I suspect that is indeed possible, but I have yet to identify the best way to create the geometry for fudged puzzles, Oskar seems to be the champ at that! He made the

Illegal Cube work, and it seems like it should require more fudging than this.

It is worth mentioning that a few puzzles have used this shape before.

First was

Timur's Skyglobe, which turns about the hexagonal faces - although fudged, this puzzle does not jumble

Then came

David Pitcher's Jumball, which turns about the irregular pentagonal faces, making it a jumbling puzzle

The dual, by the way, looks like this:

Attachment:

TruncatedTriakisTetrahedronDual.png [ 23.3 KiB | Viewed 746 times ]
It looks a lot like an icosahedron at first, but notice that there is one point in view where 6 triangles meet at a corner, instead of the usual 5. There are 28 faces and 16 points. 4 of those points are surrounded by 6 triangles, the rest by 5 triangles (exaclty one of which is regular). There are 4 regular triangles and the other 24 are all the same isosceles triangle: approx 64.55, 64.55, 52.89 degrees. Looking at it after being so used to seeing icosahedrons is a little off-putting, you can feel that the geometry is slightly off but it takes a little effort to conciously spot the illegal 6-fold symmetries (local only, the global symmetry is only 3-fold). It is a bit reminiscent of a

Geodesic dome, though I am not sure if it has a specific name - I was unable to find it on Wikipedia.

Using the

Conway Polyhedron Notation, I believe it should be called a Hexakis Truncated Tetrahedron

You should ask Oskar to make it!

Peace,

Matt Galla