Return to Question

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'.

Note added on November 15th, 2024: Just mentioning 2 earlier posts from the same ballpark: Inside-out dissections of solids and Inside-out dissections of a cube

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'.

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'.

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'.

Note added on November 15th, 2024: Just mentioning 2 earlier posts from the same ballpark: Inside-out dissections of solids and Inside-out dissections of a cube

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, 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'.

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'.