For example, take a cube. Put a corner at the center of each face of a cube, and you get an octahedron, not another cube.
If you place the corners at the centers of the faces of a tetrahedron, you get another tetrahedron. Two tetrahedral superimposed on each other form a curious and notable structure, variously known as a hyperbolic paraboloid, or duo-tet, or a merkabah.
If we look at the surfaces between two edges of superimposed duals, they are a least energy surface. Soap bubbles fit these surfaces, and soap bubbles are an accurate representation of least energy (tension vs. strength) surfaces. These least energy surfaces are also as easy as twisted sidewalks in concrete.
If the screed bars turn through ninety degrees rotation over their unit length, then the edges of the screeded shape assemble into a hyperbolic parabaloid, or duo-tet, or merkabah.