Pentaspheres

Here are all planar pentaspheres. The set is the union of the 22 pentahexes and the 12 pentominoes with the straight piece in both sets. So we get 33 different pieces.

Using only the 22 pieces, which are planar in the hexagonal grid we can make a 3x19 roof and a 4x12 roof or two size 6 tetrahedrons with the top sphere missing.

Two sets of the pieces planar in the hexagonal set have a total volume of 2x22x5 spheres and a size 10 tetrahedron can be made.

Using all 33 planar pieces some multiple constructions are possible. I got three square pyramids and three size 6 tetrahedrons each with a missing corner. For the pyramids I colored the table tennis balls to have three different puzzles.

Here are the layers for the multiple constructions and a single size 9 tetrahedron.

The truncated tetrahedrons are joined at the missing corner.

Adding the non-planar pieces we have 210 pieces, which allow for six 10x6 roofs but glueing all the pieces seems to be rather hard.

The layers are here.

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