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# Unfolding Solid Polycubes

Bastien Harismendy, Tom da Silva, Gaëlle Largeteau-Skapin, Éric Andrès

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Bastien Harismendy, Tom da Silva, Gaëlle Largeteau-Skapin, Éric Andrès. Unfolding Solid Polycubes. 2025. hal-05165047v2

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Unfolding Solid Polycubes

Bastien Harismendy, Tom DaSilva, Gaëlle Largeteau-Skapin², Eric Andres²

University of Poitiers, Laboratory XLIM, ASALI, UMR CNRS 7252, BP 30179,
F-86962 Futuroscope Chasseneuil, France

{gaëlle.largeteau.skapin, eric.andres}@univ-poitiers.fr

**Abstract.** A solid polycube is a face-connected set of unit cubes, where, unlike classical definitions for polycubes, shared faces between adjacent cubes are not removed. We prove by mathematical induction that any such polycube can be edge-unfolded into a 2D net without refinement. Our proof relies on the concept of a perfect net, defined as a net in which two designated edges are placed on the far left and far right sides. This configuration enables consistent gluing of nets in a single direction throughout the induction process, thereby guaranteeing that no overlaps occur at any step.

**Keywords:** Discrete geometry · solid polycube · edge unfolding

# 1 Introduction

Folding and unfolding encompass distinct classes of geometric problems with diverse applications. Folding problems, such as linkage folding, arise in contexts like protein folding [4] and robotic motion planning [16]. Origami, the art of folding paper, has inspired applications ranging from deployable space structures [3] to medical devices [11]. In this work, we focus on a third category: the unfolding of polyhedra, and in particular, polycubes. This problem, with historical roots dating back to Dürer’s 16th-century illustrations of polyhedral nets [10], finds modern applications in domains such as art [2], manufacturing (e.g., sheet metal fabrication) [17], and computer graphics (e.g., surface texturing) [12].

The problem of unfolding three-dimensional shapes into the plane is a long-standing topic in discrete and computational geometry, with deep connections to combinatorics, algorithm design, and theoretical geometry [13, 14]. In this paper we are considering polycubes (finite, face-connected arrangements of unit cubes in $\mathbb{Z}^3$). A central question is whether a polycube can be unfolded into a single, connected, non-overlapping planar net using cuts along its (cube) edges [1, 6, 7, 8, 9, 14].

In the classical formulation, polycubes are treated as surface-only objects: when two unit cubes are adjacent, their overlapping face are typically omitted from the surface. A longstanding conjecture posits that every such polycube can be unfolded into a 2D net via edge cuts, without refinement (without cut inside the face) [15]. However, despite partial progress, a general proof remains elusive.
In this paper, we consider a more structured variant: solid polycubes, where all faces of the constituent cubes, including those between adjacent cubes, are preserved. There are several, reasons to consider solid polycubes: in the classical scheme, where only outer surface faces are considered, cavities disappear when refolding the object. Another point of interest is when considering higher dimensional polycubes: for instance, when unfolding a tesseract into the 3D cross represented in the famous painting by Dali [5], there are 14 overlapping faces that should not be simply removed if we want to keep the possibility to refold into a tesseract.

While this model introduces additional faces, it also enables an inductive proof of unfoldability. Specifically, we show that every solid polycube can be unfolded into a 2D net using only edge cuts, by induction on the number of cubes. The key idea relies on the notion of perfect nets. A perfect net is defined as a net in which two designated edges are placed on the far left and far right sides. This allows gluing of smaller perfect nets in a consistent, non-overlapping way. Our result proves that solid polycubes can always be unfolded.

## 2 Preliminaries and basic notions

In this paper, we are considering unit cubes embedded in a 3D grid, centered on $Z^3$. Two cubes may not have the same coordinates, therefore each cube can be identified, without ambiguity, by its coordinates. The face of a cube is oriented so that its normal points outside of its cube. The face of a cube can then be identified without ambiguity by the cube it belongs to and the 3D coordinates of its center where two of them are integers and one is a half-integer ($x \pm \frac{1}{2}, x \in Z$). At most two faces may overlap. In the same way, an edge of a face can be identified without ambiguity by the face it belongs to and the 3D coordinates of its center where one is integer and two are half-integers. Up to eight edges may overlap (see Fig.1).

