Return to Answer

The general answer is sometimes Yes, various unfold/refold pairs of convex polyhedra exist, but not always, depending on what is meant by "cut." Here is an example that Erik Demaine, Marty Demaine, Anna Lubiw, and I worked out carefully:

The Latin-cross unfolding of the cube can refold into precisely 23 distinct convex polyhedra, as displayed below (all of the same surface area): the cube, two doubly covered flat quadrilaterals, seven tetrahedra, three pentahedra, each with one or more quadrilateral faces, four hexahedra, and six octahedra:

---

  ![LatinCrossRefoldings][1]
  ^{Fig. 25.30, p.408 in [*Geometric Folding Algorithms: Linkages, Origami, Polyhedra*](http://gfalop.org/), 2007.}

---

Here is a "movie" of one of the refoldings to a tetrahedron:

---

                    ![Latin2Tetra][2]

---

But more generally, in the 2012 paper [Refold Rigidity of Convex Polyhedra](http://cs.smith.edu/~orourke/papers.php#303), we showed that *every* convex polyhedron can be cut open and refolded to an incongruent convex polyhedron.

But if the cutting is restricted to follow edges of the convex polyhedron ("edge unfoldings"), then there are "refold-rigid" polyhedra. For example, each of the 43,380 edge unfoldings of a dodecahedron may only fold back to the dodecahedron.

---

**Added**. I apologize for not initially understanding Morteza's precise question, which asks whether or not, for any two given pair of convex polyhedra $P$ and $Q$ with the same surface area, is it always possible to cut open $P$ and refold to $Q$. This remains an open question as far as I know. It is a version of Open Problem 25.31 (Fold/Refold Dissections) in [*Geometric Folding Algorithms*](http://gfalop.org/).

The general answer is sometimes Yes, various unfold/refold pairs of convex polyhedra exist, but not always, depending on what is meant by "cut." Here is an example that Erik Demaine, Marty Demaine, Anna Lubiw, and I worked out carefully:

The Latin-cross unfolding of the cube can refold into precisely 23 distinct convex polyhedra, as displayed below (all of the same surface area): the cube, two doubly covered flat quadrilaterals, seven tetrahedra, three pentahedra, each with one or more quadrilateral faces, four hexahedra, and six octahedra:

---

  ![LatinCrossRefoldings][1]
  ^{Fig. 25.30, p.408 in [*Geometric Folding Algorithms: Linkages, Origami, Polyhedra*](http://gfalop.org/), 2007.}

---

Here is a "movie" of one of the refoldings to a tetrahedron:

---

                    ![Latin2Tetra][2]

---

But more generally, in the 2012 paper [Refold Rigidity of Convex Polyhedra](http://cs.smith.edu/~jorourke/papers.php#303), we showed that *every* convex polyhedron can be cut open and refolded to an incongruent convex polyhedron.

But if the cutting is restricted to follow edges of the convex polyhedron ("edge unfoldings"), then there are "refold-rigid" polyhedra. For example, each of the 43,380 edge unfoldings of a dodecahedron may only fold back to the dodecahedron.

---

**Added**. I apologize for not initially understanding Morteza's precise question, which asks whether or not, for any two given pair of convex polyhedra $P$ and $Q$ with the same surface area, is it always possible to cut open $P$ and refold to $Q$. This remains an open question as far as I know. It is a version of Open Problem 25.31 (Fold/Refold Dissections) in [*Geometric Folding Algorithms*](http://gfalop.org/).

The general answer is sometimes Yes, various unfold/refold pairs of convex polyhedra exist, but not always, depending on what is meant by "cut." Here is an example that Erik Demaine, Marty Demaine, Anna Lubiw, and I worked out carefully:

The Latin-cross unfolding of the cube can refold into precisely 23 distinct convex polyhedra, as displayed below (all of the same surface area): the cube, two doubly covered flat quadrilaterals, seven tetrahedra, three pentahedra, each with one or more quadrilateral faces, four hexahedra, and six octahedra:

---

  ![LatinCrossRefoldings][1]
  ^{Fig. 25.30, p.408 in [*Geometric Folding Algorithms: Linkages, Origami, Polyhedra*](http://gfalop.org/), 2007.}

---

Here is a "movie" of one of the refoldings to a tetrahedron:

---

                    ![Latin2Tetra][2]

---

But more generally, in the 2012 paper [Refold Rigidity of Convex Polyhedra](http://cs.smith.edu/~orourke/papers.php#303), we showed that *every* convex polyhedron can be cut open and refolded to an incongruent convex polyhedron.

But if the cutting is restricted to follow edges of the convex polyhedron ("edge unfoldings"), then there are "refold-rigid" polyhedra. For example, each of the 43,380 edge unfoldings of a dodecahedron may only fold back to the dodecahedron.

