Baiba Bārzdiņa
Riga State Gymnasium No. 1
The Forming of
Symmetrical Figures
from Tetracubes
The set of tetracubes
I
L
N
O
K
S
T
Z
Tetracubes are used usually to
form the given shapes, e. g.
Description of Problem
• The competition work is dedicated to the complicated
problem of combinatorial geometry - to find all
polycubes having four planes of symmetry and
assemblable from tetracubes.
• The aim of my work is to solve this bulky problem for
wider class of polycubes, namely for polycubes with
bases 3x3.
Let us explain that a polycube is a solid figure obtained
by combining unit cubes, joined at their faces. If the
polycube consists exactly of 4 cubes it is called a
tetracube.
Historical references
• One of the first articles in which attention has
been paid to tetracubes is the article by J. Meeus in
1973.
• A. Cibulis has popularized tetracubes in Latvia,
e. g. in the magazine “Labirints” in 1997-1999.
•The problem on symmetrical towers was offered in
the international conference “Creativity in
Mathematical Education and the Education of
Gifted Students”, Riga, 2002.
Tower with bases 3 x 3
A tower with bases 3x3 is a polycube having four
symmetrical planes and which can be inserted in a box with
bases 3x3, but which cannot be inserted in a box with bases
less than 3x3
Admissible layers
B
A
E
C
F
D
G
Plan of the problem solving
• To find all combinations of layers with the total
volume 32. I found 666 combinations by the computer
programme.
• Obtaining of permutations and their analysis.
• Necessary conditions:
- filters (Lemma on filters)
- method of invariants (colouring)
• Analysis of the remaining combinations by the
computer programme elaborated by A. Blumbergs
Filters
4 (3)
• Elementary filters
BBBB (1/1)
CAAA (2/2)
DAAA (2/2)
5 (17)
CCCBA (10/6)
CCDBA (30/4)
CDDBA (30/0)
DDDBA (10/0)
ECCAA (16/8)
ECDAA (30/6)
• More complicated filters
Lemma on filters
A tower cannot contain the following layers:
DD, DF, FD, CD, DC, BG, GB, EG, GE, FG, GF,
CF, FC, DE, ED, EF, FE, FFF, EEE, GAG, GCG,
GDG, EEG.
FF, EE, AG, DG, CG cannot be two last layers of a
tower.
To prove Lemma several nontrivial methods were used:
method of interpretation, Pigeonhole principle, and symmetry
Solutions found by the computer
programme elaborated by A. Blumbergs
Results
Some important towers
BBBB
•
•
•
•
CABCGGGGG
Only BBBB has 5 planes of symmetry
Only AAFAG contains F as the inner layer
There is a unique stable tower with height 9
Towers GADAB, GCCBBC, GAFBAG can be
assemblable only in one way
AAFAG
BADAG
GCCBBC
GAFBAG