As Light as Paper

From the light barrier to paper’s edge

13 min readJun 20, 2026

You start looking at coincidences between measurement systems and you end up wondering how deep the rabbit hole goes.

Pretty deep, it turns out. Let’s keep digging.

Following the pattern of the previous two coincidental alignments, this essay introduces an approximation before chasing metrological rabbits, mathematical tangents and astronomical digressions, all with a seasoning of historical trivia. So, if you want a short read, just skim the next section and skip to the end. If you want all the side quests, grab your drink of choice and get comfortable.

The game is a-phoot

Contrary to some questionable copywriting and journalism, the light-year is a unit of distance, not time. (On the screenwriting front, the same is true for parsec.) The speed of light in a vacuum is a universal constant. And it’s fast. Very fast. A light-year (ly) is simply the distance light can travel in a year. And it’s far. Very far.

Let’s start with a smaller unit: the light-second. That’s defined to be precisely 299 792 458 metres, which gets you just about three quarters of the way to the Moon.

Where the second is unambiguously defined, the year is open to more negotiation. Using a Julian year of 365.25 days and a standard day of 86 400 seconds, a light-year is just short of 10 trillion kilometres (~9.5×10¹⁵ m), making 10 petametres a convenient approximation to within ~6%.

>>> 365.25 * 86_400 * 299_792_458
9460730472580800.0

Our galaxy is about 100 000 light-years across. Using Indian numbering, this conveniently rolls up as 1 lakh ly (the temptation to coin a new astronomical distance, the lakhly, is strong…). Unrolling into SI units gives the galaxy’s diameter as 1 zettametre.

We are just over 25 000 ly from the galactic centre. In 2020, Reinhard Genzel and Andrea Ghez were awarded the Nobel Prize in Physics (along with Roger Penrose) for their discovery of Sagittarius A*, the black hole at the heart of the Milky Way. The discovery was based on the movement of stars around an unseen supermassive object. The light from the stars that led to the discovery began its journey 26 millennia before Alfred Nobel invented dynamite.

These scales of time are way beyond human intuitive understanding, so let’s try something more everyday. Launched in 1977, Voyager 1 is the most remote human-made object at a distance of almost 1 light-day from us (at the time of writing). Our Sun is just over 8 light-minutes away and the Moon just over a light-second. Although we can grasp the time scales I’ve just mentioned, their corresponding light distances are still beyond our mental reach. To get to something truly human in scale we have to drop well below the light-second.

The idea of the light-nanosecond as a unit of distance was popularised by Grace Hopper. She would talk of a nanosecond of wire:

Once she presented a piece of wire about a foot long, and explained that it represented a nanosecond, since it was the maximum distance electricity could travel in wire in one-billionth of a second. She often contrasted this nanosecond with a microsecond — a coil of wire nearly a thousand feet long — as she encouraged programmers not to waste even a microsecond.

That a light-nanosecond is almost but not quite 1 foot in the FPS system is quite a nice coincidence but, for most of the world, there’s a better one: 30 cm, which is the length marked on a typical metric ruler.

A 30-cm rule differs by less than 0.01% from a light-nanosecond, making it a far better approximation than a foot, which is off by ~1.6%. The next best thing to a ruler is a sheet of paper. Specifically, a sheet of A4: the dimensions of A4 are 210 mm × 297 mm, making the long side a better than 1% approximation of a light-nanosecond.

If you want to get down to 0% and eliminate any approximation error, you can always take physicist N David Mermin’s suggestion of the phoot:

You may define 0.299792458 meters to be 1 phoot, and think “phoot” (conveniently evocative of the Greek φωτος, “light”) whenever you read “foot”.

Somewhat conveniently, this makes the speed of light precisely 10 million phoot (or 10 megaphoot) per second. It’s unclear whether the plural of phoot is intended to be phoot, pheet or phoots, but going by ear, phoot is perhaps the best of the bunch.

Going the extra mille

The mile — specifically the Roman mile — used to be defined in terms of convenient decimal values: mile comes from mille passus, 1000 paces; it equates to 2000 footfalls; it is equal to 5000 Roman feet. In other words, a system based on thousands and quite reasonable multiples (1, 2 and 5 often occur in currency denominations).

