# Polycube

A polycube is a solid figure formed by joining unit cubes face to face, the three-dimensional generalization of the two-dimensional polyomino. In graphics and geometry the term also covers the resulting orthogonal polyhedra, whose faces are axis-aligned quadrilaterals meeting at 90-degree dihedral angles with integer coordinates.<sup>[1](https://www.wkbpic.com/wkbx/SA/1969/1969-07-01.pdf)</sup><sup> • </sup><sup>[2](https://www.sciencedirect.com/science/article/pii/S0097849318301717)</sup> [Martin Gardner](https://www.edgepedia.org/martin-gardner) introduced the word to American readers in his July 1969 *Scientific American* column, defining a polycube as a polyhedron formed by joining unit cubes.<sup>[1](https://www.wkbpic.com/wkbx/SA/1969/1969-07-01.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | A connected set of unit cubes joined at their faces; in d dimensions, a connected set of n cells of the d-dimensional hypercubic lattice, connected through (d−1)-dimensional faces<sup>[1](https://www.wkbpic.com/wkbx/SA/1969/1969-07-01.pdf)</sup><sup> • </sup><sup>[3](https://geometry.stanford.edu/lgl_2024/papers/vbs-apfepp-15/vbs-apfepp-15.pdf)</sup> |
| Counts (OEIS A000162) | 1, 1, 2, 8, 29, 166, 1023, 6922, 48311, ... polycubes of n cells, with reflections counted as different<sup>[4](https://oeis.org/A000162)</sup> |
| Largest known counts | 1,636,229,771,639,924 polycubes of 21 cubes (January 2024) and 12,766,882,202,755,783 of 22 cubes (February 2024)<sup>[5](https://philthompson.me/2024/Counting-Polycubes-of-Size-21.html)</sup><sup> • </sup><sup>[6](https://philthompson.me/2024/Counting-Polycubes-of-Size-22.html)</sup> |
| Enumeration status | No formula is known for the number of polycubes of size n in any dimension<sup>[3](https://geometry.stanford.edu/lgl_2024/papers/vbs-apfepp-15/vbs-apfepp-15.pdf)</sup> |
| Classic puzzle | The Soma cube, devised by Piet Hein in 1936, uses the seven non-straight polycubes of three or four cubes to build a 3×3×3 cube<sup>[7](https://www.si.edu/object/polycube-puzzle-soma-cube%3Anmah_1425322)</sup> |
| Graphics use | Polycube maps, introduced by Tarini et al. in 2004, deform 3D models onto polycubes for texture mapping, hexahedral meshing, and grid generation<sup>[8](https://arxiv.org/abs/2402.00652)</sup> |

## Definition and relation to polyominoes

A polycube of size n is a connected set of n cubes in which connectivity runs through whole faces of the cubes, exactly as a polyomino is a connected set of unit squares joined edge to edge. In the formal combinatorial literature, a d-dimensional polycube is a union of d-cells of the cubical tiling of \( \mathbb{R}^d \) whose interior is connected; in two dimensions these objects are called polyominoes and in higher dimensions polycubes.<sup>[9](https://doi.org/10.1007/s00454-022-00474-w)</sup> [Solomon W. Golomb](https://www.edgepedia.org/solomon-w-golomb), a mathematician at the [University of Southern California](https://www.edgepedia.org/university-of-southern-california) who coined the polyomino terminology in 1953, defined a polyomino as a finite rook-connected subset of the square lattice, and later work carried the definition to higher dimensions.<sup>[10](https://par.nsf.gov/servlets/purl/10353957)</sup>

Counting conventions matter. The sequence A000162 counts polycubes in which two shapes that differ by reflection are considered different, so chiral pairs are counted as two; related sequences count free polycubes (A038119), fixed polycubes (A001931), and achiral polycubes (A007743).<sup>[4](https://oeis.org/A000162)</sup><sup> • </sup><sup>[11](https://oeis.org/A038119/a038119_3.pdf)</sup>

## Counting polycubes

Enumeration is the central mathematical problem of the subject. Following Lunnon, the number of polycubes of size n that are proper in d dimensions is written DX(n, d), and no formula is known for \( A_d(n) \), the number of polycubes of size n in d dimensions, for any value of d.<sup>[3](https://geometry.stanford.edu/lgl_2024/papers/vbs-apfepp-15/vbs-apfepp-15.pdf)</sup> Counting therefore proceeds by explicit construction, one cube at a time. Redelmeier's 1981 algorithm for counting polyominoes was generalized to higher dimensions, and an improved version reduced memory consumption from exponential in the dimension to a low polynomial in both n and d.<sup>[12](https://www.sciencedirect.com/science/article/pii/S0012365X0900082X)</sup>

