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Color proportions in country flags

▲ (right) 256 country flags as concentric circles showing the proportions of each color in the flag. (left) Unique flags sorted by similarity.

Country flags are pretty colorful and some are even pretty.

Instead of drawing the flag in a traditional way (yawn...), I wanted to draw it purely based on the color proportions in the flag (yay!). There are lots of ways to do this, such as stacked bars, but I decided to go with concentric circles. A few examples are shown below.

▲ Country flags drawn as concentric rings. The width of each ring is proportional to square root of the area of that color in the flag. Only colors that occupy 1% or more of the flag are shown. (zoom)

Once flags are drawn this way, they can be grouped by similarity in the color proportions.

Check out the posters or read about the method below.

Or, download my country flag color catalog to run your own analysis.

To determine the proportions of colors in each flag, I started with the collection of all country flags in SVG from Wikipedia. The flags are conveniently named using the countries' ISO 3166-2 code. At the time of this project (21 Mar 2017), this repository contained 312 flags, of which I used 256.

I originally wanted to use the flag-icon-css collection, but ran into problems with it. It had flags in only either 1 × 1 or 4 × 3 aspect ratio, which distorted and clipped many flags. Many flags were also inaccurately drawn and had inconsistent use of colors. For example, in Turkey's flag the red inside the white crescent was slightly different than elsewhere in the flag.

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▲ Flags of 256 countries and territories drawn as concentric circles representing the proportions of colors in the flag. The flags are labeled with the country's ISO 3166-2 code. (BUY ARTWORK)

I converted the SVG files to high resolution PNG (2,560 pixels in width) and sampled the colors in each flag, keeping only those colors that occupied at least 0.01% of the flag. I apply this cutoff to avoid blends between colors due to anti-aliasing applied in the conversion. When drawing the flags as circles, I only use colors that occupy at least 1% of the flag—this impacts flags that have detailed emblems, such as Belize. I apply some rounding off of the proportions and colors with the same proportion are ordered so that lighter colors (by Lab luminance) are in the center of the circle.

There are various ways to represent the proportions of the flag colors as concentric rings—in other words, to use symbols of different size to encode area.

The accurate way is to have the area of the ring be proportional to the area of the color on the map. The inaccurate way is to encode the area by the the width of the ring. These two cases are the `k=0.5` and `k=1` columns in the figure below, where `k` is the power in `r = a^k` by which the radius of the ring, `r`, is scaled relative to the area, `a`. A perceptual mapping using `k=0.57` has been suggested by some.

▲ The concentric rings can be drawn to be either accurate in area (left, `k=0.5`) or to have their width encode the area (right, `k=1`). The hybrid approach is a mix of these two extremes. (zoom)

My goal here is not to encode the proportions so that they can be read off quantitatively. To find a value of `k`, I drew some flags and looked at their concentric ring representation. For example, with `k=0.57` the Nigerian flag's white center is too large for my eye while for `k=1` it is definitely too small. I liked the proportions for `k=1/\sqrt{2}` but wasn't happy with the fact that flags like France's, which have colors in equal areas, didn't have equal width rings.

In the end I decided on a hybrid approach in which the out radius of color `i` whose area is `a_i` is `r_i = a_i^k + \sum_{j=0}^{i-1} a_j^k` where the colors are sorted so that `a_{i-1} \le a_i`. If I use `k=0.25`, I manage to have flags like France have equal width rings but flags like Nigeria in which the proportions are not equal are closer to the encoding with `k=1/\sqrt{2}`. In this hybrid approach smaller areas, such as the white in the map of Turkey, are exaggerated. Notice that here `k` plays a slightly different role—it's used as the power for each color individually, `\sum a^k`, rather than their sum, `\left({\sum a}\right)^k`.

For the purists this choice of encoding might appear as the crime of the worst sort, representing neither correct (`k=0.5`) nor the conventionally incorrect encoding associated with `k=1`. Think of it this way—I know what rule I'm breaking.

The similarity between two flags is calculated by forming an intersection between the radii positions of the concentric rings of the flags.

▲ Example of how flag similarity is calculated using the flags of Ukraine and Sweden. (zoom)

For each intersection, the similarity of colors is determined using `\Delta E`, which is the Euclidian distance of the colors in LCH space. I placed less emphasis on luminance and chroma in the similarity calculation by fist transforming the coordinates to `(\sqrt L,\sqrt C, H)`) before calculating color differences. The similarity score is $$ S = \sum \frac{\Delta r}{\sqrt{\Delta E}} $$

Color pairs with `\Delta E < \Delta E_{min} = 5` are considered the same and have an effective `\Delta E = 1`.

▲ The order of flags using different approaches to calculating the similarity score. (zoom)

I explored different cutoffs and combinations of transforming the color coordinates. This process was informed based on how the order of the flags looked to me.

▲ Reasonable ordering for some similar flags achieved by optimizing how similarity between flags is calculated. (zoom)

I decided to start the order with Tonga, since it had the highest average similarity score to all other flags in some of my trials. The flag that is most different from other flags, as measured by the average similarity score, is Israel.

▲ (left) Order of flags when starting with Tonga. (right) Order of flags when starting with Israel, which is has the lowest average similarity score of all flags. (zoom)

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▲ Flags of 256 countries and territories drawn as concentric circles representing the proportions of colors in the flag. Flags are sorted by similarity in color proportion and labeled with the country's ISO 3166-2 code. (BUY ARTWORK)

I couldn't find a list of colors in the flags of countries, so I provide my analysis here. Every country's SVG flag was converted into a 2,560 × 1,920 PNG file (4,915,200 pixels). Colors that occupied at least 0.01% of the pixels are listed in their HEX format, followed by the number of pixels they occupy. The fraction of the flag covered by sampled colors is also shown.

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#code img_pixels sampled_pixels fraction_sampled_pixels hex:pixels,hex:pixels,...
...
cm 4366506 4364514 0.999544 FCD116:1513103,007A5E:1456071,CE1126:1395340
cn 4369920 4364756 0.998818 DE2910:4260992,FFDE00:103764
co 4364800 4364800 1.000000 FCD116:2183680,003893:1090560,CE1126:1090560
...

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#code1 code2 similarity_score
ad ae 0.0108360578506763
ad af 0.0288161214840692
ad ag 0.0510922121861494
ad ai 0.42746294322472
...
zw ye 0.473278765746989
zw yt 0.238101673130705
zw za 0.810589244643825
zw zm 0.573265751850587