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Compound of Icosidodecahedron and Dual

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This is a compound of the icosidodecahedron and its dual, the rhombic triacontahedron. Notice how each vertex of one sits above a face of the other, and that edges from each cross at right-angles in pairs. The points where they cross lie on the shared midsphere of the two polyhedra, i.e. the edges are tangent to the midsphere at those points.

The model may be constructed in Great Stella by adding the icosidodecahedron to its dual via the menu item "Poly>Add Base Model and Dual", or equivalently by going to the compound of base & dual view, and clicking the left-and-down button at the top of that view to use this compound as the new base model. Nets may then be displayed and printed.

Some nets for the rhombic triacontahedron's pentagonal and triangular peaks.
Just sticking pyramid tips onto the faces of either the icosidodecahedron or its dual won't lead to the cleanest result. But here's a clever way to build a paper model, based on advice from Fr. Magnus Wenninger's book Polyhedron Models. Pieces may be made as shown, with long tabs to hold parts together that only touch at a point in the final model. These tabs span across potential weak points in the model where edges of the two polyhedra cross, keeping them in alignment and strengthening the model. To print these hollow faces, start with the icosidodecahedron and use "Poly→Subdivide Faces" with a value of 2 so that points are printed half way along each edge. I also printed smaller pentagons to fit inside and add further strength. Here's one of each different type of net used.
The pieces go together to form these pentagonal and triangular parts.
Attach the parts together like faces of an icosidodecahedron.
Half done.
Getting there.
I put a pentagonal part in last. It all closed up nicely.

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