prime-boxes of polycubes Torsten Sillke, 1993-11-16 initial version
2000-12-20 last update
My (incomplete) list of Polycubes, for which I have computed prime rectangles,
boxes, hyperboxes. I use the hight notation, the Schroeppel notation has been
appended two times in brackets. (The pentacube numbering is from E. Kuenzell.)
News: 1997
Wolf showed that Z5 is boxable.
News: July 1997
Helmut Postl showed there is no 3x3xZ15 tiling with Y5 (problem 6).
News: Jan. 1998
Helmut Postl found packings for the pentacubes 33, 37 and 51.
He completed the N5 packings.
News: May. 1998
Michael Reid found all prime rectangles for the handed L5 pentomino.
News: June 1998
All pentacubes are boxable with two exceptions: X and 35.
The last cases have been determined by Helmut Postl.
He completed the Q5 packings.
News: July 1998
Some further list have been completed by Helmut Postl.
He completed the 81, J4, glider packings.
News: March 1999
Yura Aksyonov found an elegant proof that 5x10 is the only prime
rectangle for the handed Y5.
News: April 1999
Michael Reid completed the list of G6 rectangles by showing
that one side must be a multiple of 4.
News: April 1999
Michael Reid proofed that there is no odd rectangle for the handed P5.
News: Dec. 2000
Michael Reid showed that the area of rectangles made of X and I3
is a multiple of 3.
polyominoes with complete list a prime packings:
I5 (trivial, as it is harmonious)
P5 (everybody knows)
L5 (3*3*Z, 3*3*3*30, 3*3*3*45 are new, 3*3*...*3*15 is impossible)
Y5
U5
1 1 [ 1 1 ]
61: 2 1 [ 3 1 ]
1
71: 2 1 1
2
41: 1 1 1
1 1
31: 2 1 completed june 1994
list complete in 3-dim:
V5
N5 (completed by Helmut Postl in Jan. 1998)
1
1 1
Q5: 1 1 (completed by Helmut Postl in June 1998)
1 (completed by Helmut Postl in July 1998)
81: 2 1 1 (3*5*6 (smallest))
1
1 (completed by Helmut Postl in July 1998)
glider: 1 1 1 (e.g. 6*10, 10*10, 13*25, 3*4*5)
1
1 (completed by Helmut Postl in July 1998)
J4: 1 1 (14*36 smallest rectangle by Helmut Postl Dec. 1997)
list almost complete:
F5 the missing case in 3-dim is 5*7*7
list incomplete:
T5 (3*10*10, 5*5*12 (smallest))
W5 (5*6*6 (smallest))
1 1
U6 1 1 1 1 (2*4*6, 3*4*4)
1 1
6 1 1 1 1 (4*6)
G6: 1 1 1 (2-dim complete)
1 1
1
21: 1 (4*5*6, 4*10*10, 4*8*20, 6*6*{15,20,25})
1 1 2 (With reflections: 3*4*15, 3*5*6, 3*5*9, 4*4*10)
2
82: 1 1 1 (2*5*5 (smallest))
1 1
1 1
large-U: 1 (e.g. 4*5*6, 3*4*10)
1
1
1
J5: 1 1
few packings are known:
Z5: (6*6*25 and 6*10*10 found by Wolf 1997)
(6*6*25 has been found by Helmut Postl too)
2 1
33: 1 1 (5*8*8 has been found by Helmut Postl in Jan. 1998)
1
51: 1 2 1 (7*8*20 has been found by Helmut Postl in Jan. 1998)
2
37: 1 2 (5*N*N is tilable, all n-dim packings found for n>=4)
(4*9*20 has been found by Helmut Postl in Jan. 1998)
not boxable in three dimension (even with reflections):
2 2' [ 3 2 ] (2' : only one cube at the second level)
35: 1 1 [ 1 1 ] (shown by Helmut Postl in June 1998)
not boxable in any dimension:
X5 (N*N*...*N is not tilable, as the corner can not be filled.)
Open Problems:
1) The last tetracube tiling problem: (2nd. Update)
---------------------------------------------------
______
|\ \ Is it possible to tile a 3*2n*2m box
| \_____\ with the tetracube shown left?
| | |____
|\| | \ All other (hyper-) box tiling problems with
| *_____|_____\ only alike tetracubes are solved. See:
| |\ \ | A. L. Clarke, Packing Boxes with Congruent Polycubes,
\| \_____\ | J. of Recreational Mathematics 10 (1977/78) 177-182
* | |___|
\| |
*_____|
You can show 2 | nm.
According to my computations n and m had to be greater then 9.
I found no 3*2n*N for n = 1..9 (n=10 is halve done).
The search tree dies out before reaching 4/3 n (3rd dimension).
Mike Beeler confirmed my computations for n = 1..8.
The 3*4*Z is tileble:
build two times 2 1 1 2 2 1 1 2 2 1 1 2
. . 1 2 2 1 1 2 2 1 1 2 2 1 . .
. . 1 2 2 1 1 2 2 1 1 2 2 1 . .
2 1 1 2 2 1 1 2 2 1 1 2
use the a a b b a a b b a a b b
dissection a c c b a c c b a c c b
b c c a b c c a b c c a
b b a a b b a a b b a a
The 3*N*N is tileble:
as the 3*(4*Z bent) is tileble.
. .
. .
3 3 3 3
3 3 3 3
3 3 3 3
3 3 3 3 3 3 3 3 Try for yourself!
