1. ## Polyhedron

Does there exist a polyhedron, whose orthogonal projections on some three planes are a quadrilateral, a hexagon and an octagon?

2. Hello, atreyyu!

Does there exist a polyhedron, whose orthogonal projections on some three planes
are a quadrilateral, a hexagon and an octagon?

Get a large potato
. . and three cookie cutters: square, hexagonal, and octagonal.

Place the potato on the table.

Use the square cutter and push it through the potato from front to back.

Use the hexagonal cutter and push it through the potato from left to right.

Use the octagonal cutter and push it through the potato from top to bottom.

The remaining potato is the desired polyhedron.

3. ^ Are you sure of that?

After the first two pushes we have something like this:

When viewed from top it is a rectangle, when viewed from front it is a hexagon. We now have to push the 8-nal cutter from the left to the right.
But what size should the octogonal cutter be so that the remaining figures stay the same? If it's smaller than the rectangle, then the hexagon will not be hexagon anymore. If it looks somehting like that (view from left):

then the hexagon's corners will be cut off.

Am I right?