Minkowski Sums
Problem 228
Published on Saturday, 17th January 2009, 05:00 am; Solved by 909; Difficulty rating: 70%Let S_{n} be the regular n-sided polygon – or shape – whose vertices v_{k} (k = 1,2,…,n) have coordinates:
x_{k} = cos( ^{2k-1}/_{n} ×180° ) | |
y_{k} = sin( ^{2k-1}/_{n} ×180° ) |
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 S_{3} and S_{4} is the six-sided shape shown in pink below:
How many sides does S_{1864} + S_{1865} + … + S_{1909} have?