On Mathematics As Installation Art
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On Mathematics as Installation Art Algirdas Javtokas Department of Mathematics and Informatics, Vilnius University Naugarduko 24, 03225 Vilnius, Lithuania [email protected]
Abstract What part does the reader play in the creation and implementation of the meaning of a mathematical text? What is the role of the reader’s own personality in the interpretation of the mathematical text? The present ”mathematical” abstract text (we call it text installation) questions the role of the reader in the construction of the mathematical text. The text has all signs of a scientific paper except that it has no words; but an abstract, names of paragraphs and such words as ”Theorem”, ”Lemma”, ”Proof”. The present paper allows the reader perform the mathematical text within referential frames and make multiple connections and associations while reading. Keywords: mathematics, art, text, installation, meaning MSC 2000: 00A05, 00A30, 00A99, 03H05
1 1 1 1 1 1 1
Introduction 0 0 0 0 0 0
01 01 01 01 01 01
100 100 100 100 100 100
010 010 010 010 010 010
1 1 1 1 1 1
0 0 0 0 0 0
1 1 1 1 1 1
0 0 0 0 0 0
01 01 01 01 01 01
A
AA
100 100 100 100 100 100
010 010 010 010 010 010
text =
1 1 1 1 1 1 RR
0 0 0 0 0 0
1 1 1 1 1 1
0 0 0 0 0 0
01 01 01 01 01 01
100 100 100 100 100
010 010 010 010 010
1 1 1 1 1
0 0 0 0 0
1 1 1 1 1
0 0 0 0 0
1 1 1 1 1
RR
a
R
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1
Algirdas Javtokas :: On Mathematics as Installation Art 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0
01 01 01 01 01 01 1 0 0 01 0 01 0 01 0 01 0 01
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 01 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 000000111101
100 100 100 100 100
2 010 010 010 010 010
1 1 1 1 1
0 0 0 0 0
1 1 1 1 1
0 0 0 0 0
1 1 1 1 1
01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1
Theorem 1. 0 0 00 000 0000 1 0 0 0 01 000 000 1 0 0 0 00 0000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1
1 1 1 1
0 0 0 0
2 1 1 1 1 1
1 0 01 01 01 01
01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0
01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01
01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01
100 100 100 100
Auxiliary results 0 0 0 0 0
01 01 01 01 01 1 0 1 0 01 1 0 01 1 0 01
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 000000
a0 1 AAAA = 1 1 1 1
0 0 0 0
01 01 01 01
100 100 100 100
010 010 010 010
1 1 1 1
0 0 0 0
1 1 1 1
0 0 0 0
01 01 01 01
100 100 100 100
010 010 010 010
1 1 1 1
0 0 0 0
1 1 1 1
0 0 0 0
1 1 1 1
01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1
R R 2 +56a+klp
010 010 010 010
R
1 1 1 1
0 0 0 0
1 1 1 1
0 0 0 0
01 01 01 01
100 100 100 100
010 010 010 010
1 1 1 1
0 0 0 0
1 1 1 1
0 0 0 0
1 1 1 1
Algirdas Javtokas :: On Mathematics as Installation Art 1 1 1 1 1 1 1 1 1
0 0 0 0
01 01 01 01 1 0 0 01 0 01 0 01 0 01 0 01
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 01 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 000000111101
3
100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1
Lemma 1. W 10 11 111 0000 1 0 0 0 01 000 000 1 0 0 0 00 0000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 Proof. A 1000000 00 000 0000 1 0 0 0 01 000 000 1 0 0 0 00 0000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 11 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 11 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 O(1) ≤ f (x) I I
I
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 1 0 01 1 0 0000001111010101010 Lemma 2. W 10 11 111 0000 1 0 0 0 01 000 000 1 0 0 0 00 0000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 Proof. A 1000000 00 000 0000 1 0 0 0 01 000 000 1 0 0 0 00 0000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 11