Elizabethan England did away with this convenience and regularity, leaving 5280 feet (or 1760 yards) as an awkward legacy for future generations. The reformed system did, however, conveniently (albeit not decimally) align with furlongs, chains, rods and a few other units… units that are no longer in use, stranding the original rationale for the number on the scrapheap of history.

For countries that use the mile inherited from (or imposed by) the English system, the definition of a statute mile being 5280 feet has been an ungainly constant since then.

Kind of.

I say kind of because although the number of feet has remained constant since the 16th century, the definition of the foot has evolved. The foot of the FPS system has not wandered as far and wide in value as much as the mile, its variations being greater across geography — e.g., the Hessian foot at 0.25 m, the Roman foot at ~0.296 m (within ~1.3% of a light nanosecond), the Rhineland foot at ~0.314 m and the Tyrolian foot at ~0.334 m — than across time. For the statute foot, which can be said to have started life in England in the 1300s, the foot has been within 0.1% of its modern value; it’s not so much the actual distance as the underlying definition that has changed.

Since the 19th century, legacy measures such as the foot, pound, etc., have been defined in terms of metric/SI units. In 1959 the statute foot was adjusted to be precisely 0.3048 m, making the statute mile precisely 1609.344 m. Put simply, Imperial units are metric-based.

A star

We wandered off on foot because of light-nanoseconds. But, as mentioned before, A4 is a better approximation of a light-nanosecond than a foot (excepting the metric foot, of course).

At the end of the 18th century, French taxation law of printed publications effectively standardised a number of page formats that eventually became some of the A and B paper sizes. In Germany in the early 20th century this became more international, more complete and more standard. From the 1920s to the 1970s this DIN standard spread to many countries throughout the world. It became an ISO standard in 1975, covering A, B and C series paper sizes. D, E, F and G series exist, but these are Swedish exceptionalism rather than ISO standard.

Here is part the A series, with short and long sides in millimetres:

a_sizes = {
    'A0': (841, 1189),
    'A1': (594,  841),
    'A2': (420,  594),
    'A3': (297,  420),
    'A4': (210,  297),
    'A5': (148,  210),
    'A6': (105,  148),
}

The A series continues to smaller sizes down to A10, and larger up to 4A0 (which is −2, or 2 sizes larger than A0), but this sample from A0 (twice the size of a typical flip chart) to A6 (the size of an index card) is enough to highlight key properties, such as the aspect ratios of each size.

>>> for name, (short, long) in a_sizes.items():
...     print(f'{name}: {short} mm × {long:4} mm, ratio 1:{long/short}')

A0: 841 mm × 1189 mm, ratio 1:1.4137931034482758
A1: 594 mm ×  841 mm, ratio 1:1.4158249158249159
A2: 420 mm ×  594 mm, ratio 1:1.4142857142857144
A3: 297 mm ×  420 mm, ratio 1:1.4141414141414141
A4: 210 mm ×  297 mm, ratio 1:1.4142857142857144
A5: 148 mm ×  210 mm, ratio 1:1.4189189189189189
A6: 105 mm ×  148 mm, ratio 1:1.4095238095238096

Such ratios are often interesting. In this case, within tolerance, the ratio is the same for each size. If the numbers in the right-hand column look familiar, there’s a reason.

>>> from math import sqrt
>>> sqrt(2)
1.4142135623730951

The ratio 1:√2 is the Lichtenberg ratio and is used by the A, B and C series of ISO paper sizes. If you look through the sizes you will see that the short side length of each size is the same as the long side of the next size down, and the long side halved is the same as the short side of the next size down.

>>> for name, (short, long) in a_sizes.items():
...     print(f'{name} folded: {(long / 2):3.0f} mm × {short} mm')

A0 folded: 594 mm × 841 mm
A1 folded: 420 mm × 594 mm
A2 folded: 297 mm × 420 mm
A3 folded: 210 mm × 297 mm
A4 folded: 148 mm × 210 mm
A5 folded: 105 mm × 148 mm
A6 folded:  74 mm × 105 mm

This is the key property — and delightful regularity — of this aspect ratio: fold a sheet of A4 in half and its size is A5, fold A5 and you get A6, etc. Likewise, joining two sheets of A4 along their long side and gives you A3, two sheets of A5 make A4, and so on. This also explains the naming of the sizes larger than A0: 2A0 is equivalent to two sheets of A0 joined on their long side, while 4A0 is four times the area of A0.

