Void Cube
Encyclopedia
The Void Cube is a 3-D
mechanical puzzle
similar to a Rubik's Cube
, with the notable difference being that the center "cubes" are missing, which causes the puzzle to resemble a level 1 Menger sponge
. The central "spider" used on the Rubik's Cube is also absent, creating holes straight through the cube on all three axes. Due to the restricted volume of the puzzle it employs an entirely different structural mechanism from a regular Rubik's Cube, though the possible moves are the same. The Void Cube was invented by Katsuhiko Okamoto. Gentosha Education, in Japan, holds the license to manufacture Void Cubes.
. The lack of center cubes alters the parity considerations. A 90˚ rotation of a face either on the regular Rubik's Cube or on the Void Cube swaps the positions of eight cubes in two, odd parity, four cycles. Overall, a face turn is an even permutation (see Permutation parity). On the regular cube a 90˚ rotation of the whole cube about a principal axis swaps the positions of 24 cubes in six, odd parity, four cycles. On the regular cube a whole cube rotation is an even permutation. On the other hand, lacking center cubes, a 90˚ whole cube rotation on the Void Cube swaps 20 cubes in five, odd parity, four cycles. Thus, a whole cube rotation on the Void Cube is an odd permutation. In consequence, on the Void Cube turning the faces of the cube together with whole cube rotations can produce an arrangement where two cubes are swapped and the rest are in their original positions. This and other odd parity arrangements are not possible on the regular Rubik's Cube and afford the solver an additional challenge. These permutations are solvable with a number of simple algorithms (for example: http://www.youtube.com/watch?v=OzcYnPnIoSA).
To put the above into practical terms more simple to visualize, the set of moves that will transform a solved Rubik's cube into any of the "cat's eye" or "snake's eye" variations, where the middle pieces on 2 sides, or all 6 sides, are of different colors than the surrounding, instead result in another solution on a void cube.
Essentially, the "frame" of the mechanism comprises six identical pieces with square holes (the ones you can look through). Part of the interior of each hole is the inside surface of these pieces. Say that one of them is lying separately on a work surface, "exterior" side facing up. If you look straight down at one of these pieces (so your line of sight is parallel to that face's rotational axis) its exterior is also square.
Seen from a more-typical oblique position, however, each side of a square piece is somewhat akin to a low arch that joins neighboring corners. The low part of that arch engages mostly-hidden internal sliding pieces that (among other functions) support the edge cubelets ("cubies"). The high surface of the arch includes convex curved circular flanges that engage grooves inside the cubelets, to hold the structure together.
When all faces of the puzzle are in their normal aligned state, these six pieces are akin to the sides of an internal cube. Each one is free to rotate without any obstruction from the other five pieces. When a face is rotated, its own square piece also rotates with that face's cubelets, but that square piece's own flanges do not move relative to the cubelets.
What retains the cubelets when a face is rotated is the set of four curved flanges on the four neighboring square-hole pieces. Grooves inside the cubelets fit over those flanges. Edge cubelet grooves engage flanges on neighboring square pieces, thus keeping them together.
As described so far, however, the individual parts of the mechanism would move out of position easily. Each edge of the puzzle therefore includes a mostly-hidden sliding piece (already mentioned) with a complex shape that includes a curved dovetail surface. This surface is widest at its innermost extension, and in the center of the piece along its length.
The dovetail of this piece, acting somewhat like a wedge, keeps neighboring square pieces spaced apart.
Keeping the square pieces apart ensures that the grooves inside the cubelets stay engaged with the flanges. Close manufacturing tolerances result in sufficient friction to keep parts of the puzzle from moving on their own, but also
still allow easy movement.
Edge cubelets fit onto positioning lugs on the exteriors of these internal sliding pieces, so that rotating a face makes its edge cubelets push the sliding pieces around in a circle. The interior surfaces that face inward toward the hole drive this face's square piece so that it rotates with the cubelets.
The square pieces ensure that these internal sliding pieces stay toward the edges of the puzzle.
Edge cubelets in their normal position are retained by the flanges of neighboring square pieces. Corner cubelets are retained by a trio of short circular flanges at the ends of the interior sliding pieces. When a face is rotated, those short flanges temporarily retain edge cubelets, particularly when the face is rotated about 1/8 of a turn (roughly 45 degrees). As well, the corner cubelets are temporarily retained by the curved flanges on the neighboring square pieces.
During a rotation, flanges "change roles" as cubelets travel along their circular paths.
Three-dimensional space
Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...
mechanical puzzle
Mechanical puzzle
A mechanical puzzle is a puzzle presented as a set of mechanically interlinked pieces.- History :The oldest known mechanical puzzle comes from Greece and appeared in the 3rd century BC....
similar to a Rubik's Cube
Rubik's Cube
Rubik's Cube is a 3-D mechanical puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik.Originally called the "Magic Cube", the puzzle was licensed by Rubik to be sold by Ideal Toy Corp. in 1980 and won the German Game of the Year special award for Best Puzzle that...
