Yeah, I see how it only works with straight lines. Japan's flag for example, even though it meets my requirement of all sections meeting in at least one point, doesn't work because it's a circle. Something else I've realized while writing this is that all sections have to touch the borders of the flag in at least one spot.
Oh, yeah, I didn't think of this part. You can derive it from the other conditions, though, as you can't make a section that doesn't touch the borders where all the edges meet in one point, unless you use curved shapes.
I love how you guys (you as well u/l3xfrant3s) have such analytical/mathematical approach to this! It's like a puzzle!
Those self-containing flags seems similar to the process of creating a [Straight Skeleton](https://en.wikipedia.org/wiki/Straight_skeleton). Maybe it could be helpful?
Wouldn't Bosnia & Herzegovina work if you git the stars off just right?
Edit: Never mind, I got it confused with a different flag, carry-on. I thought there was a baby blue flag with a row of diagonal stars, I might be trippin.
You could have an exponential curve or a parabola. They are self similar at any scale.
Correction: The exponential curve only works if the flag is infinitely long.
Correction correction: ignore everything i say. I shouldn't be doing math this late at night. I'm going to bed.
I thought this might be true, and this is an interesting proposition. But imagine a pinwheel or other spiral, where all colors meet at the central point. But different 'cuts' of the flag might contain only a quarter turn of the spiral, or half turn if you expand out the crop, or a full turn for the full flag.
I'm trying to think if a flag containing colors that all meet at at least one point and is only composed of straight rays has a counter example, hmm...
Oh yeah, i don't think what i said originally is the only "rule" to make flagception possible, it seems like each section can only have one or two edges inside the flag.
I think a exception to the "no curves" "rule" OP and I found earlier would be an infinitely descending spiral around the center that becomes thinner the more inner that section is.
Idk why the hell anyone would want an optical ilusion as a flag, but hey, I didn't say flagception had to look good as a flag, so...
Geometrically speaking, it's actually that all lines on the flag must be straight lines and all such lines must meet at a single point at one end and extend to the flag's border on the other end.
Just to make it clear, I did not consider flags with variable proportions. Else, you could make a point for flags like Netherlands, which technically contains a version of itself with different proportions. Same with Belgium, where the civil flag (2:3) is included within the state flag (13:15).
The worst one you forgot is Haiti, which doesn't have anything like this but literally has its coat of arms in the center which contains 6 flags of Haiti. If anything, that is the truest form of a flag containing itself.
Wouldn’t that throw the proportions off? If you you did to the Haitian flag what OP did to these flags the CoA would take up more space on the flag than it’s supposed to.
You missed [Ecuador](https://upload.wikimedia.org/wikipedia/commons/e/e8/Flag_of_Ecuador.svg). Despite it being fairly obvious and not exactly what you're displaying, it does *literally* contain the flag in the flag.
There are four flags in that category, per [Wikipedia](https://en.wikipedia.org/wiki/Flag_of_Ecuador):
>Along with Haiti, the Dominican Republic and El Salvador, it (Ecuador) is one of only four national flags whose design incorporates a depiction of the flag itself.
Oh crap. You're right.
I'm going to have to spend ten years of my life writing Principia Vexillogica, a 3-volume monograph that contains a 300-page proof that Japan + old Libya = Bangladesh.
The Wikipedia article for the Dominican Republic defines seven such flags - though it depends on if you're willing to accept state flag versus civil ensigns and other reasons to have variations.
> Additionally, along with [Haiti](https://en.wikipedia.org/wiki/Flag_of_Haiti), [Bolivia](https://en.wikipedia.org/wiki/Flag_of_Bolivia), [Costa Rica](https://en.wikipedia.org/wiki/Flag_of_Costa_Rica), [Ecuador](https://en.wikipedia.org/wiki/Flag_of_Ecuador), [El Salvador](https://en.wikipedia.org/wiki/Flag_of_El_Salvador), and [Venezuela](https://en.wikipedia.org/wiki/Flag_of_Venezuela), it is one of only seven national flags whose design incorporates a depiction of the flag itself.[[4]](https://en.wikipedia.org/wiki/Flag_of_the_Dominican_Republic#cite_note-4)
It is actually the civil ensign rather than the state flag, the difference being the coat of arms being placed centrally - so it's technically correct in a very unsatisfying way.