The counts grow quickly. The sequence begins 1, 1, 2, 8, 29, 166, 1023 for n = 1 through 7, and reaches 346,543 at n = 10.<sup>[4](https://oeis.org/A000162)</sup> In January 2024, [Phil Thompson](https://www.edgepedia.org/phil-thompson) computed the n = 21 value, 1,636,229,771,639,924, over ten days on a 64-vCPU Amazon EC2 instance using a Rust port of Stanley Dodds's algorithm; the n = 22 value, 12,766,882,202,755,783, followed in February 2024 on a cheaper spot instance after refactoring the work into millions of smaller resumable tasks.<sup>[5](https://philthompson.me/2024/Counting-Polycubes-of-Size-21.html)</sup><sup> • </sup><sup>[6](https://philthompson.me/2024/Counting-Polycubes-of-Size-22.html)</sup> Both values were added to OEIS A000162, whose entries for n = 17 through 20 had come from Dodds in December 2023.<sup>[4](https://oeis.org/A000162)</sup>

Asymptotically, the number of polyominoes grows like \( \lambda_2^n \) for a growth constant \( \lambda_2 \) whose existence was shown by Klarner and Madras; its exact value remains unknown, with best bounds of roughly 4.0025 and 4.6496.<sup>[3](https://geometry.stanford.edu/lgl_2024/papers/vbs-apfepp-15/vbs-apfepp-15.pdf)</sup> Polycubes also appear in statistical physics, where they model percolation processes and the collapse of branched polymers.<sup>[3](https://geometry.stanford.edu/lgl_2024/papers/vbs-apfepp-15/vbs-apfepp-15.pdf)</sup>

## Puzzles

The best-known polycube puzzle is the [Soma cube](https://www.edgepedia.org/soma-cube), devised in 1936 by the Danish poet and inventor [Piet Hein](https://www.edgepedia.org/piet-hein) (1905–1996) while listening to a lecture on quantum mechanics by [Werner Heisenberg](https://www.edgepedia.org/werner-heisenberg). Its seven pieces represent all the ways three or four cubes can be arranged other than in a straight line, and they assemble into a 3×3×3 cube; the name comes from the drug SOMA in Aldous Huxley's *Brave New World*.<sup>[7](https://www.si.edu/object/polycube-puzzle-soma-cube%3Anmah_1425322)</sup> Gardner introduced the puzzle to American readers in September 1958, and Parker Brothers began marketing it commercially in the summer of 1969 with a 56-page booklet written by Hein, at a listed price of $1.57.<sup>[1](https://www.wkbpic.com/wkbx/SA/1969/1969-07-01.pdf)</sup><sup> • </sup><sup>[7](https://www.si.edu/object/polycube-puzzle-soma-cube%3Anmah_1425322)</sup> An early and popular polycube puzzle more broadly, the Soma cube was mass-marketed in plastic through the 1960s and early 1970s.<sup>[13](https://gamepuzzles.com/polyintr.htm)</sup>

Packing questions extend beyond the Soma set. There are 1,390 distinct ways to pack the eight polycubes of order 4 into a 2×4×4 box, a result due to Beeler's 1972 HAKMEM item 112.<sup>[14](https://mathworld.wolfram.com/Polycube.html)</sup> Later investigators have studied packings such as filling 4×4×4 cubes with combinations of pentacubes and planar heptacubes.<sup>[13](https://gamepuzzles.com/polyintr.htm)</sup>

## Polycubes in computer graphics

In graphics, a polycube is treated as an orthogonal polyhedron with axis-aligned quadrilateral faces, 90-degree dihedral angles, and integer coordinates, and it serves as a simplified base complex for arbitrary 3D shapes.<sup>[2](https://www.sciencedirect.com/science/article/pii/S0097849318301717)</sup> A polycube map is a bijective mapping between a general 3D model and a polycube, first proposed by Tarini et al. in 2004 for seamless texture mapping; the idea has since been applied to hexahedral mesh construction, spline fitting, and multi-block grid generation, including structured CFD gridding.<sup>[8](https://arxiv.org/abs/2402.00652)</sup>

Computing a good polycube base complex for an arbitrary mesh, balancing mapping distortion against singularity counts, is a standing algorithmic challenge. PolyCut (SIGGRAPH Asia 2013) formulated the problem as labeling each input triangle with one of six signed principal axis directions and solved it with a constrained graph-cut optimization embedded in a hill-climbing search.<sup>[15](https://www.cs.ubc.ca/labs/imager/tr/2013/polycut/)</sup> A 2024 approach constructs polycube layouts by iteratively adding or removing loops in the dual representation, and is guaranteed to return a valid layout for models of genus 0.<sup>[8](https://arxiv.org/abs/2402.00652)</sup> Polycube representations also support fabrication: mapping a shape onto a polycube and splitting it into orthogonal parallelepipeds yields a decomposition suited to 3D printing and checkable for 3-axis or 4-axis milling.<sup>[2](https://www.sciencedirect.com/science/article/pii/S0097849318301717)</sup>

