Nested Platonic Solids

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People find complex geometries fascinating when visualized. A shape known as a dodecahedron is often surrounded by simpler ones. For example, a regular icosahedron fits neatly within its boundaries. Another such shape lies inside the former, that of an octahedron, which serves as a perfect complement. It itself can enclose even smaller geometries, one being a tetrahedron - often described as a solid composed of four triangular faces meeting at vertices. Yet further still, another geometric figure resides inside it, known to mathematicians and geometers as a cube with equal square facets meeting edge on every side.