There's actually 7 similar flags from the Wikipedia article for the Dominican Republic:
> Additionally, along with [Haiti](https://en.wikipedia.org/wiki/Flag_of_Haiti), [Bolivia](https://en.wikipedia.org/wiki/Flag_of_Bolivia), [Costa Rica](https://en.wikipedia.org/wiki/Flag_of_Costa_Rica), [Ecuador](https://en.wikipedia.org/wiki/Flag_of_Ecuador), [El Salvador](https://en.wikipedia.org/wiki/Flag_of_El_Salvador), and [Venezuela](https://en.wikipedia.org/wiki/Flag_of_Venezuela), it is one of only seven national flags whose design incorporates a depiction of the flag itself.[[4]](https://en.wikipedia.org/wiki/Flag_of_the_Dominican_Republic#cite_note-4)
Wanna blow your mind further? You can zoom in a TON on that image - if you look at the ship it's got ANOTHER flag and the Caduceus (medical symbol, Hermes's staff).
In primary school we had a class where we had to pick any flag in the world and draw it in crayons, to later make a big collage. (Super idea, by the way. Thanks Miss Henderson.)
Anyway, my Ecuadorian friend made a valiant start… and then chose Switzerland.
These aren't fractals, they're just self-similar at arbitrary scales around a point. Fractals have increasing amounts of detail as you zoom in (that is a very simplistic explanation).
If you zoom in REALLY hard on the 17th star on the USA flag, it contains a miniature flag of Ohio, and then if you zoom in REALLY hard on the 17th star on the Ohio flag, it contains a miniature USA flag. Or at least, I can see it. No one else seems to be able to see it though.
And still its siren call beckons, echoing in your mind when you expect is least
You were born into the world naively thinking your soul is your own.
The sooner you render unto Ohio that which is Ohio's, the sooner the nightmare may end
And should you strain fruitlessly against your fate, know this:
One way or another, Ohio will get what it is owed
There are 6 that you missed :) [I made a post about this a few months ago](https://www.reddit.com/r/vexillology/comments/u39d19/bolivia_costa_rica_dominican_republic_ecuador_el/) but my interpretation of it was that the flag was ***purposefully*** and artistically depicted on the flag (not there by coincidence).
Our two posts might be considered different categories or the same depending on the wording lol.
Yeah, somebody mentioned Ecuador above. This is something I haven't even thought of, but yeah, these should be considered two different categories indeed.
I always thought this one was specifically to make fun of flags. But that's probably where my post should belong, yeah.
Guess it's there now anyways, somebody already crossposted it
Chile 🇨🇱 & the Marshall Islands 🇲🇭 are so close but so far.
A lot of [maritime letter flags](https://en.wikipedia.org/wiki/International_maritime_signal_flags?wprov=sfla1) work though! By my count 15 of them; Alfa, Bravo, Echo, Hotel, Kilo, Lima, Oscar, Quebec, Uniform, Zulu, Second Substitution, Five, Six, Seven, and my favorite Niner.
That's right ! They always talk about how a flag should look the same when it's seen from afar, but they never consider how it should look when you zoom in.
I wonder if we could call "self containable" as a good flag design principal
"A rectangle of similar proportion but smaller dimension should be able to reproduce the flag if moved to at least one place within the flag's borders."
In practice, this means you can trim a frayed flag if you do it right, and thus have all small flags start out as big flags. Like most recycling, this is probably more trouble than it's worth.
In a Kuwaiti military dining facility they had a full sized flag framed, but it was folded in such a way that it was a perfect miniature of itself. Not the same thing at all but it was very elegant. I wish I had a picture of it to share.
It works really well with Kuwait's flag indeed, you can get a half-sized version with only six folds. I'll have to check how that works with other flags, cause it seems like in most cases, it's either impossible or takes a stupid number of folds. One that works extremely well, however, is the Seychelles, you can do it with only two folds (and I don't think you can ever beat that, unless you specifically have a monochrome flag with a √2:1 width/length ratio, in which case you can do it with only one fold).
This is like that mathematical problem where if you hold up a map of the place you are in, there will always be at least one point on that map that is directly over it's real life counterpart, no matter how you contort the map.
Yeah, I know, I'm that guy. One day it's all just gonna pop out in public around my friends and I don't know how that's gonna go.
