ISO 216–style boxes

Originally posted on reddit.com/r/themotte

As you probably are aware, international paper sizes (defined by, inter alia, ISO 216) are configured so that the next size in the series can be obtained by cutting the current size in half across its longest dimension.

Size	Area (m²)	Dimension (m)
Size	Area (m²)	1	2
A0	2⁰ = 1	2^−1/4 ≈ 0.8409, 37/44, 5/6	2^1/4 ≈ 1.189, 44/37, 6/5
A1	2⁻¹ = 1/2	2^−3/4 ≈ 0.5946	2^−1/4 ≈ 0.8409
A2	2⁻² = 1/4	2^−5/4 ≈ 0.4204	2^−3/4 ≈ 0.5946
A3	2⁻³ = 1/8	2^−7/4 ≈ 0.2973	2^−5/4 ≈ 0.4204
A4	2⁻⁴ = 1/16	2^−9/4 ≈ 0.2102	2^−7/4 ≈ 0.2973

Larger sizes are denoted with prefixes: 2A0, 4A0, etc. Intermediate sizes are denoted with different letters: B0 starts at 2^1/2 m², C0 at 2^1/4 m², and D0 at 2^3/4 m², sometimes going all the way to H0 at 2^7/8 m².

What if there were an analogous series of boxes?

Size	External volume (m³)	External dimension (m)
Size	External volume (m³)	1	2	3
A0	2⁰ = 1	2^−1/3 ≈ 0.7937, 50/63, 4/5	2^0/3 = 1	2^1/3 ≈ 1.260, 63/50, 5/4
A1	2⁻¹ = 1/2	2^−2/3 ≈ 0.6300	2^−1/3 ≈ 0.7937	2^0/3 = 1
A2	2⁻² = 1/4	2^−3/3 = 0.5	2^−2/3 ≈ 0.6300	2^−1/3 ≈ 0.7937
A3	2⁻³ = 1/8	2^−4/3 ≈ 0.3969	2^−3/3 = 0.5	2^−2/3 ≈ 0.6300
A4	2⁻⁴ = 1/16	2^−5/3 ≈ 0.3150	2^−4/3 ≈ 0.3969	2^−3/3 = 0.5

These boxes would have inset “corner castings”, so that they could be fastened together (like intermodal containers) without having their combined dimensions messed up (unlike intermodal containers). For example, a standard “high&h;cube” intermodal container (8 ft × 9.5 ft × 40 ft, or approximately 2.438 m × 2.896 m × 12.19 m) could be replaced with a set of three 16B0 boxes (totaling approximately 2.245 m × 2.828 m × 10.69 m or 7.365 ft × 9.278 ft × 35.07 ft). With inset corner castings, the interior volume available in each of those 16B0 boxes could be around 1.885 m × 2.608 m × 3.204 m.

Alternatively, these dimensions could be used for steles, on whose faces text is engraved. The six faces of a B3 stele (2^−2.5 m³, 2^−3.5/3 m × 2^−2.5/3 m × 2^−1.5/3 m) have approximately the same total surface area as the two faces of an A0 sheet (2 × (2^−3.5/3 m × 2^−2.5/3 m + 2^−3.5/3 m × 2^−1.5/3 m + 2^−2.5/3 m × 2^−1.5/3 m) ≈ 1.924 m²). Dropping three sizes reduces surface area by a factor of four, so a B9 stele (2^−8.5 m³, 2^−9.5/3 m × 2^−8.5/3 m × 2^−7.5/3 m, approximately 4.4 in × 5.5 in × 7.0 in) has approximately the same area as an A4 sheet.

If you&a;re willing to settle for a very rough approximation, cardboard boxes (rated for net weight of 40 lb, not the gross weight of 67 200 lb that intermodal containers must meet) with the following dimensions are available:

Volume (in³)	Equivalent size	Internal dimension (in)
Volume (in³)	Equivalent size	1	2	3
60 × 2⁰ = 60	A10	3	4	5
60 × 2¹ = 120	A9	4	5	6
60 × 2² = 240	A8	5	6	8
60 × 2³ = 480	A7	6	8	10
60 × 2⁴ = 960	A6	8	10	12
60 × 2⁵ = 1920	A5	10	12	16
60 × 2⁶ = 3840	A4	12	16	20
60 × 2⁷ = 7680	A3	16	20	24