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We record some general questions based on

  • Inside-out dissections of solids

  • Inside-out dissections of a cube

  • Can every convex polyhedral solid be inside-out dissected to a congruent polyhedral solid? If the answer is “yes”, can the number of intermediate pieces required be bounded in terms of the complexity of the polyhedron being dissected? What if we require the intermediate pieces to be all convex?

Obviously, if any tetrahedron can be inside outed (to a tetrahedron congruent to itself), any polyhedron that admits partition into tetrahedrons also can be.

  • If we consider only inside-out dissections (if they exist) of a given convex polyhedral solid into any polyhedron of same volume such that the number of intermediate pieces (required to be a convex) is to be a minimum, is it guaranteed that starting with any initial convex polyhedral solid, the resulting polyhedral solid is also convex? What could be said about those convex solids for which the resulting solid is congruent to itself?

Note: Variants of both questions can be asked with 'totally inside-out' replacing 'inside-out'.

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  • $\begingroup$ You probably know the 2D question was asked on MO: Inside-out polygonal dissections. $\endgroup$
    – Joseph O'Rourke
    Commented Jun 12 at 20:14
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    $\begingroup$ Yes. That one question was the source of all this. Thank you! $\endgroup$
    – Nandakumar R
    Commented Jun 13 at 4:19
  • $\begingroup$ Are inside-out dissections related to hinged dissections? en.wikipedia.org/wiki/Hinged_dissection $\endgroup$
    – Gerry Myerson
    Commented Nov 13 at 0:09
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    $\begingroup$ @GerryMyerson: Most of the hinged dissections I've seen are in-out dissections, but I don't know if this is universal. There is a collection here, but I am not seeing a counterexample. $\endgroup$
    – Joseph O'Rourke
    Commented Nov 13 at 17:05

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There has been progress on both the 2D (link) and the 3D questions:

"On inside-out Dissections of Polygons and Polyhedra." Reymond Akpanya, Adi Rivkin, Frederick Stock. arXiv.

"In this work we study inside-out dissections of polygons and polyhedra. We first show that an arbitrary polygon can be inside-out dissected with 2n+1 pieces, thereby improving the best previous upper bound of 4(n−2) pieces. Additionally, we establish that a regular polygon can be inside-out dissected with at most 6 pieces. Lastly, we prove that any polyhedron that can be decomposed into finitely many regular tetrahedra and octahedra can be inside-out dissected."

Fig.7

In particular, a regular tetrahedron can be inside-out dissected with $34$ pieces.

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  • $\begingroup$ It is quite remarkable that even to inside out dissect a regular tetrahedron, one might need 13 intermediate pieces. It seems that a general tet can be inside outed to a congruent tet but it might be very hard to minimize the number of intermediate pieces! Thank you. $\endgroup$
    – Nandakumar R
    Commented Nov 13 at 10:30
  • $\begingroup$ @NandakumarR: Typo correction: $34$ intermediate pieces. $\endgroup$
    – Joseph O'Rourke
    Commented Nov 13 at 17:21
  • $\begingroup$ Thank you. And one wishes the totally inside out versions of the question also receive attacks - for example, the least number of intermediate pieces to totally inside-out a square, say. $\endgroup$
    – Nandakumar R
    Commented Nov 15 at 7:19

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