linguistic:voice_change, structural:expansion, information:filtering, information:classification
Aspect ratios

The geometric mean has been used in choosing a compromise aspect ratio in film and video: given two aspect ratios, the geometric mean of them provides a compromise between them, distorting or cropping both in some sense equally. Concretely, two equal area rectangles (with the same center and parallel sides) of different aspect ratios intersect in a rectangle whose aspect ratio is the geometric mean, and their hull (smallest rectangle which contains both of them) likewise has the aspect ratio of their geometric mean.

In the choice of 16:9 aspect ratio by the SMPTE, balancing 2.35 and 4:3, the geometric mean is 2.35 × 4 3 ≈ 1.7701 {\textstyle {\sqrt {2.35\times {\frac {4}{3}}}}\approx 1.7701}, and thus 16: 9 = 1.77 7 ¯ {\textstyle 16:9=1.77{\overline {7}}}... was chosen. This was discovered empirically by Kerns Powers, who cut out rectangles with equal areas and shaped them to match each of the popular aspect ratios. When overlapped with their center points aligned, he found that all of those aspect ratio rectangles fit within an outer rectangle with an aspect ratio of 1.77:1 and all of them also covered a smaller common inner rectangle with the same aspect ratio 1.77:1. The value found by Powers is exactly the geometric mean of the extreme aspect ratios, 4:3 (1.33:1) and CinemaScope (2.35:1), which is coincidentally close to 16: 9 {\textstyle 16:9} (1.77 7 ¯: 1 {\textstyle 1.77{\overline {7}}:1}). The intermediate ratios have no effect on the result, only the two extreme ratios.

Applying the same geometric mean technique to 16:9 and 4:3 approximately yields the 14:9 (1.55 5 ¯ {\textstyle 1.55{\overline {5}}}...) aspect ratio, which is likewise used as a compromise between these ratios. In this case 14:9 is exactly the arithmetic mean of 16: 9 {\textstyle 16:9} and 4: 3 = 12: 9 {\textstyle 4:3=12:9}, since 14 is the average of 16 and 12, while the precise geometric mean is 16 9 × 4 3 ≈ 1.5396 ≈ 13.8: 9, {\textstyle {\sqrt {{\frac {16}{9}}\times {\frac {4}{3}}}}\approx 1.5396\approx 13.8:9,} but the two different means, arithmetic and geometric, are approximately equal because both numbers are sufficiently close to each other (a difference of less than 2%).