, with the notable difference being that the center "cubes" are missing, which causes the puzzle to resemble a level 1 Menger sponge
Menger sponge
In mathematics, the Menger sponge is a fractal curve. It is a universal curve, in that it has topological dimension one, and any other curve is homeomorphic to some subset of it. It is sometimes called the Menger-Sierpinski sponge or the Sierpinski sponge...
. The central "spider" used on the Rubik's Cube is also absent, creating holes straight through the cube on all three axes. Due to the restricted volume of the puzzle it employs an entirely different structural mechanism from a regular Rubik's Cube, though the possible moves are the same. The Void Cube was invented by Katsuhiko Okamoto. Gentosha Education, in Japan, holds the license to manufacture Void Cubes.
Solution
The Void Cube is slightly more difficult than a regular Rubik's Cube due to parityParity (physics)
In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...
. The lack of center cubes alters the parity considerations. A 90˚ rotation of a face either on the regular Rubik's Cube or on the Void Cube swaps the positions of eight cubes in two, odd parity, four cycles. Overall, a face turn is an even permutation (see Permutation parity). On the regular cube a 90˚ rotation of the whole cube about a principal axis swaps the positions of 24 cubes in six, odd parity, four cycles. On the regular cube a whole cube rotation is an even permutation. On the other hand, lacking center cubes, a 90˚ whole cube rotation on the Void Cube swaps 20 cubes in five, odd parity, four cycles. Thus, a whole cube rotation on the Void Cube is an odd permutation. In consequence, on the Void Cube turning the faces of the cube together with whole cube rotations can produce an arrangement where two cubes are swapped and the rest are in their original positions. This and other odd parity arrangements are not possible on the regular Rubik's Cube and afford the solver an additional challenge. These permutations are solvable with a number of simple algorithms (for example: http://www.youtube.com/watch?v=OzcYnPnIoSA).
To put the above into practical terms more simple to visualize, the set of moves that will transform a solved Rubik's cube into any of the "cat's eye" or "snake's eye" variations, where the middle pieces on 2 sides, or all 6 sides, are of different colors than the surrounding, instead result in another solution on a void cube.
Internal mechanism
The parts of a Void Cube are:- 20 cubelets, sometimes called "cubies":
- 8 corner cubelets
- 12 edge (middle) cubelets
- 6 internal support pieces with the square hole through them
- 12 mostly-hidden internal sliding pieces with several functions
Essentially, the "frame" of the mechanism comprises six identical pieces with square holes (the ones you can look through). Part of the interior of each hole is the inside surface of these pieces. Say that one of them is lying separately on a work surface, "exterior" side facing up. If you look straight down at one of these pieces (so your line of sight is parallel to that face's rotational axis) its exterior is also square.
Seen from a more-typical oblique position, however, each side of a square piece is somewhat akin to a low arch that joins neighboring corners. The low part of that arch engages mostly-hidden internal sliding pieces that (among other functions) support the edge cubelets ("cubies"). The high surface of the arch includes convex curved circular flanges that engage grooves inside the cubelets, to hold the structure together.
When all faces of the puzzle are in their normal aligned state, these six pieces are akin to the sides of an internal cube. Each one is free to rotate without any obstruction from the other five pieces. When a face is rotated, its own square piece also rotates with that face's cubelets, but that square piece's own flanges do not move relative to the cubelets.
What retains the cubelets when a face is rotated is the set of four curved flanges on the four neighboring square-hole pieces. Grooves inside the cubelets fit over those flanges. Edge cubelet grooves engage flanges on neighboring square pieces, thus keeping them together.
As described so far, however, the individual parts of the mechanism would move out of position easily. Each edge of the puzzle therefore includes a mostly-hidden sliding piece (already mentioned) with a complex shape that includes a curved dovetail surface. This surface is widest at its innermost extension, and in the center of the piece along its length.
The dovetail of this piece, acting somewhat like a wedge, keeps neighboring square pieces spaced apart.
Keeping the square pieces apart ensures that the grooves inside the cubelets stay engaged with the flanges. Close manufacturing tolerances result in sufficient friction to keep parts of the puzzle from moving on their own, but also
still allow easy movement.
Edge cubelets fit onto positioning lugs on the exteriors of these internal sliding pieces, so that rotating a face makes its edge cubelets push the sliding pieces around in a circle. The interior surfaces that face inward toward the hole drive this face's square piece so that it rotates with the cubelets.
The square pieces ensure that these internal sliding pieces stay toward the edges of the puzzle.
Edge cubelets in their normal position are retained by the flanges of neighboring square pieces. Corner cubelets are retained by a trio of short circular flanges at the ends of the interior sliding pieces. When a face is rotated, those short flanges temporarily retain edge cubelets, particularly when the face is rotated about 1/8 of a turn (roughly 45 degrees). As well, the corner cubelets are temporarily retained by the curved flanges on the neighboring square pieces.
During a rotation, flanges "change roles" as cubelets travel along their circular paths.