The Forming of The Forming of Symmetrical Figures Symmetrical Figures

from Tetracubesfrom Tetracubes

Baiba Bārzdiņa

Riga State Gymnasium No. 1

The set of tetracubes

I L N T

O K S 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 referencesHistorical 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 3Tower 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

A B C D

E F 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• Elementary filters

4 (3)

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 filtersA 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

• 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

BBBB

BADAG GCCBBC GAFBAGAAFAG

CABCGGGGG