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If a regular hexagonal prism instersects a plane at an angle, what will be the special properties of the hexagonal cross-section?

In particular is there any relationship between the angles?

You can't create any hexagonal shape this way, only certain ones but which ones? (In the same way, a square prism will always give a diamond shaped cross-section.)

Another way of saying this is what is the property of a hexagonal shadow cast by a regular hexagon from an infinitely far light source.

zooby
  • 4,343
  • Hint: parallel projection along $\vec v=(v_x,v_y,v_z)$ onto the $x$-$y$ plane is the mapping $(x,y,z)\mapsto\left(x-\frac{v_x}{v_z}z,y-\frac{v_y}{v_z}z,0\right)$. – amd Aug 09 '17 at 18:36
  • Geometric hints: 1. Parallel planes intersect a third non-parallel plane in a pair of parallel lines whose distance is no smaller than the distance between the original planes. 2. The sum of the interior angles of a hexagon is $4\pi$ (a.k.a. $720^\circ$). – Andrew D. Hwang Aug 09 '17 at 20:26

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