Twelve heptagonal antiprisms, meeting at triangular faces, can make a ring with only a small gap remaining. It’s a “near miss,” made using Stella 4d, which you can try for free right here.
A Near-Miss to a Ring of Twelve Heptagonal Antiprisms
RobertLovesPi.net
Polyhedra, tessellations, and more.
RobertLovesPi.net 
suddenly it seemed to go backward, but i couldn’t make that perception stick
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