B corp

The short side of B0 is defined to be 1000 mm. From this you can work out not only its long side, but also the sizes of the rest of the B series. For a given B size n (e.g., for B1, n = 1), the length of the short side is given by 1000 mm / √2ⁿ and the long side by 1000 mm / √2ᵐ, where m = n − 1.

b_sizes = {
    f'B{n}': (1000 / sqrt(2) ** n, 1000 / sqrt(2) ** (n - 1))
    for n in range(7)
}

The generated sizes are to within 1 mm tolerance (e.g., the 177 mm side in B5 and B6 is normally quoted as 176 mm), which is better than required for ISO paper sizes.

>>> for name, (short, long) in b_sizes.items():
...     print(f'{name}: {short:4.0f} mm × {long:4.0f} mm')
          
B0: 1000 mm × 1414 mm
B1:  707 mm × 1000 mm
B2:  500 mm ×  707 mm
B3:  354 mm ×  500 mm
B4:  250 mm ×  354 mm
B5:  177 mm ×  250 mm
B6:  125 mm ×  177 mm

As with the A series, the dials on the B and C series go to 10, but they only go physically larger than B0 and C0 by one size, 2B0 and 2C0, respectively.

Given that we’re exploring the intention behind measures, we can see the B series has an obvious connection to metric units with the short side of B0 defined as 1 m. But what about the A series? What’s so significant about A0’s 841 mm × 1189 mm? Neither 841 nor 1189 seem particularly anchored in metric convenience — or, indeed, any convenience. And that’s because they’re not; 841 × 1189, however, is. Whereas B0 is based on length, A0 is based on area: with rounding, the area of A0 is 1 m².

>>> for name, (short, long) in a_sizes.items():
...     print(f'{name}: {short * long / 1e6:1.3f} m²')

A0: 1.000 m²
A1: 0.500 m²
A2: 0.249 m²
A3: 0.125 m²
A4: 0.062 m²
A5: 0.031 m²
A6: 0.016 m²

If we say that A = ½, then the area of paper size An is Aⁿ m².

>>> A = 0.5
>>> for n in range(7):
...     print(f'A{n}: {A**n:1.3f} m²')

A0: 1.000 m²
A1: 0.500 m²
A2: 0.250 m²
A3: 0.125 m²
A4: 0.062 m²
A5: 0.031 m²
A6: 0.016 m²

C suite

The C series size is commonly used for envelopes, corresponding to ~9% larger than the corresponding A sizes they can contain.

c_sizes = {
    name.replace('A', 'C'): (short * 1.09, long * 1.09)
    for name, (short, long) in a_sizes.items()
}

The proportion 9% gives the C sizes no more than a millimetre off their standard values.

>>> for name, (short, long) in c_sizes.items():
...     print(f'{name}: {short:4.0f} mm × {long:4.0f} mm')

C0:  917 mm × 1296 mm
C1:  647 mm ×  917 mm
C2:  458 mm ×  647 mm
C3:  324 mm ×  458 mm
C4:  229 mm ×  324 mm
C5:  161 mm ×  229 mm
C6:  114 mm ×  161 mm

That 9% might seem a little arbitrary, until you realise it comes from the geometric mean of A and B sizes… OK, I admit realise is doing a lot of work in that sentence. It’s unlikely you’re every going to realise this, given that the geometric mean is one of the lesser known members of the average family. Although it may be unfamiliar, it has a number of well-defined uses, not least in finance.

The geometric mean determines a central value of a sample based on relative proportions between values rather than their absolute differences. It is calculated as the nth root of the product of n numbers, i.e., ⁿ√(x₁ × x₂ × … × xₙ), in contrast to the more familiar arithmetic mean, which is the sum of n numbers divided by n, i.e., (x₁ + x₂ + … + xₙ) / n.

As we are concerned only with the geometric mean of two values — an A side and a B side — calculating the C sizes involves the regular square root, i.e., c = √(a × b).

c_sizes = {
    f'C{n}': (sqrt(a_short * b_short), sqrt(a_long * b_long))
    for n, ((a_short, a_long), (b_short, b_long))
    in enumerate(zip(a_sizes.values(), b_sizes.values()))
}

This gives slightly more accurate C sizes.

