Untitled, wood, hinges, and glue
[8 F, 14 E, 8 V] The gyrobifastigium arises from gluing together two identical triangular prisms with square sides along one square side of each. However, one of the prisms must be turned a quarter turn (90 degrees). All the faces of the gyrobifastigium are regular polygons but the polyhedron is not a Platonic solid, an Archimedean solid, a prism, or an antiprism. A surprising feature of this polyhedron is that copies of it can fill three-dimensional space with no gaps or overlaps.
I attempted to build the entire object with hinges and wood, but realized that connecting the final sides with interior hinges would be challenging. So instead, I created the two separate triangular prisms each without one square side, and then glued them together along a single wooden square.