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Polycube

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Polycube

The seven free tetracubes
A chiral pentacube
Puzzle with a unique solution

A polycube is a solid figure formed by joining one or more equal cubes face to face. Polycubes are the three-dimensional analogues of the planar polyominoes. The Soma cube, the Bedlam cube, the Diabolical cube, the Slothouber–Graatsma puzzle, and the Conway puzzle are examples of packing problems based on polycubes.[1]

Contents

  • Enumerating polycubes 1
  • Symmetries of polycubes 2
  • Properties of pentacubes 3
  • References 4
  • External links 5

Enumerating polycubes

Like polyominoes, polycubes can be enumerated in two ways, depending on whether chiral pairs of polycubes are counted as one polycube or two.[2] For example, 6 tetracubes have mirror symmetry and one is chiral, giving a count of 7 or 8 tetracubes respectively.[3] Unlike polyominoes, polycubes are usually counted with mirror pairs distinguished, because one cannot turn a polycube over to reflect it as one can a polyomino. In particular, the Soma cube uses both forms of the chiral tetracube.

Polycubes are classified according to how many cubical cells they have:[4]

n Name of n-polycube Number of one-sided n-polycubes
(reflections counted as distinct)
(sequence A000162 in OEIS)
Number of free n-polycubes
(reflections counted together)
(sequence A038119 in OEIS)
1 monocube 1 1
2 dicube 1 1
3 tricube 2 2
4 tetracube 8 7
5 pentacube 29 23
6 hexacube 166 112
7 heptacube 1023 607
8 octocube 6922 3811

Polycubes have been enumerated up to n=16.[5]

Symmetries of polycubes

As with polyominoes, polycubes may be classified according to how many symmetries they have. Polycube symmetries (conjugacy classes of subgroups of the achiral octahedral group) were first enumerated by W. F. Lunnon in 1972. Most polycubes are asymmetric, but many have more complex symmetry groups, all the way up to the full symmetry group of the cube with 48 elements. Numerous other symmetries are possible; for example, there are seven possible forms of 8-fold symmetry [3]

Properties of pentacubes

Twelve pentacubes are flat and correspond to the pentominoes. Of the remaining 17, five have mirror symmetry, and the other 12 form six chiral pairs. The types of the flats are 5-1-1, 4-2-1, 3-3-1, 3-2-1. The 3-D types are 4-2-2, 3-2-2, 2-2-2.[6]

A polycube may have up to 24 orientations in the cubic lattice, or 48 if reflection is allowed. Of the pentacubes, two flats (5-1-1 and the cross) have mirror symmetry in all three axes; these have only three orientations. Ten have one mirror symmetry; these have 12 orientations. Each of the remaining 17 pentacubes has 24 orientations.

Two typical puzzles with pentacubes are to fill a 5×5×5 box with 25 different pentacubes, and to pack all 29 pentacubes into a 9×9×3 box. In the first puzzle, the first piece may be placed in over 2,000 ways. More puzzles may be found at sourceforge.net and by following the links below.

References

  1. ^ Weisstein, Eric W. "Polycube." From MathWorld
  2. ^ "Enumeration of Specific Classes of Polycubes", Jean-Marc Champarnaud et al, Université de Rouen, France PDF
  3. ^ a b Lunnon, W. F. (1972). "Symmetry of Cubical and General Polyominoes". In Read, Ronald C. Graph Theory and Computing. New York: Academic Press. pp. 101–108.  
  4. ^ Polycubes, at The Poly Pages
  5. ^ "Dirichlet convolution and enumeration of pyramid polycubes", C. Carré, N. Debroux, M. Deneufchâtel, J. Dubernard, C. Hillairet, J. Luque, O. Mallet; November 19, 2013 PDF
  6. ^ Aarts, Ronald M. "Pentacube." From MathWorld

External links

  • An actual wooden hexacube built by Kadon
  • Polycube Symmetries
  • Polycube solver Program (with Lua source code) to fill boxes with polycubes using Algorithm X.

Recreational mathematics

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