Torsten Sillke, BI, 1993-02-26
Building a 3-cube with tricubes last update: 2000-04-04
-------------------------------
The tricubes: face-connected
0) o o o Tower o : cube on the first level
x : cube on the second level
1) o L y : cube on the third level
o o
The 3-cube with tricubes only: (Torsten Sillke, 1992)
==============================
There are two different tricubes the ... and :.
All combinations of nine tricubes will tile the cube with
the exception of 8*... & 1*:.
3
You can dissect the cube into 3* 3 3 & 1* 3 3 as
was shown by O'Beirne's in Martin Gardner "Knotted Doughnuts ..." Chap 3.
3 3 is tileable with 2*... and 2*:.
3 is tileable with 3*... , 2*:. & ... , and 3*:.
3 3
This dissection gives you solutions for all combinations exept 8*... & 1*:.
Impossible proof for 8*... & 1*:. (Torsten Sillke)
Slice the 3*3*3-cube int 3 layers each containing 0 (mod 3) cubes. The
tricube ... has only the possibilities (0,0,0) and (1,1,1) for used
cubes of each layer (mod 3). Assume the were a tiling then :. has to
have the vector (0,0,0), as the sum is (0,0,0). So we conclude that :.
lays entierly in one layer. As you can slice in three different directions,
and each time :. lays in one layer only, :. is contained in the intersection
of three othogonal layers. But this is impossible. qed
Number of solutions: (rotation and reflection counted as different)
:. | 0 1 2 3 4 5 6 7 8 9
... | 9 8 7 6 5 4 3 2 1 0
solutions| 21 0 360 1104 4980 11640 29274 36960 36144 5828 (111)
There are 3 essential different solutions for 9*...
All parallel (3 oriantations), one face turned (12 oriantations),
and the middle layer turned (6 oriantations).
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
The tricubes: face-edge-connected
figures 0 and 1 plus
2) o o Hook
o
3) o o Diagonal L
x
5) x o 6) o x 5=Worm Sandwich
o o 6=Worm Head
7) o o Wing
o
4) o Steps
o
o
8) x o Claw
o
Def: x-free
A tiling is said x-free if there is no configuration:
a b 7 2 2
b a with a<>b. Example for a xing: 7 2 7
Packing the 3-cube with pseudo-tricubes:
========================================
With the tiles 0..8 it is possible to build a 3x3x3 cube x-free.
There is a solution consists of six symmetric parts
0, 1, (2,7), (3,5,6), 4, 8.
\ \ \
0 8 4 0 3 7 0 3 5
8 4 7 5 8 6 3 5 1
4 2 2 2 6 7 6 1 1
\ \ \
This 3-cube packing problem has been invented by many people.
I don't know who was the first. One of them was E. Rubik.
He calls them Mini Bricks (copyright 1974) see http://www.rubiks.com
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
The tricubes: face-edge-corner-connected
figures 0 to 8 plus
10) o
x
y
11) o o
x
12) o o
x
13) o x
o
14) o
o
x
15) o
x
o
Golomb call the tricubes 0 to 8 and 10 to 15 3-dim pseudo-trominoes.
They are listed in figure 126 in his book [Golomb, 1994].
References:
- S. W. Golomb;
Polyominoes, Princeton Univ. Press, 1994 (2nd edition).
- E. Rubik; Mini Bricks (C. 1974)
http://www.rubiks.com
- Michael Keller;
Polyform update. World Game Review 7 (Oct 1987) 10-13.
Says that Nob Yoshigahara has solved a problem posed by O'Beirne:
How many ways can 9 L-trominoes make a cube? Answer is 111.
Gardner, Knotted, chap. 3, mentioned O'Beirne's solution.
Says there are solutions with n L-trominoes and 9-n straight
trominoes for n < 1 and there are 4 solutions for n = 0 (3 is correct).
Tiling the 3-cube with equal (congruent as well) tricubes:
----------------------------------------------------------
It is long known that (...) and (:.) tile the 3-cube.
The only other tricube is (2) which tiles the cube.
Even if you allow unconnected figures like
16) o . o / quasi-tricube (Golomb)
o . .
no other solution will appear.
Three reasons exclude all impossible tricubes:
A) if the tricube is mono-colored (chessboard coloration).
This is the case for: 4, 5, 6, 7, 8
B) if the tricube matches at most two corner or edge-cube (pigeonhole).
This is the case for: 3, 4, 10, 11, 12, 13, 14, 15
C) if it is impossible to match the center cube.
This is the case e. g. for: 16
:: Counting (Coloring) restriction:
The L3 and the Hook behave similar as they are
equal according to the parity function (Z/Z2, Z/Z2, Z/Z2).
:: 3-cube
:: o o L or L3
o
a solution with three 1x2x3 blocks
a b b
a a c
d d c
The 3-cube has 4 types of cubes (vertex, edge, face, center).
Each L3 covers at most one vertex or center cube. As the 9 pieces
must cover these 9 cubes each L3 must cover exactly one vertex
or center cube. Possible (allowed) (v,e,f,c) configurations
[ 1 1 . . ] [a] [ 8] vertex cube count
[ 2 1 1 . ] [b] [12] edge cube count
[ . 1 1 2 ] * [c] = [ 6] face cube count
[ . . 1 1 ] [d] [ 1] center cube count
This matrix has only rank 3. [1, -1, -1, 1] is in the Null space.
