## Minkowski Sums

### Problem 228

Published on Saturday, 17th January 2009, 05:00 am; Solved by 813

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

| xk =   cos( 2k-1/n 180° )
| yk = sin( 2k-1/n 180° )

Each Sn 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?