Flag Saga Part 1: Symmetry

Note: For some reason browsers in Windows stubbornly refuse to display flag emoji which I have used throughout this post. At some point I'll write some code to address this but until then you may see two letter flag designations where the emoji should be if you're reading this on a Windows PC.

In the wee hours of the morning I got curious about the reflectional and rotational symmetry of national flags so my wife stayed up with me in bed to look through nearly 200 national flags on my iPhone and categorize each one based on their number of axes of reflectional symmetry and their degree of rotational symmetry.

We placed each flag in one of the following four categories:

1. Flags with no symmetry (e.g. the United States 🇺🇸)

2. Flags with exactly one axis of reflectional symmetry (e.g. France 🇫🇷)

3. Flags with exactly 2 axes of reflectional symmetry (e.g. Israel 🇮🇱) Note: these flags also necessarily have 2-fold rotational symmetry.

4. Flags with 2-fold rotational symmetry but no reflectional symmetry (e.g. the United Kingdom 🇬🇧)

The Swiss flag🇨🇭is the only one that did not fit into any of those categories. It has 4 axes of reflectional symmetry and 4-fold rotational symmetry. This is made possible by the fact that the Swiss flag has a square aspect ratio. It is one of only three square national flags in the world.

The second square flag is Vatican City’s🇻🇦 which has no symmetry.

Finally, there’s the Wiphala flag which represents the native people of the Andes and has technically been the second official flag of Bolivia since 2009. The Wiphala flag is the only flag other than Switzerland that has a diagonal axis of reflectional symmetry but since it has exactly 1 axis of reflectional symmetry, it fits into the second category above.

Also of note: Nepal🇳🇵has the only flag that is not shaped as a quadrilateral and the only flag whose sides are not all right angles.

There are some interesting near-symmetries that would be easy to miscategorize if you aren’t looking very carefully. Kyrgyzstan’s flag🇰🇬 is very nearly symmetrical but for the asymmetric rays of sunlight surrounding the symbol in the center. Bangladesh 🇧🇩 and Palau 🇵🇼 almost have 2 axes of symmetry but the circles on their flags are slightly off-center.

The orientation of stars can also often make or break a flag’s symmetry. Some flags have 5-pointed stars that are all oriented the same way with one point facing straight up (e.g. Honduras 🇭🇳). This can be compatible with vertical axes of symmetry but it will break horizontal axes of symmetry every time. Other flags orient stars in such a way that they preserve as much symmetry as possible, with the points of stars facing in opposite directions on different sides of the flag (e.g. Micronesia 🇫🇲).

And then there’s East Timor 🇹🇱... Its flag’s star seems to be placed at an angle chosen with the explicit goal of frustrating people with OCD. I cannot discern any rhyme or reason to its star’s orientation. (more on this in The Flag Saga Part 2)

Bonus fact: if it weren’t for the rectangular edges, Japan’s flag 🇯🇵 would be the only one with true circular symmetry.

tl;dr: This thanksgiving season I’m thankful to have a wife who will enthusiastically stay up late at night to help me satiate my esoteric midnight curiosities.

P.S. What’s your favorite national flag? Leave it in the comments.

For more vexillological fun, click here for part 2 of The Flag Saga.