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Problem 228

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Minkowski Sums

Let S_n be the regular n-sided polygon – or shape – whose vertices v_k (k = 1,2,…,n) have coordinates:

$x_k= \text{cos(}\frac{2k-1}{n} \times 180^\circ \text{)}$ $y_k= \text{sin(}\frac{2k-1}{n} \times 180^\circ \text{)}$

Each S_n is to be interpreted as a filled shape consisting of all points on the perimeter and in the interior.

The Minkowski sum, S+T, of two shapes S and T is the result of adding every point in S to every point in T, where point addition is performed coordinate-wise: (u, v) + (x, y) = (u+x, v+y).

For example, the sum of S₃ and S₄ is the six-sided shape shown in pink below:

How many sides does S₁₈₆₄ + S₁₈₆₅ + … + S₁₉₀₉ have?

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闵可夫斯基和

记S_n为正n边形——或者叫图形——且其顶点v_k(k = 1,2,…,n)的坐标为：

$x_k= \text{cos(}\frac{2k-1}{n} \times 180^\circ \text{)}$ $y_k= \text{sin(}\frac{2k-1}{n} \times 180^\circ \text{)}$

每个S_n都是实心图形，包含有其边界上和内部的所有点。

两个图形S和T的闵可夫斯基和是将S所包含的每个点加在T内的每个点所得的结果，这里所谓两个点相加即将它们的坐标相加：(u, v) + (x, y) = (u+x, v+y)。

例如，S₃和S₄的闵可夫斯基和是如下粉色的六边形：

闵可夫斯基和S₁₈₆₄ + S₁₈₆₅ + … + S₁₉₀₉是一个多少边形？

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