## Tiling and folding

Polycubes connect to tiling theory. Gambini and Vuillon investigated minimal polycubes by volume that tile \( \mathbb{R}^3 \) by translation, constructing cube unions with the same numbers of faces as the five Fedorov polyhedra, the convex polyhedra that tile space.<sup>[16](https://journals.sagepub.com/doi/abs/10.3233/FI-2016-1381?download=true)</sup>

Folding links the flat and solid versions of the problem. Aichholzer, Demaine, Eppstein, and colleagues studied when a polyomino can be folded into a polycube, characterizing which polyominoes fold into a single unit cube under several folding models and giving a linear-time dynamic programming algorithm to fold a tree-shaped polyomino into a constant-size polycube.<sup>[17](https://arxiv.org/abs/1712.09317)</sup> One basic question remains open: it is unknown whether every polycube has an edge unfolding, that is, a cutting along cube edges that flattens the surface into a single nonoverlapping polygon.<sup>[18](https://andrewwinslow.com/papers/polyunfold-jcdcggg16.pdf)</sup>

## References

1. [Martin Gardner, "Mathematical Games: Pleasurable problems with polycubes," Scientific American, July 1969](https://www.wkbpic.com/wkbx/SA/1969/1969-07-01.pdf)
2. [Fanni, Cherchi, Muntoni, Tola, Scateni, "Fabrication oriented shape decomposition using polycube mapping," Computers & Graphics 77 (2018)](https://www.sciencedirect.com/science/article/pii/S0097849318301717)
3. [Barequet & Shalah, "Automatic Proofs for Formulae Enumerating Proper Polycubes"](https://geometry.stanford.edu/lgl_2024/papers/vbs-apfepp-15/vbs-apfepp-15.pdf)
4. [OEIS A000162: Number of 3-dimensional polyominoes (polycubes) with n cells](https://oeis.org/A000162)
5. [Phil Thompson, "Counting Polycubes of Size 21" (2024)](https://philthompson.me/2024/Counting-Polycubes-of-Size-21.html)
6. [Phil Thompson, "Counting Polycubes of Size 22" (2024)](https://philthompson.me/2024/Counting-Polycubes-of-Size-22.html)
7. [Smithsonian National Museum of American History, "Polycube Puzzle, SOMA Cube"](https://www.si.edu/object/polycube-puzzle-soma-cube%3Anmah_1425322)
8. [Snoep, Speckmann, Verbeek, "Polycube Layouts via Iterative Dual Loops," arXiv:2402.00652](https://arxiv.org/abs/2402.00652)
9. ["Topology and Local Geometry of the Eden Model," Discrete & Computational Geometry (2022)](https://doi.org/10.1007/s00454-022-00474-w)
10. ["High-Dimensional Holeyominoes," NSF Public Access repository](https://par.nsf.gov/servlets/purl/10353957)
11. ["Counting free polycubes," OEIS A038119 supplementary PDF](https://oeis.org/A038119/a038119_3.pdf)
12. [Aleksandrowicz & Barequet, "Counting polycubes without the dimensionality curse," Discrete Mathematics 309 (2009)](https://www.sciencedirect.com/science/article/pii/S0012365X0900082X)
13. [Kadon Enterprises, "More about polyominoes and polycubes"](https://gamepuzzles.com/polyintr.htm)
14. [Eric W. Weisstein, "Polycube," MathWorld](https://mathworld.wolfram.com/Polycube.html)
15. [Livesu, Vining, Sheffer, Gregson, Scateni, "PolyCut: Monotone Graph-Cuts for PolyCube Base-Complex Construction," ACM TOG 32(6), 2013](https://www.cs.ubc.ca/labs/imager/tr/2013/polycut/)
16. [Gambini & Vuillon, "Tiling the Space by Polycube Analogues of Fedorov's Polyhedra," Fundamenta Informaticae 146(2), 2016](https://journals.sagepub.com/doi/abs/10.3233/FI-2016-1381?download=true)
17. [Aichholzer et al., "Folding Polyominoes into (Poly)Cubes," arXiv:1712.09317](https://arxiv.org/abs/1712.09317)
18. [Winslow, "Some Polycubes Have No Edge Zipper Unfolding," arXiv:1907.08433](https://andrewwinslow.com/papers/polyunfold-jcdcggg16.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › Metric, convex, and discrete geometry*

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