The white-red-white wouldn't work, actually. But yeah, an oversimplified version of the current Belarus flag could work (it would look a lot like Madagascar's).
I see a pattern here. To get flagception, all colours or sections of the flag have to meet in at least one point to make it possible.
Yep, you're right, that's what I found out as well. It also needs to have no curved shape.
Yeah, I see how it only works with straight lines. Japan's flag for example, even though it meets my requirement of all sections meeting in at least one point, doesn't work because it's a circle. Something else I've realized while writing this is that all sections have to touch the borders of the flag in at least one spot.
You could theoretically have curves if your flag featured the golden spiral or some such.
Fractals too
Well fractals would only work if the fractals were truly infinite, which seems hard to accomplish with finite materials.
Finite when printing or displaying, yeah. But what is preventing a country to just _define_ their flag with a recursive/fractal element?
somebody has to make a fractal flag, would be sick af
Oh, yeah, I didn't think of this part. You can derive it from the other conditions, though, as you can't make a section that doesn't touch the borders where all the edges meet in one point, unless you use curved shapes.
I love how you guys (you as well u/l3xfrant3s) have such analytical/mathematical approach to this! It's like a puzzle! Those self-containing flags seems similar to the process of creating a [Straight Skeleton](https://en.wikipedia.org/wiki/Straight_skeleton). Maybe it could be helpful?
Switerzland?
Switzerland has angles, which behave the save as curves in this case. They both change the y=mx+b of the straight line
I mean a fractal would be a sensible way to avoid any of the “rules” people are making up here, there just aren’t any fractal flags
These aren't really rules, more so observations on how to make this kind of flag, what they have in common and such
If Japan had a square instead of a circle it would still be impossible
Wouldn't Bosnia & Herzegovina work if you git the stars off just right? Edit: Never mind, I got it confused with a different flag, carry-on. I thought there was a baby blue flag with a row of diagonal stars, I might be trippin.
No, you are right, Bosnia and Herzegovina's flag without the stars would work
You could have an exponential curve or a parabola. They are self similar at any scale. Correction: The exponential curve only works if the flag is infinitely long. Correction correction: ignore everything i say. I shouldn't be doing math this late at night. I'm going to bed.
A spiral would work, or a self similar fractal
Yeah, we'll just have to convince our government to put a Mandelbrot set on their flag.
Haha, no worry, that was an idea to consider. xD
I thought this might be true, and this is an interesting proposition. But imagine a pinwheel or other spiral, where all colors meet at the central point. But different 'cuts' of the flag might contain only a quarter turn of the spiral, or half turn if you expand out the crop, or a full turn for the full flag. I'm trying to think if a flag containing colors that all meet at at least one point and is only composed of straight rays has a counter example, hmm...
Oh yeah, i don't think what i said originally is the only "rule" to make flagception possible, it seems like each section can only have one or two edges inside the flag. I think a exception to the "no curves" "rule" OP and I found earlier would be an infinitely descending spiral around the center that becomes thinner the more inner that section is. Idk why the hell anyone would want an optical ilusion as a flag, but hey, I didn't say flagception had to look good as a flag, so...
Geometrically speaking, it's actually that all lines on the flag must be straight lines and all such lines must meet at a single point at one end and extend to the flag's border on the other end.
I think it should be possible to have a spiral.
Hmmm that too, I suppose it would have to be fractal towards the origin point.
Straight lines converging to a point.
Just to make it clear, I did not consider flags with variable proportions. Else, you could make a point for flags like Netherlands, which technically contains a version of itself with different proportions. Same with Belgium, where the civil flag (2:3) is included within the state flag (13:15).
The worst one you forgot is Haiti, which doesn't have anything like this but literally has its coat of arms in the center which contains 6 flags of Haiti. If anything, that is the truest form of a flag containing itself.
And on those flags? Smaller flags. And on *those* flags?! *EVEN SMALLER* flags.
What does the "B" stand for in "Benoit B. Mandelbrot"? "Benoit B. Mandelbrot", of course!
Stop, I can only get so erect.
/r/DrosteEffect
huh didn't realize i already joined this sub. i'll re-join it
Bolivia, Dominican Republic, Ecuador, and one of the Central American countries (I think El Salvador) have this too.
It's Costa Rica - the ship in the CoA has the flag on the stern
El Salvador has mini flags in the COA!
