Minkowski Sums

Let $S_n$ be the regular $n$-sided polygon – or shape – whose vertices

$v_k$ ($k = 1, 2, \dots, n$) have coordinates:
\begin{align}
x_k &= \cos((2k – 1)/n \times 180^\circ)\\
y_k &= \sin((2k – 1)/n \times 180^\circ)
\end{align}

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 $S_{1864} + S_{1865} + \cdots + S_{1909}$ have?

The Minkowski sum of two regular polygons, $S_k$ and $S_l$, shares the same vertices as $S_p$ where $p$ is the least common multiple (LCM) of $k$ and $l$.

Intuitively, the vertices of the resultant regular polygon will be all the possible sums of two vertices, one from each of the two original regular polygons, which creates the count of vertices that is the LCM of the number of vertices of the two original polygons.

So the Minkowski sum of two regular polygons doesn’t necessarily create a polygon with $k + l$ vertices, but rather with $\mbox{LCM}(k,l)$ vertices. Therefore, the number of sides in the Minkowski sum $S_{1864} + S_{1865} + \cdots + S_{1909}$ would be the LCM of all integers from 1864 to 1909.

The key is to find the prime factors of all these numbers, and take the largest power of each of these primes. That would give us the LCM.

The largest prime number less than $\sqrt{1909}$ is 43, so we need to find the prime factors of 1864 to 1909 only up to 43.

Primes up to 43 are: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43$.

Among these, the primes that divide at least one number from 1864 to 1909 are: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41$. $43$ will not divide any of the numbers in this range.

Multiply the largest power of these primes found in the range 1864-1909:

* $2^{10} = 1024$
* $3^5 = 243$
* $5^3 = 125$
* $7^2 = 49$
* $11^2 = 121$
* $13^1 = 13$
* $17^1 = 17$
* $19^1 = 19$
* $23^1 = 23$
* $29^1 = 29$
* $31^1 = 31$
* $37^1 = 37$
* $41^1 = 41$

This gives the least common multiple of integers from 1864 to 1909 to be:

$1024 \times 243 \times 125 \times 49 \times 121 \times 13 \times 17 \times 19 \times 23 \times 29 \times 31 \times 37 \times 41 = 932568679041686476800$,

which is the number of sides in the Minkowski sum $S_{1864} + S_{1865} + \cdots + S_{1909}$.

More Answers: