Very Minkowski
Consider the Euclidean plane equipped with the rectilinear metric . In this metric, the distance between two points is defined as
We call the resulting geometry as Minkowski geometry . For example, in Minkowski geometry, the points and are units apart.
Recall that a circle is given by a point and a radius ; a circle is the set of points that have distance exactly from .
Consider a circle in Minkowski geometry. It can be proven that the ratio of its circumference to its diameter is constant. Find this ratio.
The answer is 4.00.
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Note that π is defined as the ratio between the circumference and diameter of a circle. Also, a circle is defined as the locus of all points equidistant from a central point.
In Minkowski geometry, since the distance formula between two points is the sum of the differences of corresponding coordinates. Thus, a Minkowski circle would be in the shape of a Euclidean square oriented 4 5 ∘ with respect to the coordinate axes. The circumference of this circle would be 8 ∣ a + b ∣ (measuring using Minkowski, not Euclidean), whilst the diameter would be 2 ∣ a + b ∣ . Thus, the value of π would be 4 .