Algirdas Javtokas :: On Mathematics as Installation Art 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0
01 01 01 01 01 1 0 0 01 0 01 0 01 0 01 0 01
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 0000001111010101010
100 100 100 100
4 010 010 010 010
1 1 1 1
0 0 0 0
1 1 1 1
0 0 0 0
1 1 1 1
01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1
Lemma 3. W 10 11 111 0000 1 0 0 0 01 000 000 1 0 0 0 00 0000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 Proof. A 1000000 00 000 0000 1 0 0 0 01 000 000 1 0 0 0 00 0000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 11 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 11 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 W
ii
AA
O = f AAA = a AA
a
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 A
AA
A
O AA
≥ O12
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 1 0 01 1 0 0000001111010101010 Lemma 4. W 10 11 111 0000 1 0 0 0 01 000 000 1 0 0 0 00 0000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1
Algirdas Javtokas :: On Mathematics as Installation Art
5
Proof. A 1000000 00 000 0000 1 0 0 0 01 000 000 1 0 0 0 00 0000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 II
I
AAA = a I I
I
A
AA
R AA
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 OOO = I I
I
i
1 0 01 100 010 1 0 1 0 01 100 010 1 0000101010 A
II a AA
A
≥ AA
x
a ≥ f (x2 − 1) ≥ AA A
Rb a
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 A
O(12 ) AAA 1 0 01 1 0 1 0 01 1 0 01 1 0 01 1 0 01 1 0 01 1 0 1 0 01 1 0 01 1 0 01 1 0 01 1 0 01
3 1 1 1 1
a, b, c
100 010 1 0 1 0 01 100 010 1 0 1 0 01 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 01010111101 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 0000001111010101010
100 010 1 01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1
Proof of the main theorem 0 0 0 0
01 01 01 01 1 0
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 01 100 010 1 0 1 0
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 01 100 010 1 0 1 0
100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 01 100 010 1 0 11
Algirdas Javtokas :: On Mathematics as Installation Art 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0
01 01 01 01 01 1 0 0 01 0 01 0 01 0 01 0 01 1 0 0 01 0 01 0 01 0 01 0 01 1 0 0 01
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 01010111101 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 0000001111010101010 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 aaa I I
I
II
I
A
aaa = AA
100 100 100 100
6 010 010 010 010
1 1 1 1
0 0 0 0
1 1 1 1
0 0 0 0
1 1 1 1
01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 01 100 010 1 0 11 100 010 1 A
aa AA
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 .. I = 1 0 01 1 0 01 1 0 1 0 01 1 0 01 1 0 01 1 0 01 1 0 01 1 0 1 0 01 1 0 01 1 0 01 1 0 01 1 0 01 1 0
Pb
a Ui
.. AW A .
100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 0000001111010101010 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 010111111111111110111101 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 00000 01 100 010 1 0 1 0 01 100 010 1 0 1 0
100 010 1 0 1 0 1 01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 01 100 010 1 0 11
Algirdas Javtokas :: On Mathematics as Installation Art 1 1 1 1 1
0 0 0 0 0
01 01 01 01 01 1 0 1 0 01
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 0000001111010101010 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 I2 =
PR
100 100 100 100
7 010 010 010 010
1 1 1 1
0 0 0 0
1 1 1 1
0 0 0 0
1 1 1 1
01 100 010 1 0 11 100 010 1
.