---

**Added**. I apologize for not initially understanding Morteza's precise question, which asks whether or not, for any two given pair of convex polyhedra $P$ and $Q$, is it always possible to cut open $P$ and refold to $Q$. This remains an open question as far as I know. It is a version of Open Problem 25.31 (Fold/Refold Dissections) in [*Geometric Folding Algorithms*](http://gfalop.org/).

The general answer is sometimes Yes, various unfold/refold pairs of convex polyhedra exist, but not always, depending on what is meant by "cut." Here is an example that Erik Demaine, Marty Demaine, Anna Lubiw, and I worked out carefully:

The Latin-cross unfolding of the cube can refold into precisely 23 distinct convex polyhedra, as displayed below (all of the same surface area): the cube, two doubly covered flat quadrilaterals, seven tetrahedra, three pentahedra, each with one or more quadrilateral faces, four hexahedra, and six octahedra:

---

  ![LatinCrossRefoldings][1]
  ^{Fig. 25.30, p.408 in [*Geometric Folding Algorithms: Linkages, Origami, Polyhedra*](http://gfalop.org/), 2007.}

---

Here is a "movie" of one of the refoldings to a tetrahedron:

---

                    ![Latin2Tetra][2]

---

But more generally, in the 2012 paper [Refold Rigidity of Convex Polyhedra](http://cs.smith.edu/~orourke/papers.php#303), we showed that *every* convex polyhedron can be cut open and refolded to an incongruent convex polyhedron.

But if the cutting is restricted to follow edges of the convex polyhedron ("edge unfoldings"), then there are "refold-rigid" polyhedra. For example, each of the 43,380 edge unfoldings of a dodecahedron may only fold back to the dodecahedron.

---

**Added**. I apologize for not initially understanding Morteza's precise question, which asks whether or not, for any two given pair of convex polyhedra $P$ and $Q$ with the same surface area, is it always possible to cut open $P$ and refold to $Q$. This remains an open question as far as I know. It is a version of Open Problem 25.31 (Fold/Refold Dissections) in [*Geometric Folding Algorithms*](http://gfalop.org/).

The general answer is sometimes Yes, but not always, depending on what you mean but "cut." For Yes, here is an example that Erik Demaine, Marty Demaine, Anna Lubiw, and I worked out carefully:

The Latin-cross unfolding of the cube can refold into precisely 23 distinct convex polyhedra, as displayed below (all of the same surface area): the cube, two doubly covered flat quadrilaterals, seven tetrahedra, three pentahedra, each with one or more quadrilateral faces, four hexahedra, and six octahedra:

---

  ![LatinCrossRefoldings][1]
  ^{Fig. 25.30, p.408 in [Geometric Folding Algorithms: Linkages, Origami, Polyhedra](http://gfalop.org/), 2007.}

---

Here is a "movie" of one of the refoldings to a tetrahedron:

---

                    ![Latin2Tetra][2]

---

But more generally, in the 2012 paper [Refold Rigidity of Convex Polyhedra](http://cs.smith.edu/~orourke/papers.php#303), we showed that *every* convex polyhedron can be cut open and refolded to an incongruent convex polyhedron.

But if the cutting is restricted to follow edges of the convex polyhedron ("edge unfoldings"), then there are "refold-rigid" polyhedra. For example, each of the 43,380 edge unfoldings of a dodecahedron may only fold back to the dodecahedron.

The general answer is sometimes Yes, various unfold/refold pairs of convex polyhedra exist, but not always, depending on what is meant by "cut." Here is an example that Erik Demaine, Marty Demaine, Anna Lubiw, and I worked out carefully:

The Latin-cross unfolding of the cube can refold into precisely 23 distinct convex polyhedra, as displayed below (all of the same surface area): the cube, two doubly covered flat quadrilaterals, seven tetrahedra, three pentahedra, each with one or more quadrilateral faces, four hexahedra, and six octahedra:

---

  ![LatinCrossRefoldings][1]
  ^{Fig. 25.30, p.408 in [*Geometric Folding Algorithms: Linkages, Origami, Polyhedra*](http://gfalop.org/), 2007.}

---

Here is a "movie" of one of the refoldings to a tetrahedron:

---

                    ![Latin2Tetra][2]

---

But more generally, in the 2012 paper [Refold Rigidity of Convex Polyhedra](http://cs.smith.edu/~orourke/papers.php#303), we showed that *every* convex polyhedron can be cut open and refolded to an incongruent convex polyhedron.

But if the cutting is restricted to follow edges of the convex polyhedron ("edge unfoldings"), then there are "refold-rigid" polyhedra. For example, each of the 43,380 edge unfoldings of a dodecahedron may only fold back to the dodecahedron.

---

**Added**. I apologize for not initially understanding Morteza's precise question, which asks whether or not, for any two given pair of convex polyhedra $P$ and $Q$, is it always possible to cut open $P$ and refold to $Q$. This remains an open question as far as I know. It is a version of Open Problem 25.31 (Fold/Refold Dissections) in [*Geometric Folding Algorithms*](http://gfalop.org/).