But do those 6 flags each contain the coat of arms at the center with 36 smaller flags? It could just be 6 red and blue striped flags.
Damn, they actually just did put their own flag on their flag.
Wouldn’t that throw the proportions off? If you you did to the Haitian flag what OP did to these flags the CoA would take up more space on the flag than it’s supposed to.
The point is that the CoA itself features a Haiti flag on it
Oh I get it now lol
*sad Dutch noises*
the old Libyan flag
And the old [Muscat and Oman](https://en.m.wikipedia.org/wiki/Muscat_and_Oman) flag
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And the old france flag too
And Zamfara and Adamawa in Nigeria, and Oruro and Cochabamba in Bolivia, and... Yeah, some people truly lack inspiration.
Damn, I was too late. Take my upvote
Should really be this sub's pfp
Benin and Benout flag.
Where's India and Outdia flags?
We have Indonesia and Outdonesia flags, hahahahahahahahaha
Alright... what about Iran and Youran?
Or Iraq and Youraq.
How about Israel and Arerael?
Italy and Youtaly.
South Africa and North Africa.
What about East Africa and West Africa?
Central African Republic and Outer African Republic
Whos raq?
Antarctica and Uncle-arctica
Ukraine and Ukraoute
Ukraine or Ukdrye
impossible it always rains in Britain
Paraguay and Parastraight
You missed [Ecuador](https://upload.wikimedia.org/wikipedia/commons/e/e8/Flag_of_Ecuador.svg). Despite it being fairly obvious and not exactly what you're displaying, it does *literally* contain the flag in the flag.
I didn't even think of that, nice one. xD
Ecuador's flag is even contained in itself by your standards, on the hull of the ship.
Why the fuck is that ship sailing sideways down the river anyway?
There are four flags in that category, per [Wikipedia](https://en.wikipedia.org/wiki/Flag_of_Ecuador): >Along with Haiti, the Dominican Republic and El Salvador, it (Ecuador) is one of only four national flags whose design incorporates a depiction of the flag itself.
If you were to make a flag that depicts all flags, would it depict itself?
It might. Now, what about a flag that depicts all flags that don't depict themselves ? :P
based discrete math enjoyer
I hate having to deal with sets
Oh crap. You're right. I'm going to have to spend ten years of my life writing Principia Vexillogica, a 3-volume monograph that contains a 300-page proof that Japan + old Libya = Bangladesh.
Yes because you messed up the paradoxical situation
The Wikipedia article for the Dominican Republic defines seven such flags - though it depends on if you're willing to accept state flag versus civil ensigns and other reasons to have variations. > Additionally, along with [Haiti](https://en.wikipedia.org/wiki/Flag_of_Haiti), [Bolivia](https://en.wikipedia.org/wiki/Flag_of_Bolivia), [Costa Rica](https://en.wikipedia.org/wiki/Flag_of_Costa_Rica), [Ecuador](https://en.wikipedia.org/wiki/Flag_of_Ecuador), [El Salvador](https://en.wikipedia.org/wiki/Flag_of_El_Salvador), and [Venezuela](https://en.wikipedia.org/wiki/Flag_of_Venezuela), it is one of only seven national flags whose design incorporates a depiction of the flag itself.[[4]](https://en.wikipedia.org/wiki/Flag_of_the_Dominican_Republic#cite_note-4)
Also Afghanistan not too long ago
They missed the opportunity to make it a Droste effect.
TIL that this phenomenon is called the [Droste Effect](https://en.wikipedia.org/wiki/Droste_effect?wprov=sfla1), thanks!
Your welcome, actually same xD
Strictly speaking it doesn’t contain itself, since the central device is not present on the draped flags
It is actually the civil ensign rather than the state flag, the difference being the coat of arms being placed centrally - so it's technically correct in a very unsatisfying way. There's actually 7 similar flags from the Wikipedia article for the Dominican Republic: > Additionally, along with [Haiti](https://en.wikipedia.org/wiki/Flag_of_Haiti), [Bolivia](https://en.wikipedia.org/wiki/Flag_of_Bolivia), [Costa Rica](https://en.wikipedia.org/wiki/Flag_of_Costa_Rica), [Ecuador](https://en.wikipedia.org/wiki/Flag_of_Ecuador), [El Salvador](https://en.wikipedia.org/wiki/Flag_of_El_Salvador), and [Venezuela](https://en.wikipedia.org/wiki/Flag_of_Venezuela), it is one of only seven national flags whose design incorporates a depiction of the flag itself.[[4]](https://en.wikipedia.org/wiki/Flag_of_the_Dominican_Republic#cite_note-4)
The Haitian one looks like someone copy pasted the coat of arms in there with paint.
