Class 14: Hinged Dissections  Summary and Q&A
TL;DR
Two polyhedra can be dissected into each other if their volumes and Dehn invariants match, but the algorithmic details and efficient solutions for 3D dissections are still open problems.
Questions & Answers
Q: What is the Dehn invariant in 3D dissections?
The Dehn invariant is obtained by summing the lengths and angles of the edges of a polyhedron, with the condition that rational multiples of pi do not change the invariant.
Q: Are there efficient algorithms for determining if two polyhedra can be dissected into each other?
The decidability of matching Dehn invariants was proven in 2008, but there is no known efficient algorithm for determining the dissections or finding the actual dissections.
Q: Can hinged dissections be implemented in 3D?
Hinged dissections rely on the ability to dissect polyhedra, so if an efficient algorithm is found for 3D dissections, it can be extended to hinged dissections in 3D as well.
Q: What is the current state of research in 3D dissections?
The details and efficient algorithms for 3D dissections are still open problems, with the focus on determining if two polyhedra can be dissected into each other and finding the actual dissections.
Summary & Key Takeaways

3D dissections involve dissecting one polyhedron into another, and the volumes and Dehn invariants must match for a valid dissection.

The Dehn invariant is obtained by summing the lengths and angles of the edges of a polyhedron.

Efficient algorithms for determining whether two polyhedra have matching volumes and Dehn invariants, as well as finding the actual dissections, are still unknown.