**Definition 1.** Faces that have the same coordinates are called overlapping faces. If $f$ is such a face, we denote $\bar{f}$ the face that overlaps $f$. A face without overlap is called a free face.

**Definition 2.** Two cubes are said to be face-connected if they have faces that overlap. A solid polycube is a face-connected set of unit cubes.

Contrary to classical definitions of polycubes, when two cubes are face-connected, the connecting faces are not removed.

**Definition 3.** Edges that have the same coordinates are called overlapping edges. This set contains at least *e* and the overlapping edge denoted *ê* that belongs to the same cube (red and blue edges in Fig. 1). Edges *e* and *ê* are called twin edges, and the set of both these edges is denoted [*e*]. If the edge *e* belongs to a face *f* that is not free, then we denote $\bar{e}$ the overlapping edge belonging to $\bar{f}$ (red and green edges in Fig. 1). The edge $\bar{e}$ is called the face-overlap edge of *e*.
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** The two orange faces $f$ and $\bar{f}$ are overlapping. The overlapping edge set is formed of all the edges with same coordinates (blue, red, green and yellow). The red edge $e$ and the green edge $\bar{e}$ are called face-overlap edges. The red edge $e$ and the blue edge $\bar{e}$ are called twin edges and form the set $[e]$.</figcaption></figure>

### 3 Unfolding a Solid Polycube using Perfect Nets

We are now considering the embedding of all the faces of a solid polycube into a 2D grid while preserving their orientation. The normals of the faces in the net (if we imagine the net being embedded in 3D) should all point in the same direction. Then, similarly to 3D, an embedded face can be uniquely identified by its 2D integer coordinates since no two net faces may have the same coordinates. An edge is uniquely identified by the face it belongs to and by 2D coordinates with an integer and a half-integer coordinate. The orientation of an edge results directly from the orientation of the face.

**Definition 4.** Two faces in 2D are said to be edge-connected if they have an edge with the same coordinates, called overlapping net edges, that correspond to overlapping edges in the solid polycube.

No two faces may share the same coordinates, meaning that they must not overlap, except possibly along shared edges. Note that in a net, two faces may have edges that overlap geometrically without being edge-connected in the original 3D polycube, i.e. the overlapping edges in 2D do not belong to the same edge equivalence class in the 3D structure. In this case both faces are not considered edge-connected in the net.

**Definition 5.** The net $\overline{P}$ of a solid polycube $P$ is an edge-connected set of all the faces of $P$ embedded in a 2D grid without overlap.

Our goal is to construct a net obtained by unfolding a solid polycube. To achieve this, we introduce the notion of perfect nets and prove that they can be combined to form larger nets corresponding to larger polycubes.
**Definition 6.** Let $P$ be a polycube $P$ and $a$ and $b$ two of its edges. We denote $\overline{P}_{a,b}$ the net of $P$ such that the embedded edges $a$ and $b$ in the net are positioned vertically and such that there is no face of $P$ embedded left of $a,b$ (abscissa less than $\min(a_x, b_x)$) nor right of $a,b$ (abscissa greater than $\max(a_x, b_x)$). We say that $\overline{P}_{a,b}$ is a perfect net for $a$ and $b$ and $(a,b)$ is called a perfect couple (see left of Fig. 2).

An edge in the net is positioned vertically if its abscissa is a half-integer and its ordinate an integer. Note that for a perfect couple, the left edge is always oriented upwards (in the +Y direction), while the edge on the right has the opposition orientation (as illustrated in Fig. 2).

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** An example of a perfect net $\overline{P}_{a,b}$ for edges $a$ (orange, left) and $b$ (yellow, right). The right image show a rotated version,i.e. the perfect net $\overline{P}_{b,a}$.</figcaption></figure>

The first property of such nets that we wish to highlight is a symmetry property:

**Lemma 1.** Let $P$ be a solid polycube. If there exists a perfect net $\overline{P}_{a,b}$ then there exists a perfect net $\overline{P}_{b,a}$.

