the forming of symmetrical figures from tetracubes baiba bārzdiņa riga state gymnasium no. 1
TRANSCRIPT
The Forming of The Forming of Symmetrical Figures Symmetrical Figures
from Tetracubesfrom Tetracubes
Baiba Bārzdiņa
Riga State Gymnasium No. 1
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
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
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