Polycube
Encyclopedia
Polyform
In recreational mathematics, a polyform is a plane figure constructed by joining together identical basic polygons. The basic polygon is often a convex plane-filling polygon, such as a square or a triangle. More specific names have been given to polyforms resulting from specific basic polygons, as...
whose base form is a cube. Polycubes are the three-dimensional analogues of the planar polyomino
Polyomino
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling with a connected interior....
es. The Soma cube
Soma cube
The Soma cube is a solid dissection puzzle invented by Piet Hein in 1933 during a lecture on quantum mechanics conducted by Werner Heisenberg. Seven pieces made out of unit cubes must be assembled into a 3x3x3 cube...
and the Bedlam cube
Bedlam cube
The Bedlam cube is a solid dissection puzzle invented by British puzzle expert Bruce Bedlam.- Design :The puzzle consists of thirteen polycubic pieces: twelve pentacubes and one tetracube. The objective is to assemble these pieces into a 4 x 4 x 4 cube...
are examples of packing problem
Packing problem
Packing problems are a class of optimization problems in mathematics which involve attempting to pack objects together , as densely as possible. Many of these problems can be related to real life packaging, storage and transportation issues...
s based on polycubes.
Enumerating polycubes
Like polyominoPolyomino
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling with a connected interior....
es, polycubes can be enumerated in two ways, depending on whether chiral
Chirality (mathematics)
In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...
pairs of polycubes are counted as one polycube or two. For example, 6 tetracubes have mirror symmetry
Reflection symmetry
Reflection symmetry, reflectional symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.In 2D there is a line of symmetry, in 3D a...
and one is chiral
Chirality (mathematics)
In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...
, giving a count of 7 or 8 tetracubes respectively. 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
Soma cube
The Soma cube is a solid dissection puzzle invented by Piet Hein in 1933 during a lecture on quantum mechanics conducted by Werner Heisenberg. Seven pieces made out of unit cubes must be assembled into a 3x3x3 cube...
uses both forms of the chiral tetracube.
| n | Name of n-polycube | n-polycubes (reflections counted as distinct) |
n-polycubes (reflections counted together) |
|---|---|---|---|
| 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 |
Kevin Gong has enumerated polycubes up to n=16. See the external links for a table of these results.
Properties of pentacubes
Twelve pentacubes are flat and correspond to the pentominoesPentomino
A pentomino is a polyomino composed of five congruent squares, connected along their edges ....
.
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.
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.
See also
- polyformPolyformIn recreational mathematics, a polyform is a plane figure constructed by joining together identical basic polygons. The basic polygon is often a convex plane-filling polygon, such as a square or a triangle. More specific names have been given to polyforms resulting from specific basic polygons, as...
- polyhexPolyhex (mathematics)In recreational mathematics, a polyhex is a polyform with a regular hexagon as the base form.As with polyominoes, polyhexes may be enumerated as free polyhexes , fixed polyhexes and one-sided polyhexes In recreational mathematics, a polyhex is a polyform with a regular hexagon (or 'hex' for...
- polyominoPolyominoA polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling with a connected interior....
- polyominoidPolyominoidIn geometry, a polyominoid is a set of equal squares in 3D space, joined edge to edge at 90- or 180-degree angles. The polyominoids include the polyominoes, which are just the planar polyominoids. The surface of a cube is an example of a hexominoid, or 6-cell polyominoid. Polyominoids appear to...
- tessellationTessellationA tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
External links
- Kevin Gong's enumeration of polycubes
- Polycubes, at The Poly Pages
- Polycube Symmetries
- Polycube solver Program (with Lua source code) to fill boxes with polycubes using Algorithm X.
- play pentacubes online