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Ratios of flags

Last modified: 2003-07-18 by phil nelson
Keywords: ratio of flags | flag ratios | naval ensign | off-centered | scandinavian cross | golden section |
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Summary of Ratios of Flags

My "holy book" (Whitney Smith, Flags Through the Ages and Across the World) quotes for the United States' flags a 10:19 ratio, really closer to the 1:2 traditional ratio of the British flags, from which the "Star Spangled Banner" comes.
In fact there are three main "threads" in the world of flags:

British flags have a 1:2 ratio (United Kingdom, Australia, Bahamas, Canada, Ireland and with the little correction of 10:19 United States and of course Liberia);

French flags have a 2:3 ratio (France, Italy, Cameroon, Ivory Coast, Algeria, Spain and the most of Latin-American flags);

German flags have a 3:5 ratio;

moreover some nations have unusual ratios, as Denmark (28:37) or Belgium (13:15).

Alessio Bragadini

The British ensigns (including the Union Jack) ratio varied with the standard breadth of the textile industry, but always retaining a length of 18:

Year Ratio of Length to Breadth
1687-17xx 11:18
17xx-1837 10:18 (5:9)
1837-present 9:18 (1:2)

On the other hand, I'm not sure how strict this regulations would be followed in civilian rebellions taking place in faraway Australia, nor how fast they were inforced throughout the empire...

I don't know at what point after 1837 the proportions were actually regulated. Possibly not until the reorganisation of Squadron colours in 1864.
David Prothero, 03 June 1999

The Golden Section

Many flags, picture frames, book covers, etc., are proportioned in accordance with what artists and mathematicians call "the golden section." This relationship exists when the length and width of a rectangle are divided into extreme and mean ratio, or when the parts follow (or approximate) the formula:

L2 = W (L + W)

Another way to look at it is if the length and width roughly equal 62 and 38 percent of their sum respectively.
Lou Stewart, 1998 January 30

If you solve the equation Lou Stewart gave analytically,

L * L = W * ( L + W )

you'll find a solution:

L = alpha * W,


alpha = (1 + sqrt(5))/2 = 1.618... [sqrt being the square root]

Mathematically, there's another solution to this equation, namely

alpha = -0.618...

but I don't think we're looking for a flag with a negative length.
So, the ratio is 1.618...:1.

This ratio was already known to the Greeks, and the Acropolis reflects this ratio in many ways (correct me if I'm wrong).
Filip Van Laenan who studied *applied* mathematics, and this surely is applied ;-) 1998 January 30

Let's try it this way:
the first format to think of is 1:1 (A:B), then we put the B as a new A, and A+B as a new B. So next we'll get 1:2, and next 2:3 and 3:5 and 5:8 and 8:13 and 13:21 and 21:34 and 34:55 and 55:89 and so on... We'll get closer and closer to 1.618 or something like that, the golden section. It has been used a lot in art and Kepler spoke of 'divina proportio'. It is mostly a proportion that 'looks nice'. Many mathematicians and physicists have written about it.
Ole Andersen, 1998 January 30

1.1618... is phi the golden ratio and is, like pi, irrational. However, if we look at the Fibonacci series we'll see that the difference between each number gets closer and closer to the ratio first over then under. The series is 1 1 2 3 5 8 13 21 ... where each number is the sum of the previous two. You will also note that the numbers are not far from many flag ratios 5:8 8:13 13:21 etc.
Rich Hansen, 1998 January 30

A more interesting approach uses the Golden Ratio's connection to the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13,..., in which each new member of the sequence is the sum of the preceding two. Then, you can generate successive (and closer) approximations to the Golden Ratio as 1/1, 2/1, 3/2, 5/3, 8/5, 13/8,... where the numerator is just the member of the sequence that is one ahead of the denominator. (The two approximations above are derived from this method, just a bit further along in the sequence).

An awful lot has been written about the Golden Ratio (also called 'The Divine Proportion'). It crops up in nature: shapes of shells, arrangements of sunflower seeds; in architecture (the most aesthetic shapes are ones having proportions equal to the Golden Ratio), and flags! It was known to the Ancient Greeks (look at a picture of the Parthenon), and probably earlier. Interesting it should show up in flag design too!

anonymous, 1999 February 03

At least one flag used the "golden proportions" as an integral part of its design: see Saarland 1947-1956.
Dave Martucci, 1998 January 30

No. There wasn't such flag manufactured in Saarland. I don't know where the document which mentioned this was to be found, but all the projects of laws of 1947 mentionned a flag with the proportions 1 x 1,5, not 1 x 1,61803398875. If such flag was proposed in Saarland, this was really absurd and ridiculous: how can you draw precisely such a flag, and above all how can you manufacture such a flag: it is impossible and not practical! If the flag existed it was only a proposition, not a real flag.
Pascal Vagnat, 1999 February 05

New Brunswick's isn't a national flag, but its 5:8 ratio is the closest approximation you can get to the golden ratio with one-digit numbers. The designer probably considered this when choosing the ratio. Any other flags with this ratio were probably designed with the golden ratio in mind.
Dean Tiegs, 1999 February 05

AFAIK artists Bruno Tuukkanen and Eero Snellman had the Golden Ratio in their mind when they designed the Finnish flag. Ratio 11:18 = 0.6111... which differs very little from 0.6180...
Ossi Raivio, 1999 February 06

Vertical Proportions of Flags

The proportions of vertical stripes on French naval flag are 30:33:37, to enable good visual effect of flag when flying.
Portugal, too, obviously, has an off-centered pattern and I suppose the Scandinavian cross flags have the same reason for the vertical bar shifted right.

Zeljko Heimer, 1995 September 23

Bangladesh, North Korea, Nauru, Turkey and the Japanese Ensign all shift their designs to the hoist. Whitney Smith's book mentions that Bangladesh does this so that the flag will look proper while flying. There is no reason given for the others and in the case of Nauru especially I suspect that the star is toward the hoist for some other reason.

Nathan Augustine, 1995 September 27

Why keep the right proportions?

Since we are talking about flag proportions, I was wondering if the proportions are ever symbolic in and of themselves, or always more or less arbitrary. (Let's leave oddballs like Nepal and Qatar out for the moment.) This question arises from the question of why it's so important to keep the proportions right. For instance, Ron pointed out that many of the errors are caused by standardizations of the flag manufacturing process. Earlier, someone said that all the flags of the former Soviet Union kept to the proportions of the old Hammer and Sickle. Similarly, looking at my flag chart, all the flags of the former Yugoslavia seem to be more or less the same proportions. I'm willing to bet that this is a result less of nostalgia for the old days and more of the fact that it was easier to leave the settings on the flag-making machines as is....

Thus, I ask again, why is it important to keep proportions straight? Colors and symbols have meanings which it would wrong to alter, but if proportions are chosen arbitrarily...

Josh Fruhlinger, 1996 January 29

One pair of flags that differ only in their proportions are those of Indonesia (2:3) and Monaco (4:5). Of course, I don't know whether the proportions have significance in themselves, but they have significance in relation to each other in that they are the only way to distinguish the two flags.

I'm having a hard time thinking of a real-world situation in which these two countries' flags could be confused, though. (Shipwrecked sailors wash up on an unfamiliar shore; "What country are we in?" "Must be Indonesia -- look at that flag." "Yes, and that big building up there must be the famous Djakarta Casino!")

Bruce Tindall, 1996 January 29