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Faceless octahedral 4x4x4
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An octaehdral 4x4x4 without the triangular faces one would expect.

This transformation is basically a 4x4x4 which was heavily truncated on all eight corners and afterwards extended on all remaining faces. The truncation was so radical in this case that almost nothing was left of the corner pieces. They are still present just to keep the puzzle functional. Extending the remaining pieces to fill the gap was not possible since these pieces would have blocked each other.
This is a physical implementation of Gelatibrain 4.2.4

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