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I hope this is not too trivial for this forum. I was wondering if someone has come across this polytope.

You start with the rhombic dodecahedron, subdivide it into four parallellepipeds, enter image description here

and then fill the space between the four parallellepipeds with a tetrahedron, six parallellepipeds and four prisms (hopefully I counted correctly), so as to obtain a convex polytope.

Does this have a name? could someone provide a link to a picture?

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4 Answers 4

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Is this it?
           Rhombic Dodecahedron + Tetrahedron

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  • $\begingroup$ wow thanks. i think so, although it would help a lot if it could be seen from other perspectives, or even exploded according to the above subdivision. $\endgroup$ Feb 18, 2013 at 8:47
  • $\begingroup$ This doesn't look right to me. I would expect faces that are adjacent to equilateral triangles to all be rectangles. In your left picture, they look more like rhombi... $\endgroup$ Feb 24, 2013 at 21:45
  • $\begingroup$ @André: I don't doubt you. I formed this as the Minkowski sum of a rhombic dodecahedron and a regular tetrahedron, but perhaps I did not scale one with respect to the other appropriately. Cannot check now... $\endgroup$ Feb 25, 2013 at 0:44
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It's the Minkowski sum of the rhombic dodecahedron with a regular tetrahedron. (The rhombic dodecahedron is it self the Minkowski sum of four segments).

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  • $\begingroup$ indeed. have you casually seen a picture of it with the decomposition written in the question? $\endgroup$ Feb 16, 2013 at 16:13
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It seems to ressemble the "Self-Dual Icosioctahedron #4" :

http://dmccooey.com/polyhedra/SelfDualIcosioctahedron4.html

Some code:

sage: P = polytopes.rhombic_dodecahedron()
sage: Q = polytopes.tetrahedron()
sage: R = P.minkowski_sum(Q)
sage: R.f_vector()
(1, 28, 54, 28, 1)
sage: R.plot()
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The above mentioned Minkowski sum is just an example of Alicia Boole Stott's expansion operation(s), cf. https://en.wikipedia.org/wiki/Expansion_(geometry)

--- rk

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