Very Minkowski

Geometry Level 5

Consider the Euclidean plane equipped with the rectilinear metric. In this metric, the distance between two points \((a,b), (x,y)\) is defined as

\[d((a,b), (x,y)) = |x-a| + |y-b|.\]

We call the resulting geometry as Minkowski geometry. For example, in Minkowski geometry, the points \((1,5)\) and \((3,2)\) are \(|3-1| + |2-5| = 2+3 = 5\) units apart.

Recall that a circle is given by a point \(x\) and a radius \(r\); a circle is the set of points that have distance exactly \(r\) from \(x\).

Consider a circle in Minkowski geometry. It can be proven that the ratio of its circumference to its diameter is constant. Find this ratio.


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