Final answer:
To address the question on Platonic solids, one must note that a cube is a type of Platonic solid and has 6 faces, 12 edges, and 8 vertices. Euler's formula for polyhedra can be applied (F + V - E = 2). For the index of refraction, Snell's Law is used with the angles of incidence and refraction to determine the index of refraction for the gas within a hollow cube.
Step-by-step explanation:
To find the missing values for each cube and to make observations about the Platonic solid, we first identify the known measurements and properties of these solids. Platonic solids are highly symmetric, three-dimensional shapes, which are made up of faces with the same number of edges meeting at each vertex. They have identical vertices, edges, and angles, and the only Platonic solids are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
When dealing with a cube, which is one of the Platonic solids, we know that it has 6 faces, 12 edges, and 8 vertices. As we observe a cube, we can say two things about any Platonic solid: (1) The number of faces plus the number of vertices minus the number of edges always equals 2. This is known as Euler's formula for polyhedra (F + V - E = 2). (2) Each face has the same number of edges (in this case, 4 for a cube), and the same number of such edges meet at each vertex (in this case, 3 for a cube).
If you're tasked to determine the index of refraction of the gas inside a hollow cube, the information required would include the angle of incidence and the angle of refraction of a light beam as it enters and exits the plastic and gas. You would use Snell's Law, which relates the angles of incidence and refraction to the indices of refraction of the two media.