The deltoidal icositetrahedron is a Catalan solid, a category of polyhedra which constitute the duals of Archimedean solids. This particular one is composed of 24 kites, and is the dual of the rhombicuboctahedron. Its geometry is also related to the cube and octahedron, as it can be seen as 6 groups of 4 units (cubic) or 8 groups of 3 units (octahedral).
This polyhedron requires 24 uncut rectangles in dimensions 3:4, preferably a crisp, medium-weight paper like elephant hide, Stardream, or Tant. Makes a great ornament for the holidays!