*Proof.* The perfect net $\overline{P}_{a,b}$ can be rotated by $180^\circ$ to obtain the perfect net $\overline{P}_{b,a}$ (see Fig. 2). $\square$

The following proposition formalizes how two perfect nets can be combined when two solid polycubes are joined along a face:

**Proposition 1.** [*Gluing property*] Let $P$ and $Q$ be two solid polycubes that can be joined into a solid polycube $R$ by overlapping a face $f \in P$ and a face $\bar{f} \in Q$. Let $\alpha \in f$ and $\bar{\alpha} \in \bar{f}$ be two face-overlapping edges in $R$. If there exists a perfect net $\overline{P}_{a,\alpha}$ and a perfect net $\overline{Q}_{\bar{\alpha},b}$ then a perfect net $\overline{R}_{a,b}$ can be constructed.
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** The perfect net $\overline{P}_{a,\alpha}$ on the left, the perfect net $\overline{Q}_{\bar{\alpha},b}$ on the right. Joining $\alpha$ and $\bar{\alpha}$ merges both nets into the perfect net $\overline{R}_{a,b}$.</figcaption></figure>

*Proof.* The proof is illustrated in Fig. 3. The perfect property ensures that joining $\alpha$ and $\bar{\alpha}$ does not cause an overlap in the resulting glued perfect net.

Note that the orientation of $\alpha$ and $\bar{\alpha}$ is not problematic since $R$ is a solid polycube, and since nets preserve the orientation of faces, both edges are oriented correctly so that they can be glued (as illustrated by Fig. 5). $\square$

To initiate the induction for constructing the nets, we start by stating that solid polycubes made of one or two cubes can be unfolded and have perfect nets for any pair of edges as long as they don't belong to the same face.

**Lemma 2.** For a cube $C$, there exists a perfect net $\overline{C}_{a,b}$ for all couple of edges $(a, b)$ of $C$ where $a$ and $b$ are not on the same face.

*Proof.* The proof is simply done by enumeration. Fig. 4 shows perfect nets for all the couples ((edge1, Face 1), (edge $i$, Face $j$)), where (edge $i$, Face $j$) is an edge of the cube that is not on Face 1. By symmetry, all other configurations can be constructed. $\square$

**Lemma 3.** For a polycube $P$ formed of two cubes $A$ and $B$, and two arbitrary edges $a \in A$ and $b \in B$, there exists a perfect net $\overline{P}_{a',b'}$ where $a' \in [a]$ and $b' \in [b]$.

*Proof.* Let $f \in A$ and $\bar{f} \in B$ be the overlapping faces that connect $A$ and $B$. If $a \in f$, resp. $b \in \bar{f}$, we consider $a' = \dot{a}$, resp. $b' = \dot{b}$, otherwise, $a' = a$ and $b' = b$.

Let $\alpha$ be an edge of $f$ and $\bar{\alpha} \in \bar{f}$. Lemma 2 ensures that there exists a perfect net $\overline{A}_{a',\alpha}$ and $\overline{B}_{\bar{\alpha},b'}$. We then use proposition 1 to combine them to obtain $\overline{P}_{a',b'}$. $\square$
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Perfect nets with edge 1 on Face 1, and the other edges of the cube (that are not on Face 1).</figcaption></figure>

Fig. 5 illustrates Lemma 3 for a two-cube solid polycube composed of cubes $F$ and $G$: $F$ has a perfect net with edges (8F2, 9F5), while $G$ has a perfect net with edges (10G6, 1G1). Edges 9F5 and 10G6 are overlapping, allowing both nets to be assembled without overlap into a perfect net of the combined polycube.

We now state our main theorem which establishes that for any solid polycube and any two edges belonging to distinct cubes within it, there exists a perfect net for those edges.

**Theorem 1.** Let $P$ be a solid polycube and let $a$ and $b$ be two edges belonging to two distinct cubes $A$ and $B$ of $P$. Then, there exists a perfect net $\overline{P_{a',b'}}$ with $a' \in [a]$ and $b' \in [b]$.

*Proof.* Let us suppose that the theorem is verified for all solid polycubes of $n$ cubes or less. We know already that it is verified for 2 cubes (see Lemma 3), and without the restriction about edges belonging to two different cubes, for one cube (see Lemma 2).

Let $Q$ be a solid polycube consisting of $n$ cubes, with $n > 1$, and let $f$ be a free face of $Q$ to which we can attach a new cube $C$ such that $Q \cup C = P$ is also a solid polycube of $n + 1$ cubes. Let $A$ and $B$ be two distinct cubes of $P$, and two edges $a \in A$ and $b \in B$ two edges.