But as we are interested in integral nonnegative solutions only
we find only two solutions as c+d = 1.
This system has the solutions [a,b,c,d] = [3,5,1,0] or [4,4,0,1].
Each tiling of the 3-cube of type [4,4,0,1] gives a solution of
the other type by swapping the central 2xL3.
The 6 solutions having an 'a'-face (these are 2x3-block-free packings):
Pieces 'c' and then 'd' are forced. The 'a,c,d'-subsection is symmetric.
a a 11 a a 12 a a 11 a a 10 a a 12 a a 12
a c a a c a a c a a c a a c a a c a
d a a d a a d a a d a a d a a d a a
14 12 11 14 12 12 14 14 11 14 14 10 14 12 12 14 12 12
c c 11 c c 10 c c 11 c c 10 c c 10 c c 10
d d 10 d d 10 d d 10 d d 12 d d 10 d d 11
14 12 12 14 b b 14 b b 14 b b 14 14 b 14 14 10
14 13 10 14 b b b b 10 b b b b b b 13 13 10
13 13 10 b b 10 b b 10 b 12 12 b b 10 13 11 11
:: 3 3 3 Solutions:
3 3 3
3 3 .
a a d with 'a', 'b', 'c', 'd' 2x3 blocks
b b d
c c .
:: 3 3 3 Impossible:
3 3 3
3 . 3 as the number of vertex and center cubes is 9.
:: 3 3 3 Solutions: (total 144 arragments)
3 . 3
3 3 3
a d d with 'a', 'b', 'c', 'd' 2x3 blocks
a . c
b b c
A: a a B: x x x B' b b b C: b b b a D: a b b b E: a c c c a E' b b b a a
a a x x x a c c c d a a a a c c a a b a a a x x x a
a a b b b a a c c c d d d d d c d d d b b a a x x x
combinations: 4A, 2A+C, 2C, 2A+D, 2D, B+E', B'+E, A+B+B'
v e v d f d v e v classifying the solutions
e . e f . f e . e by (v,e,d,f) vectors
v e v d f d v e v
[ 1 1 1 1 ] [a] [ 8] vertex cube count
[ 1 . 1 2 ] [b] [ 8] horizontal-edge cube count
[ . 1 1 . ] * [c] = [ 4] vertical-edge cube count
[ 1 1 . . ] [d] [ 4] face cube count
This matrix has only rank 3. [1, -1, 1, -1] is in the Null space.
The integral non-negative solutions are: [4,0,4,0], [3,1,3,1], [2,2,2,2],
[1,3,1,3], [0,4,0,4]. Each counting solution has a tiling solution.
:: o o Hook
o
A solution for with x-ings gives the dissection:
b b a
a a c
d d c (the 1x2x3 boxes 'b', 'c', 'd' have x-ings)
There are 132 solutions allowing x-ings.
Each Hook covers at most one vertex or center cube. As the 9 pieces
must cover these 9 cubes each Hook must cover exactly one vertex
or center cube.
This gives the same system of equations as for the L3.
One possible way to classify the solutions is by their central axis:
. . x . . 1 . 1 1 . . 1
x x . -> 1 3 . 1 3 . 1 3 1
. . . . 1 . . . . . . .
solutions: 44 66 22
:: x-free boxes of the Hook
There are x-free tilings of the 3-cube possible.
No rectangles are possible. (Same reason as for the N-Tetromino)
Even figures of height 3 are most likely to build as a closed chain.
:: 2 2 2 fold up: a b c d
2 2 2 a b c d
a b c d
:: 2 2 2 fold up: a b c d e f
2 2 2 a b c d e f
2 2 2 a b c d e f
:: 3 3 3 Impossible:
3 3 3
3 . 3 as the number of vertex and center cubes is 9.
3 3 3 fold up: a b c d e f g h
3 3 3 a b c d e f g h
3 3 . a b c d e f g h
3 3 3 fold up: a b c d e f g h total 656 arangments
3 . 3 a b c d e f g h
3 3 3 a b c d e f g h
v e v d f d v e v classifying the solutions
e . e f . f e . e by (v,e,d,f) vectors
v e v d f d v e v
This gives the same system of equations as for the L3.
A special nice solution which can be modified to form a cube.
1 a a 2 2 1 b b 2 now c c forming
a . a 1 . 1 b . b replace c c a 3-cube
a a 1 1 2 2 2 b b 'b' by 'c' c c with 9 Hooks
3-cube x-free:
There is another x-free solution having a
. a a
a . a face.
a a .
3
3
1 2 3 b b a a fold up this cube an add the center piece.
1 2 b b a a
1 2 4 b b a a
4 4
6 3 1
7 0 3
7 4 3
7 6 8
5 0 1
5 4 1
5 6 2
0 2 8
4 2 8
7 a a
a a a
a 5 3
7 4 3
6 4 3
6 5 4
6 7 b
b b b
b b 5
7 a a
a a a
a 4 3
5 7 3
6 5 3
6 5 4
6 7 b
b b b
b b 4