Here is this polyhedron’s dual.
I made these using Stella 4d, a program you can try for free at http://www.software3d.com/Stella.php.
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RobertLovesPi.net 
Beautiful work, both of them
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Thank you!
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Dear Robert,
Your models are impressing, indeed! In fact, recently I was searching in the Internet for a ready answer to my question: does a polyhedron with two heptagonal bases and 2×7 pentagons on the two lower “floors” exist? I hope you have the answer to my question. I understand, that there is an additional problem: will the vertices of the pentagons be coplanar? I don’t know. Eh, I found your construction. Also very interesting. However, I have a next question: would you be able to modify the picture of the “barrel” so to obtain maximally near to the “regularity” triangles and quadrupoles? If you succeed, you will offer a new very original present to the designers and… mathematicians.
Thank you advance!
Yours sincerely,
Vladimir (Bulgaria)
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If I can find such a polyhedron, you’ll see it here first!
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Thank you for the fast reply! I will be waiting. In any case, the answer will be unexpectable: will we obtain planar pentahedra? I am remaining in waiting.
Vladimir
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To be the first? Thank you sincerely, that one of the very rear occasions to be the first!
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Thank you! It will be a happy chance for me, but for all our readers, too.
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Drop the net
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Sorry, but so far I have been unabe to re-create this polyhedron so I can examine its net.
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