– case 1: $a \in Q$ and $b \in C$.

The proof follows the same reasoning as Lemma 3.

If $a \in f$, resp. $b \in \bar{f}$, we consider $a' = \dot{a}$, resp. $b' = \dot{b}$, otherwise, $a' = a$ and $b' = b$. Let $\alpha$ be an edge of $f$. $Q$ is composed of $n$ cubes and, by hypothesis, there exists a perfect net $\overline{Q_{a',\alpha}}$. $C$ is a single cube and we know from
<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Two cubes with perfect nets $\overline{F_{8F2,9F5}}$ for the cube F and $\overline{G_{10G6,1G1}}$ for the cube G.</figcaption></figure>

Lemma 2 that there is a perfect net $\overline{C_{\bar{\alpha},b'}}$ since by construction $\bar{\alpha}$ and $b'$ do not belong to the same face. Proposition 1 allows to build the perfect net $(Q \cup C)_{a',b'} = \overline{P_{a',b'}}$.

- case 2: $a \in Q$ and $b \in Q$.

Let us consider $P \setminus A$. This object can have up to six different connected components (face-connected to the six faces of A). Let us denote $P'$ the connected component containing C. If $P \setminus A$ has only one connected component or if $b \in P'$ then we consider the two solid polycubes $P'$ and $P \setminus P'$. If $P \setminus A$ has more than one connected component and $b \in P'' \neq P'$, i.e. b is not in the same connected component as C, then we consider $P''$ and $P \setminus P''$.

In either cases, we obtain two solid polycubes $P1$, with $a \in P1$, and $P2$, with $b \in P2$. By construction, the polycubes $P1$ and $P2$ can be joined with an overlapping face $g$ of A (see Fig. 6), and both these polycubes are composed of less than $n$ cubes. They therefore have perfect nets for every arbitrary pair of edges that do not belong to the same face.

If $a \in g$, resp. $b \in \bar{g}$, we consider $a' = \dot{a}$, resp. $b' = \dot{b}$, otherwise, $a' = a$ and $b' = b$. Let $\alpha$ be an edge of $g$, then, by hypothesis, there exists two perfect nets $\overline{P1}_{a',\alpha}$ and $\overline{P2}_{\bar{\alpha},b'}$ of $P1$ and $P2$. Proposition 1 allows to construct the perfect net $\overline{P1} \cup \overline{P2}_{a',b'} = \overline{P_{a',b'}}$.

- case 3: $a \in C$ and $b \in Q$

This case is symmetric to case 1 and can be handled in the same way using Lemma 1 to obtain a perfect net.

□
<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Case 2: $A \in Q$ and $B \in Q$. Top: $B$ belongs to the same connected component as $C$. Bottom: $B$ belongs to a different connected component as $C$. In both cases we can define $P1$ and $P2$ to build the net of $P$.</figcaption></figure>

# 4 Conclusion and Future Work

We studied solid polycubes, face-connected assemblies of unit cubes, in which all faces, including internal ones, are retained. Our main result shows that every solid polycube admits an edge-unfolding into a single, connected, non-overlapping planar net, without refinement. The proof proceeds by induction on the number of cubes and leverages the notion of perfect nets, thereby establishing existence for any number of cubes.

This result is non-constructive: it guarantees the existence of such an unfolding but does not yield an explicit algorithm to produce it. Developing a version of this proof, capable of producing an unfolding for any given solid polycube, remains an important direction for future research.

Another key perspective is to explore whether the extra internal faces used in the solid model can be eliminated after or during unfolding, while maintaining connectedness and avoiding overlap. However, considering the solid polycube net, the removal of overlapping faces poses a significant difficulty. For example, a 3×3×3 polycube has 162 faces in total, 108 of which overlap and must therefore be discarded without disconnecting the net. This may represent a challenge.

It may also be possible to develop an inductive proof for unfolding classical polycubes, similar to the approach proposed for solid polycubes, i.e., without relying on internal faces.

Addressing either of these problems maybe key toward resolving the long-standing conjecture that all surface-only polycubes can be unfolded via edge cuts without refinement.
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