What is the roundest country?

Posted on July 26, 2016

We can define roundness in many ways. For example, as you may know, the circle is the shape that given a fixed perimeter maximizes the area. This definition has many problems. One of the problems is that countries generally have chaotic perimeters (also known as borders), so they tend to be much longer than they seem to be.

For that reason, we have to define roundness some other way. Given a country, I will represent it as a plane region, more precisely a compact set \(C \subset \mathbb{R}^2\), and I will define its roundness as

\[ roundness(C) = \max_{ x \in \mathbb{R}^2, r \in \mathbb{R}_{>0}

} \frac{ area(C \cap D(x,r)) }{ \max \{ area(D(x,r)), area(C) \} } \]

where \(D(x, r)\) is the disk of center \(x\) and radius \(r\).

Note that:

For all \(C\), \(0 \leq roundness(C) \leq 1 \).

If \(C\) is a circle, then taking \(D(x,r) = C\) maximizes the formula, so \(roundness(C) = 1\).

If \(roundness(C) = 1\), then \(C\) is a circle, because it overlaps completely with \(D(x,r)\), for some \(x\) and \(r\).

Let’s see the results and then we will see the code I used to get them. You can scroll to see all the results, and sort by any column you like.

Rank Country Roundness Image 1 Sierra Leone 0.934 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2 Nauru 0.923 9wvPL69wj08AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEPCxUgAAAAlwSFlzAAAPYQAAD2EBqD%2BnaQAABhBJREFUeJztXNt24yAMDIfr//8wKwlsg%2BukaXWJe5Z52mYTGIuRkLCtx2NhYWFhYWFhYWFhYWFhYWFh4b9CqJ9m8FPUEp3/W7RDds5H73z5K7TRyC6Dlcna6Q/QrmjkuBs4IP%2Bb067Jo5Gnj/L5k1uBZBG/qrimy4/vAPK9VOsVuVpu6JNk4wM%2BxpxmjvSN2/hkLSl2pjl15Og9fhQn5uEm4q6d8AWZGkqORPzgjT4ZP0ubIgWa9LmT1YBXNYRq/M3nxN0ixczo1Td3ouSTHxE3GSz7tu%2B9%2BfWDaPiET1LcvQ7JT38yEzX3SaTsv%2Bx7b/wuj7Tpws2SwNKC2W%2BcCazty/6XnU9WjGK/9yOgHYcI2LJBGWZPgSv8pu89GwFUUYa/8SrYtF7Nh97D9/o6Gfvx8Jk54Ku5cCcRCVRgbD9IImpZmsQHbiPkNcEPEsleZtDzHLTv5XlVWQCJ7GuWnEoAieh7UXQPA4lsJihKpOOj%2BsnnBVA21kF6ZEIFG3snHk1DdxEYXnpoGj0ocD5YO42YBxFKgzOyJoWoxDyIz0o7bWOtEfOqU/EUQkE9a8S8rKK5jgQGUYh5wenE0QYI/yGIqw/EoZuFgZNLyw/TZ40wOszgovQMxWVNdbQphJNTMINSEjbMATWY8HhVM0fvszgnOVxxpeoF6R1Zcg4Qh1biOEE0qiYIoFFb0g/Sh1j8qLAVVs3t8JjIi5kag51Ohn6GFwvVZGSlAu6EGKV2AxrHQtKYnAqVL2RoE0nTesqYumDuZSNpjKsyM3lM7mwkTbbxAslku3K1I6sZqOgikFWTxnSq%2BwvI7AhtCPma4glwRTP7pLCpw0jSrSDn66NFIAnneAtoHb4Wia6ZpFsuyd3IWhYt4dDvgZyHmwU3MauXhzvISNx6oxWGZpKGmIdCZFZ2xpLuE/JE3ehaVFobiC8vwrZNRf3wYEDGgpzn%2BO2S9Q8PTjPyPJFsbHF4sKOtLSv9IIVZSrp7EWttKSO1lDRYOj2YmbC3q7Q62j7GshNetlGl1dH2scQ41CMXNEtLEd1CnDmJtM3hQUdny/F9JG0r6Z7lcEjjYplKetsSuKRNJb1NxpWH0eFBw5YDM0lbpqVHDswkbXZ4gNilyCOdTCW9S5GzucA2blhpDRUSaxuXO5l/B4coWN4flZ5KeTbb9i/W%2BmbVJw9OGAIVK59OlqSPypAXZ4P2owcjDnXw4mx1XuoJzTfm2u3DjLMQPqzypeHcgFngRW8WqIe8nXlmkVwyCnpD3s7NdyA5NSoCBnVwT5bh%2Bo3OeYc7Lewziyz/DNclJnVw1xaWyqSyHUIzPxsGpzC5eTGIUECP0VvU46M6BIIs2NnAFQfHkVhYuHCDOnGwS5RIHCAREBnnFYaH/mQsZOGKgzqEtAjDaCcgR1on5fVBvyY/jCKW6kSn7IqHOuTCK5haN%2BodCyk4T4ZcTzMB2dUhuY/BWJqueNzzE13Q5LJi1NvVEURdp3qvmIBs50lVuIoOgs/bnrGHJvHVzHqvE23qkBUHonrZtwwG9CJDWhyIoPW2WVcHvlIkP3hRMnU/4E068hN/UbWhqUPrpKKqCKSpIwi%2BIT0jir0vPoDinIYTdhQnz5rewK56mwCWi9KsE3bu0nhFepwiyLI24IxOE6aeC0wQZ53XugdA6hjkIh9xLhrePQHrIrFmCMh5aIWgB6wFahbpe1eQc7RoodeK5SIgbOQsKLVX6EUczMaUCHJOkk79Cr2KA4mwJiwuwoVbdVfcjyhYcwJn5lX/DPt9gblv1Y8APmHbxHI4jETav2lUVZx1383pPCX8olvpR/r7zYeR4Yf9tajvlH1LwvNhJPVyS2/xCNR3Sn8LvMCX%2BwKto9Q3ncZqwFaA0WLbvkKBXO88cesMGnO5amZaa%2B9cWWr5EGesFYnfiTmYMm4NQksJhFKwSWj7Pn5drxT8njV2MEUqPp9sTjZtLId2rHtP0w9y3ghuzOOFWkLH9B%2Bf59xQQ8qd%2BbfbjHCpxsTWIvZC6APuxblhZ%2B6vmd%2BRc8dTF70x5wbsy3sS%2Bu05N0wu%2Bkc4LywsLCwsLCwsLCwsSOIfouFwyXH2Rw0AAAAASUVORK5CYII%3D 3 Zimbabwe 0.915 %2BQrQrzCPUG4AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACLTCbwAAAAlwSFlzAAAPYQAAD2EBqD%2BnaQAABaRJREFUeJztXNGSgzAINJOQ5P9/mAMSrW29VhtQH9iX6820uiVkIQidJofD4XA4HA6Hw%2BFwOBwOh8PhcDgcDofDBjhBDBlyTLFkvJrNHiCEJ9SrCX0HVmGaygSpIpn6/qQxhRAz0UQEAZbbk85kYoCaYlz7R8oFb%2BvYhTgHphtTrqVZulT6T5jXW/LOYtZc4Ykd0NdotPnb3ExKgDkneCOFQjYs/nIj1kVs%2BekdSMSBlSXdhDakEEsJ6fs7xe4pXS8oQhnJpfOeN8c7BBwOJiIMaS8TBBbzK52Ec4x2/92kCfVK1mTmCP112uPTywdFzS%2BRQAray13hmJpl8u3E/p3h%2B5s1QSpW%2BqvCt//h04V4xzNjJdm2WQmrxO6jn6d1IrqSYnGCpc5vC9D3EseVVH65J2bxLmwRXpneJmbOFL1/302VP3seaWjhmBZ3SLtqszVJzwn7sdmZ/XEwlaii7gel5zdgJM7IO3/YQJk0hDgXDVofgbyaqJOwybXOMDRbh91R5WK8arsyxDHIWRWVOMuenqK1qZHdAhVvUwIl41om%2BAeZ6WouKLk1Gut0K2QkzfWktbPVD4ysGfD5QHgUnG9ZbsXKsQuVDcOSb7gTkS2iHwysDY2rPFoNKRgmH6zOaJDdkB3sTE2GJs4GG90wN2Xp0PeNqR0ERpcv8QUwv9UEIHAVdPDiW4jHz5ivgF6ybd%2B%2BKbMgRTxUKdgNUrw6KHq9xMlFxYztHzlSoXwVC0ElDcXBWlnkdJmrtk8FfczRrFbLl41jSwiz38qZM/MhLnfjG8UtTmXG/WM%2Brj7qEWl8r/wPzqlhUJZwUzbBLmqRlWjDD9pkM8cfXb8PSOzUaVRMYcPU2S5RlzA%2BbpQNmRhdvg%2BQXHfc/fJb6WhUSD%2BgyFMmhRukV1sb7sPQEvTxpcT4Yutitw%2B7fZJChhCfMy9D8Sht2w/LxyT14/VXr6pn2TX4oTr/VZGn52qEleKVMMdanbWMa//Q8LgtULJb22MXnV3zdOy2Iv14qK601XFla0PSnayWPq3CuRXpvDxx0SK9EhAr0tJeJi/0SD8sbXfI7/FbKxKsqnZ2Sd5cSVGLBA%2BnNouIOEdePbMslWiz3GOpSujtmuXoZpblLXVexYQ9Lkdzo2J97e6hmbCT6rWmDLPDOB1I2dSjx/E1cK6PmQl1j4m6m6a0i5rJRyetrKmtUFOsdqJ0uQzXxd4vy/mB4cmWro8Gl2/9XdpXXUDZh0UckA50qyeU/NDMJrdphV%2BLLkBeRjPn6x3o6rR5n5tt8xoADZxEMiazzFeaPFD9qQvFAdVw%2BAJpm1DtQGAkSn%2BzXSkIeLdUZe8rItKGXSps5aLcgsA729DQzdRkGZYRHe1DOWyZGlq8Os8DIAqXK20AwtTQIiAo3gEqItLSXrAeEsjzKVTFs6XiIQ2VppjvoNPgJWc5bTnaAPRNo0M6PC5oiuYgWYU0Sai9czDak1GdCRvuEzijd1rcGQ/NtfyLEmI9oQ1ZQF6o0srElcIzHLqhhKrQ30Wc85lTczVojDIZHYT%2BA20fBVuXk4cqmfVosyVY9r1so4a3x%2BfHUK6YA/1hYGsFSRYvmO%2BDL3O1n8AzMudElVcUGauOGQ/Pi5VwFededgqHx2a5j/KMgZz/wPOq7SgT6057y1julZwbBZ51a/2R31ccC%2Bk75Vwna90bjSwDzUL7m3ALZZgu5zw12vxc5puluXoey8EhYjuwk1CW%2BUkBUWZEue9W5/ygAWy/h5A3YxxCTcsI8Hv33JWQoWb%2BEZJEXtCcFhFKlT26/MBHvdVPT0xMe6Xcc7t7iGn1ox9nZ3a70HquayOfcwHiSxFztjNcFgg/Yg4x/XeC8tryUO7JeYUecMRBUuqvrwyEe1EqzLYHuPgnMn7EzV3D4XA4HA6Hw%2BFwOBwOh8PhcDgcDodjN/4A7v8nk73/tTcAAAAASUVORK5CYII%3D 4 Vatican 0.908 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5 Poland 0.903 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6 Scarborough Reef 0.901 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7 Ivory Coast 0.899 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8 Suriname 0.897 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9 Swaziland 0.896 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10 Uruguay 0.894 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11 Romania 0.894 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12 Sudan 0.891 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13 Ethiopia 0.888 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14 Botswana 0.886 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15 San Marino 0.886 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16 Kosovo 0.881 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17 Monaco 0.880 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18 Andorra 0.879 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22 Nicaragua 0.866 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23 Nigeria 0.862 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24 Gabon 0.861 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25 Bajo Nuevo Bank (Petrel Is.) 0.860 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