While doodling, I noticed that platonic solids can be mapped onto the plane with many of the edges presented as circles and radial lines, with a few edges as curves and any outer point represented as a circle around the entire image. Superimposed duals look especially nice in this representation. These are rendered with my calligraphic lines library to highlight the vertices.
Centered Solid  Dual  Superimposed Duals 
icosahedron 
dodecahedron 

dodecahedron 
icosahedron 

octahedron 
cube 

cube 
octahedron 

tetrahedron 
tetrahedron 