3 3 3 3 3 3 3 3 . .
3 3 3 3 3 3 3 3 . .
3 3 3 3 3 3 3 3
My Conjecture:
- the 3*n*N box (one side open) is not tileble for all n>0.
2) Rectangles with the One-sided Y5, P5, L5.
--------------------------------------------
There are three pentominoes, which tile rectangles and are handed.
These are: L, P, and Y.
Show: the only prime rectangles for the one-sided pentominoes are:
L: 2*5 -> see update (solved)
P: 2*5 -> see update (solved)
Y: 5*10 -> see update (solved)
The handed L4 is long known (D. A. Klarner, AMM 70:7 (Sep. 1963) 760-61 E1543).
If you allow for the bent tromino only the halve turn, so you have
o o o
o o and o only, then the prime rectangles are 2*3 and 3*2.
There is no odd rectangle possible. This follows from Conway's proof
of tiling a triangle with o
o o = T2.
Shearing T2 gives the two orientations of the tromino. This has been
noticed by Noam Elkies.
UPDATE: Michael Reid found (until May 1998) all
prime rectangle of the handed L5: 2x5, 13x55, 15x39, 17x35, 19x25.
He showed that there is no odd rectangle for the
handed P5 of width <= 21.
UPDATE: Yura Aksyonov (1999-03-06)
He found a elegant proof that 5x10 is the only prime.
UPDATE: Michael Reid (1999-04-14)
He found a group proof that 2x5 is the only prime.
3) T4 rectangles with one hole
------------------------------
Is there a rectangle n*m with n*m = 1 (mod 4) which can be tiled with
T4 leaving one hole.
My Conjecture: No
D. W. Walkup, Covering a Rectangle with T-tetrominoes, AMM 72:9 (1965) 986-88
4) n-bone conjecture
--------------------
The n-bone are n spheres in a row glued together.
So the 5-bone looks like: ooooo
John H. Conway solved the 3-bone conjecture:
It is impossible to tile with 3-bones a triangular arangement of spheres.
See: W. P. Thurston, Conway's Tiling Groups, AMM 97 (1990) 757-773
You see easily that with 2-bones there is a tiling of the triangle
or tetrahedron iff the number of spheres is odd.
On the right is a 2
example tiling of 2 1
a triangle with 2-bones 0 0 1
My extended conjecture is: (the n-bone conjecture)
There is no tiling of the d-simplex with n-bones, where d>=2 and n>=3.
The simplex had to be finite of course.
5) N-pentasphere tetrahedron
----------------------------
Is there a easy proof, that 24 o o
o o o
do not tile an tetrahedron with 8 spheres on the edge.
(last year I made a list of possible packings of the thetrahedron
with polyspheres [written in german]. This tile needed the longest time.)
My program takes 100h on a HP850 to show this.
Is it possible to tile an tetrahedron with 9 spheres on the edge?
6) Y5 packings with minimal period
----------------------------------
A periodic tiling of 3x3xZ with the Y5 has a length which is a multiple
of 15. I constructed a tiling with period 30. Find one with period 15.
UPDATE: Helmut Postl solved this (letter from july 1997).
He showed that period 15 is impossible by coloring arguments.
He find constructions for all periods of length 15*n and n>1.
It is possible to have a symmetric glide reflections of period 15.
-- problem solved --
7) F5 box 5x7x7
---------------
The last open problem for the F-Pentomino box packings is:
Can the 5x7x7 box be packed?
8) Pu prime rectangles (with u = 2n-1) [Helmut Postl]
--------------------------------------
The P(2n-1) is a 2xn rectangle without one corner.
Helmut Postl thinks that the only prime rectangle
is the trivial one 2x(2n-1) for n >= 5.
For the P3, P5, P7 other prime rectangles have been found.
UPDATE: odd prime rectangles are possible for general L-pieces.
Michael Reid (1999-01)
-- problem solved --
References:
-) S. W. Golomb, Tiling with Polyominoes,
JoCT 1 (1966) 280-296
-) D. A. Klarner, Packing a Rectangle with Congruent N-ominoes,
JoCT 7 (1969) 107-115; Zbl 174.41
-) F. G"obel, D. A. Klarner, Packing Boxes with Congruent Figures,
Kon. Ned. Akad. Wet. Amst. A (=Indagationes Math.) 72 (1969) 465-472
-) M. Gardner, Mathematical Magic Show,
New York (1977), Chap. 13: Polyominoes and Rectification
-) A. L. Clarke, Packing Boxes with Congruent Polycubes,
J. of Recreational Mathematics 10 (1977/78) 177-182
-) S. W. Golomb, Polyominoes which tile rectangles
JoCT A 51 (1989) 117-124; Zbl 723.05041
-) K. A. Dahlke, A Heptomino of order 76,
JoCT A 51 (1989) 127-8 & 52 (1989) 321; Zbl 715.05014
-) K. A. Dahlke, A Y-hexomino has order 92,
JoCT A 51 (1989) 125-6; Zbl 715.05013
-) I. Stewart, Another Fine Math You've Got Me Into,
Freeman, 1992, Chap. 2: Tile and Error
-) S. W. Golomb, Polyominoes,
1994, 2nd Ed., Chap. 8: Tiling Rectangles with Polyominoes
--
Dr. Torsten Sillke
mailto:sillke@mathematik.uni-bielefeld.de
http://www.mathematik.uni-bielefeld.de/~sillke/PENTA/qu-prime