD).

The international standards for paper sizes, ISO 216, ISO 217 and ISO 269, prescribe a quite different aspect ratio, one with a unique mathematical property. Cutting this shape in half along a line connecting the midpoints midpoints of the long sides produces pieces having the same aspect ratio, square root of 2 to one (approximately 1.414:1, whereas American 8.5*11 has an aspect ratio of about 1.294:1).

How big does a computer screen have to be to display an A4 page at its true size? That will depend on the aspect ratio of the display, and the rotation of the page. A4 paper is 210 mm x 297 mm, which is very nearly 36.4cm (14.3") diagonal. If the page is to be displayed in portrait position (as opposed to landscape position), a 4:3 display will have to be 29.7cm high and "high" is typically the "3" part of the formula, implying a 39.6cm width and overall, a 49.5 cm diagonal! The 16:9 ratio will need to be even wider : 29.7*16/9=52.8 cm width and a 60.6 cm diagonal!

Clearly, this reckoning has been unreasonable. Let's reconsider the question in two ways: how big must the display be for the page to fit turned on its side (landscape), and how much of the page height shows when it fits in width.
  • Landscape fits:
    • If the 4:3 display is at least 29.7 cm wide, the page will fit, since 21 cm is less than 3/4 of 29.7; our display will have a diagonal of [29.72 + (29.7*3/4)2]1/2= 37.13 cm (=14.6").
    • The 16:9 display will have to be bigger, because 29.7*9/16= 16.7cm, less than 21cm; our display will be over 37cm wide and have a diagonal of 42.8 cm (=16.86"). In other words, even a 15.6" display isn't big enough in 16:9.
  • Height cropping: this measure is a bit different from those considered so far, inasmuch as we are "fixing" display size rather than having a paper dimension determine it. We'll fix the diagonal at that of a 4:3 display 21 cm wide, wide enough to fit the page: 26.25cm. (or about 10.33").
    • The 4:3 display will be 15.75 cm high, revealing just over half of the page; if we assume margins of 1" top and bottom, we'd see 15.75/24.6, nearly two thirds of the page body.
    • The 16:9 display will be more complicated to compute: we need to get the height (or width) from the diagonal and the aspect ratio. Using "x" to represent height/9, (h*16)2+(h*9)2=(26.25)2 yields (after some math) a height of 12.87 cm, so the display will be 22.9 cm wide and 12.87 cm high. 12.87 cm is a mere 43% of the page height, and just over half the page body height.

One last question, pertaining to netbook/notebook format computers. If the computer is to be no bigger than A4 for easy packing with papers and reports in a briefcase or other tote, how big a 16:9 screen can it have? Let's take a half-inch frame as an assumption, and have a 29.7-2.54=27.16cm width, consequently a 15.28 cm height, and a 31.16 cm diagonal (12.26").

For perspective: According to the Wikipedia article on the history of 35mm film, the Society of Motion Picture Engineers set a standard aperture ratio of 0.800 in by 0.600 in, in 1929, and Edison had also used an aspect ratio of 1.33:1 in the earliest days of cinema.

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