The title says it all: Is there a (regular) icosahedron containing a rational point on each of its face?

For other Platonic solids, the affirmative answer is easy. Indeed, regular tetrahedra, cubes, and octahedra may have all their vertices rational. A dodecahedron cannot, but 8 of its vertices form a cube, and each face contains an edge of that cube as its diagonal.

The title says it all: Is there a (regular) icosahedron containing a rational point on each of its faces?

For other Platonic solids, the affirmative answer is easy. Indeed, regular tetrahedra, cubes, and octahedra may have all their vertices rational. A dodecahedron cannot, but 8 of its vertices form a cube, and each face contains an edge of that cube as its diagonal.

The title says it all: Is there a (regular) icosahedron containing a rational point on each its face?

For other Platonic solids, the affirmative answer is easy. Indeed, regular tetrahedra, cubes, and octahedra may have all their vertices rational. A dodecahedron cannot, but 8 its vertices form a cube, and each face contains an edge of that cube as its diagonal.

The title says it all: Is there a (regular) icosahedron containing a rational point on each of its face?

For other Platonic solids, the affirmative answer is easy. Indeed, regular tetrahedra, cubes, and octahedra may have all their vertices rational. A dodecahedron cannot, but 8 of its vertices form a cube, and each face contains an edge of that cube as its diagonal.