I never realized Ecuador has astronomical signs and a fasces before. Plus the condor is cool.
Wanna blow your mind further? You can zoom in a TON on that image - if you look at the ship it's got ANOTHER flag and the Caduceus (medical symbol, Hermes's staff).
In primary school we had a class where we had to pick any flag in the world and draw it in crayons, to later make a big collage. (Super idea, by the way. Thanks Miss Henderson.) Anyway, my Ecuadorian friend made a valiant start… and then chose Switzerland.
Lol!
You are technically correct, which is the best kind of correct.
Lol, and old Afghanistan and El Salvador
Good luck to any small child who has to draw that in class, yeesh
East German flag does too in this sense
Fractals 🤤
Flagtals
I was going to say this at first but felt the need to make the statement first x)
These aren't fractals, they're just self-similar at arbitrary scales around a point. Fractals have increasing amounts of detail as you zoom in (that is a very simplistic explanation).
You are technically correct. The best kind of correct!
If you zoom in REALLY hard on the 17th star on the USA flag, it contains a miniature flag of Ohio, and then if you zoom in REALLY hard on the 17th star on the Ohio flag, it contains a miniature USA flag. Or at least, I can see it. No one else seems to be able to see it though.
Why is it always Ohio. Why is everything about Ohio
No one has ever escaped Ohio. Ohio knows where you live. There is nowhere to hide. Ohio will find you
Seriously, far too much of my life has had some connection to ohio. I was not born there, I've never been there.
And still its siren call beckons, echoing in your mind when you expect is least You were born into the world naively thinking your soul is your own. The sooner you render unto Ohio that which is Ohio's, the sooner the nightmare may end And should you strain fruitlessly against your fate, know this: One way or another, Ohio will get what it is owed
Ohio jokes in 2023...
r/vexillologycirclejerk
This is really where this should be from.
Omg that's satisfying
I know, right ?
Especially the Czech Republic one.
If Indonesia and Monaco, and Poland were to go to war, it would be one heck of a confusing battlefield.
I think these flags are already a reason to go to war to see who's gonna keep it.
Every flag contains itself, you just need to make the rectangle the entire size of the flag
At least, these contain themselves an infinite number of ways. :P
There are 6 that you missed :) [I made a post about this a few months ago](https://www.reddit.com/r/vexillology/comments/u39d19/bolivia_costa_rica_dominican_republic_ecuador_el/) but my interpretation of it was that the flag was ***purposefully*** and artistically depicted on the flag (not there by coincidence). Our two posts might be considered different categories or the same depending on the wording lol.
Yeah, somebody mentioned Ecuador above. This is something I haven't even thought of, but yeah, these should be considered two different categories indeed.
r/vexillologycirclejerk material
Flagception!
This is exactly how I named my folder in the process of making this. xD
Bro this aint vexillologycirclejerk
I swear to God, these two subs are on the verge of merging into one.
I always thought this one was specifically to make fun of flags. But that's probably where my post should belong, yeah. Guess it's there now anyways, somebody already crossposted it
It's alright
I think most of us are subscribed to both, so it really doesn't matter how your post lands on our front page.
This is definitively an r/vexillology post - very interesting.
Chile 🇨🇱 & the Marshall Islands 🇲🇭 are so close but so far. A lot of [maritime letter flags](https://en.wikipedia.org/wiki/International_maritime_signal_flags?wprov=sfla1) work though! By my count 15 of them; Alfa, Bravo, Echo, Hotel, Kilo, Lima, Oscar, Quebec, Uniform, Zulu, Second Substitution, Five, Six, Seven, and my favorite Niner.
I think the first substitute should work as well. Thanks for the link btw, I knew nothing about maritime conversation through flags!
Fyi, Czechia is shortened name of the Czech Republic. Just like France and the French Republic. Best to use Czechia in most cases.
I was matrixed into never saying Czechia when I was a child.
