Tribal War

The following is a mathematical explanation of Global Super Powers. In particular, I will focus upon the financial costs of maintaining territorial integrity, as history has proven that the most expensive things a nation can do are to plan, build and deploy a military.

First, assume a nation's territory is encapsulated within a circle. Any shape can be used, but for the purposes of simplicity, a circle will suffice.

The first thing to understand is the surface area of the resultant nation. This can be summarized as the sum total of of its demographical, strategic, and economic resources. Anything within the circle belongs to the nation. The surface area of a circle of radius r is equal to 𝜋 * r².

The second is the circumference of the nation. This is essentially the sum total of the nation's borders. It directly defines the requisite military force the nation must construct to maintain its territorial integrity. The longer the border, the more the nation must spend on its military. The circumference of a circle of radius r is 2 * 𝜋 * r.

Now, let us calculate the ratio of the circumference to the surface area. This is equal to ^{(2 * 𝜋 * r)} / _{(𝜋 * r²)}, or ²/_r.

There is a key realization in the above mathematical equation, which is that the two above values (circumference and surface area) do not both increase at the same rate. The circumference increases linearly with increasing radius, while the surface area increases quadratically with increasing radius.

What does this mean? Well, if the radius of the nation is doubled, the circumference rises to 4 * 𝜋 * r, but the surface area rises to 4 * 𝜋 * r², and the ratio between the two doubles to ¹/_r.

In other words, the larger the nation, the less proportion of its resources must be spent upon the defense of its borders. And there is no upward limit of this calculation. The larger the nation, the easier it is to defend from foreign incursions.

Another key realization is the shape of a nation. Again, for the purposes of simplicity, a rectangular nation will be used.

Imagine a square nation of length r per side. The surface area of the square is r², while its circumference is 4 * r. As with the above, surface area is the sum total of the nation's assets, while circumference is its cumulative borders.

Now imagine a rectangle of the same area, but twice the length 2 * r. For the rectangle to have the same surface area, the height would necessarily need to be 0.5 * r, so that 2 * r * 0.5 * r = r².

But the surface area of such a rectangle would be the sum of its sides, or (2 + 2 + 0.5 + 0.5) * r, which is 5 * r. So the surface area would remain the same, but the circumference would rise by 25%, or ⁵ / ₄.

In other words, the more elongated a nation's shape is, the greater a circumference it has as a ratio of its surface area, and the more of a border it must spend its resources defending.

In conclusion, both the size and shape of a nation play a central role in its feasibility, as well as in its chances of success.

I hope you have found this post interesting, and I'll see you out there.