This cage is made of 36 identical rhombi, and has six openings. I made it with Stella 4d, a program available at this website.
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RobertLovesPi.net 
How were you able to ‘construct’ the cage with 36 Rhomi? I’ve built a similar one with only hexagons, but I call it a Foil Cube.
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I wish I could remember all the details, but I didn’t make notes. I can tell you that the “holes” were actually made of polygonal faces, making the whole thing a polyhedron, and I created the six “holes” by rendering all non-rhombic faces invisible. I hope that helps.
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