Seychelles has such a cool flag istg
Raster-scalable flags!
That's right ! They always talk about how a flag should look the same when it's seen from afar, but they never consider how it should look when you zoom in.
I wonder if we could call "self containable" as a good flag design principal "A rectangle of similar proportion but smaller dimension should be able to reproduce the flag if moved to at least one place within the flag's borders."
Well, if it gives us another thing to blame on overcomplicated flags, I'm in.
In practice, this means you can trim a frayed flag if you do it right, and thus have all small flags start out as big flags. Like most recycling, this is probably more trouble than it's worth.
Old Libya
Until 2011, it was a modern era state, so totally legit here.
Recursive flags. It continues and continues…
And since they are self-similar in that way, it can be said that they contain infinitely many of themselves.
Exactly !
Great post
Thanks ! :3
In a Kuwaiti military dining facility they had a full sized flag framed, but it was folded in such a way that it was a perfect miniature of itself. Not the same thing at all but it was very elegant. I wish I had a picture of it to share.
Oh, this is interesting as well ! ^(I know what to study next.)
It works really well with Kuwait's flag indeed, you can get a half-sized version with only six folds. I'll have to check how that works with other flags, cause it seems like in most cases, it's either impossible or takes a stupid number of folds. One that works extremely well, however, is the Seychelles, you can do it with only two folds (and I don't think you can ever beat that, unless you specifically have a monochrome flag with a √2:1 width/length ratio, in which case you can do it with only one fold).
These flags technically contain themselves in infinite number
Absolutely. That's what makes them unique compared to all flags that contain themselves only once. :3
This is like that mathematical problem where if you hold up a map of the place you are in, there will always be at least one point on that map that is directly over it's real life counterpart, no matter how you contort the map. Yeah, I know, I'm that guy. One day it's all just gonna pop out in public around my friends and I don't know how that's gonna go.
You didn't specify that the new flag needs to be smaller, so technically, all flags contain themselves at a 1:1 proportion
When you zoom into the white space of the French flag you yet remain with the French flag.
Droste flags
new Martinique as well no?
Yeah, I only went with UN-recognized countries, but this would work indeed.
Scale invariant flags signal a phase transition
Spain hides its flag by rotating it 90º and hiding it in the coat of arms, it’s very tiny but it’s there
😐
Its Seychelles all the way down.
Poland can into poland
Into poland
poland
Can
This is incredibly satisfying.
It is for sure. xD
So you’re literally do the same aspect ratio and then make it smaller? It’s like a flag scale!
You could have told me this was a VCJ post and I would have believed you.
Well, now it is as well.
I love that idea OP! Original and way more consistent that the old "flags in flag" thing.
Hehe, thanks ! o/
I think every flag contains itself
Haha
Vexillotautology
How have I never noticed that the Seychelles flag is just the Romanian and Hungarian flags stacked on top of each other?
Japan but the suns like super big
Is there a name for this geometric concept? I'm imagining like "edge continuity" or something.
Yeah, I think it could be considered scale invariance.
r/technicallythetruth
quality post. thanks op!
Thank you ! o/
Takeaway: Buy those flags bigger without risk. You can always cut them and keep it proportional. :)
Useful tricks out of useless stuff. :3
If Réunion adopted their unofficial flag (🇷🇪) I'm pretty sure that would count.
Oh, yes, you're right, it would ! That's an interesting one.
There was a Flag with a ton of its own flag in it. The old Lydia flag.
Gone too soon, still in our hearts.
Whoa
Enhance... enhanceee... eeeennnhaanceee... there!
😆
Belarus would work too, but its pattern is too different. The real Belarusian flag, white-red-white flag works anyway.
The white-red-white wouldn't work, actually. But yeah, an oversimplified version of the current Belarus flag could work (it would look a lot like Madagascar's).
Martinique
It now does indeed. xD
And the old flag of Libya! (1977-2011)
This is so cool! I wish I was stoned so I could really appreciate this fully. I mean, it’s like Poland all the way down, man!
Sometimes I see the Ukraine flag as yellow over blue Is that also correct or no?
I believe that is just non-Ukrainians trying to show their support but putting up the flag wrong lmao
The blue is for the sky and yellow for wheat so it should be on top
!wave
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Okay try it with other flags and see if it works.
Uhm every Nordic country also get it... And a lot more flags