• assemblings of triangles and squares
3 triangles and 4 squares: impossible! for the assembling we create an edge by superimposing TWO sides of adjacent faces; the number of sides MUST thus be EVEN (here this number is odd: 3x3 + 4x4 = 25)
|8 triangles and 2 squares
square antiprism (assemble two pieces to create a ring of head to foot triangles)
||2 triangles and 3 squares
||4 triangles and 3 squares
|4 triangles and 5 squares
• shadows of polyhedra
The following polyhedra may produce a square shadow :
- cube, truncated cube, cuboctahedron, regular tetrahedron and octahedron, rhombic dodecahedron,
- prisms with square section, triangular prism with square lateral faces,
- pyramids with square base (projection of the apex inside the base) and square diamonds (assemblings of two such pyramids),
The following polyhedra may produce a regular hexagonal shadow :
- prisms with hexagonal section, cube, regular octahedron, rhombic dodecahedron,
- pyramids with regular hexagonal base (projection of the apex inside the base) and hexagonal diamonds (assemblings of two such pyramids with regular bases).
There are many others! To create new ones for example just add faces which don't modify the shadow.
• cross sections of a cube
• To get a square we need to use a plane parallel to a face of the cube.
• To get an equilateral triangle (resp. a regular hexagon) we need to use a plane orthogonal to a diagonal of the cube which intersects three edges stemming from he same vertex (resp. which goes through the midpoints of six consecutive edges building a "skew ring").
• But we cannot get a regular pentagon. To prove it we use the pigeons holes theorem: If more than n pigeons live in n pigeons holes then there is at least one hole with more than one pigeon. Our "holes" are the three directions of the cube's faces and our "pigeons" are the five sides of the pentagon. Since the intersection of a plane with two parallel planes is a pair of parallel lines, there are two parallel sides in at least one hole. Impossible! the regular pentagon has no parallel sides.
The pentagonal cross sections of a cube have two pairs of parallel sides (in our "holes" there are at most two sides). Likewise the quadrilateral cross sections are trapezoids (at least one "hole" with two sides).