>>> for name, (short, long) in c_sizes.items():
...     print(f'{name}: {short:4.0f} mm × {long:4.0f} mm')

C0:  917 mm × 1297 mm
C1:  648 mm ×  917 mm
C2:  458 mm ×  648 mm
C3:  324 mm ×  458 mm
C4:  229 mm ×  324 mm
C5:  162 mm ×  229 mm
C6:  115 mm ×  162 mm

You can see that the proportion by which C is larger than A is around 9%. For example, the short side of C0 is 917 mm, which is approximately 1.09 × 841 mm, the short side of A0.

The 9% might still seem a little magical, but it emerges from a pile-up of square roots. The area of A0 is 1 m² and the area of B0 is 1.414 m², which relates the areas of corresponding A and B sizes by a factor of √2. This in turn means their sides are related by the square root of this, the square root of a square root, or the fourth root, i.e., ⁴√2. Taking the geometric mean of corresponding sizes throws in another square root, i.e., ⁸√2. We can also write this as raising 2 to the power of ⅛th. Either way, the value is just over 1.09.

>>> 2 ** (1/8)
1.0905077326652577

An irrational proof

In a tangent to a digression before a conclusion, let’s take a look at √2, given that it keeps turning up. A previous essay touched on irrationality in connection with π and e, but √2 also got a mention:

In addition to being irrational numbers — they cannot be expressed as the ratio of two integers — π and e are transcendental numbers. Where an algebraic number can be expressed as the solution to a polynomial equation, a transcendental number cannot. All transcendental numbers are irrational, but not all irrational numbers are transcendental. For example, √2 is irrational but not transcendental because it can be found as the solution to x² − 2 = 0.

For a number to be rational, it means that it can be expressed as the ratio of two non-zero integers p and q such that the number equates to the fraction p/q. Furthermore, we normally suppose that p/q cannot be simplified further, i.e., p and q share no common factors.

The proof that √2 is irrational dates back to Ancient Greece. It is normally attributed to the Pythagoreans — who were upset by this discovery — and sometimes specifically to Hippasus — who receives the blame for it.

We can prove √2’s irrationality for ourselves by assuming that it is rational, then seeing how that works out for us. If we are led into a contradiction, we know it is not rational.

We start by assuming p/q = √2, where p and q are non-zero integers that share no common factors.
Given that p/q = √2, then p² = 2q².
Because the right-hand side, 2q², must be even (the result of multiplication by 2 is, by definition, wholly divisible by 2), p must also be even because p² is even (if p were odd, p² would also be odd). Therefore, p is divisible by 2, i.e., p = 2k for some value k.
If we substitute 2k back into p² = 2q² then we get 4k² = 2q², which leads to 2k² = q².
This means that q² is also even and, therefore, q must itself be even.
This would mean p and q are both even — and therefore both divisible by 2 — which contradicts the initial statement that p and q do not share any common factors.
Therefore, √2 is irrational.

Contrary to the belief of the Pythagoreans that all numbers must be rational, rationality is the exception rather than the rule. There are infinitely more irrational numbers than rational, but we don’t have a systematic (i.e., rational) way of writing them. The rational numbers are countably infinite; real numbers are uncountably infinite; rational numbers form an infinitesimal subset of the real numbers. This is the problem of finding the hay in a haystack.

Twice as light

Moving from irrationality back to coincidence, it is both a convenience and a wonder that the A4 approximation to a light-nanosecond is so close.

The most obvious coincidence is that, to 2 significant figures, the distance light travels in a second (3.0×10⁸ m) is 9 orders of magnitude (10⁹) larger than long side of an ISO A4 piece of paper (3.0×10⁻¹ m). Thus, A4 is 1 light-nanosecond long and 1 light-second is 1 giga-A4. A billion sheets of A4 laid end to end gets you about three quarters of the way to the Moon. To get all the way you’ll need another 6 lakh reams of paper.

But there’s another coincidence in play, one that was centuries in the making. The metre started life in the 18th century as a fraction of the world, then as a standardised platinum–iridium bar and then, in 1960, as a number of wavelengths of a particular emission. From 1983, the metre has been defined in terms of lightspeed.

ISO paper sizes are defined in terms of the metre, e.g., the area of A0 is 1 m² and the short side of B0 is 1 m. A4’s relationship to the speed of light is, therefore, doubly coincident.