A Cluster of Thirteen Rhombic Dodecahedra, and Three Other Related Polyhedra

13 Rhombic Dodeca

One of the thirteen rhombic dodecahedra in this cluster cannot be seen, for it is hidden in the middle. The other twelve are each attached to a face of the central rhombic dodecahedron.

If one then creates the convex hull of this cluster — the smallest convex polyhedron which can contain it — this is the result:

Convex hull before TTMFR

This polyhedron has fifty faces:  the six square faces of a cube, the eight triangular faces of an octahedron, the twelve rhombic faces of a rhombic dodecahedron, and twenty-four rectangles to fill the gaps between the other faces.

This fifty-faced polyhedron also has an interesting dual, with 48 faces, all of which are kites. Half of these 48 kites are of one type, and arranged into eight panels of three kites each, while the other half are arranged into six panels of four kites each:

48 kites

Returning to the fifty-faced polyhedron, two images above, here is what happens if one tries to make each face as regular as possible:


In this polyhedron, the six squares are still squares, the eight triangles are still regular, and the twelve rhombi are still rhombi, although these rhombi are wider than before. The 24 rectangles, however, have now been transformed into isosceles trapezoids.

[Software credit:  see http://www.software3d.com/Stella.php for more information about Stella 4d, the program I use to make these rotating images. A free trial download is available at that website.]

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