R
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 A+ AA A = aa 1 0 01 1 0 01 1 0 1 0 01
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 1 0 0000001111010101010 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 11 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 R
I2 = a + b + c
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 A
AA A 1 0 01 1 0 01 1 0 1 0 01 1 0 01 1 0 01 1 0 01 1 0 01
Ra a
+1
100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 0000001111010101010 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 010111111111111110111101 A
AA 1 0 01 1 0 1 0 01 1 0 01
A AA A =
AA
AAA A+ a = a AA
100 010 1 0 1 0 01 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01
100 010 1 0 1 0 1 01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1
AAAA
100 010 1 0 1 0 01 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01
100 010 01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1
Algirdas Javtokas :: On Mathematics as Installation Art
8
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 1 0 01 1 0 00000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 11 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 1 010111111111111111111111111110111101 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 11 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 1 0 01 1 0 00001111111111111111111111111111111110 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 11 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 1 0 01 1 0 00001111111111111111111111111111111110
4 1. 1 2. 1 3. 1 4. 1 5. 1 6. 1 7. 1 8. 1
References 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1000000000000000000000000000000 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100000000000 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 10100 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1100100 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 10 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 00000000000000000010 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100010 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 10001010101010101
0 01 100 010 1 0 1 0 1 0 01 100 010 1 0 1 0 1 0 01 100 010 1 0 1 0 1 0 01 100 010 1 0 1 0 1 0 01 100 010 1 0 1 0 1 0 01 100 010 1 0 1 0 1 0 01 100 010 1 0 1 0 1 0 01 100 010 1 0 1 0 1
Abstract What part does the reader play in the creation and implementation of the meaning of a mathematical text? What is the role of the reader’s own personality in the interpretation of the mathematical text? The present ”mathematical” abstract text (we call it text installation) questions the role of the reader in the construction of the mathematical text. The text has all signs of a scientific paper except that it has no words; but an abstract, names of paragraphs and such words as ”Theorem”, ”Lemma”, ”Proof”. The present paper allows the reader perform the mathematical text within referential frames and make multiple connections and associations while reading. Keywords: mathematics, art, text, installation, meaning MSC 2000: 00A05, 00A30, 00A99, 03H05
1 1 1 1 1 1 1
Introduction 0 0 0 0 0 0
01 01 01 01 01 01
100 100 100 100 100 100
010 010 010 010 010 010
1 1 1 1 1 1
0 0 0 0 0 0
1 1 1 1 1 1
0 0 0 0 0 0
01 01 01 01 01 01
A
AA
100 100 100 100 100 100
010 010 010 010 010 010
text =
1 1 1 1 1 1 RR
0 0 0 0 0 0
1 1 1 1 1 1
0 0 0 0 0 0
01 01 01 01 01 01
100 100 100 100 100
010 010 010 010 010
1 1 1 1 1
0 0 0 0 0
1 1 1 1 1
0 0 0 0 0
1 1 1 1 1
RR
a
R
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1
Algirdas Javtokas :: On Mathematics as Installation Art 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0
01 01 01 01 01 01 1 0 0 01 0 01 0 01 0 01 0 01
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 01 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 000000111101
100 100 100 100 100
2 010 010 010 010 010
1 1 1 1 1
0 0 0 0 0
1 1 1 1 1
0 0 0 0 0
1 1 1 1 1
01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1
Theorem 1. 0 0 00 000 0000 1 0 0 0 01 000 000 1 0 0 0 00 0000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1
1 1 1 1
0 0 0 0
2 1 1 1 1 1
1 0 01 01 01 01
01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0
01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01
01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01
100 100 100 100
Auxiliary results 0 0 0 0 0
01 01 01 01 01 1 0 1 0 01 1 0 01 1 0 01
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 000000
a0 1 AAAA = 1 1 1 1
0 0 0 0
01 01 01 01
100 100 100 100
010 010 010 010
1 1 1 1
0 0 0 0
1 1 1 1
0 0 0 0
01 01 01 01
100 100 100 100
010 010 010 010
1 1 1 1
0 0 0 0
1 1 1 1
0 0 0 0
1 1 1 1
01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1
R R 2 +56a+klp
010 010 010 010
R
1 1 1 1
0 0 0 0
1 1 1 1
0 0 0 0
01 01 01 01
100 100 100 100
010 010 010 010
1 1 1 1
0 0 0 0
1 1 1 1
0 0 0 0
1 1 1 1
Algirdas Javtokas :: On Mathematics as Installation Art 1 1 1 1 1 1 1 1 1
0 0 0 0
01 01 01 01 1 0 0 01 0 01 0 01 0 01 0 01
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 01 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 000000111101
3
100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1
Lemma 1. W 10 11 111 0000 1 0 0 0 01 000 000 1 0 0 0 00 0000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 Proof. A 1000000 00 000 0000 1 0 0 0 01 000 000 1 0 0 0 00 0000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 11 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 11 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 O(1) ≤ f (x) I I
I
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 1 0 01 1 0 0000001111010101010 Lemma 2. W 10 11 111 0000 1 0 0 0 01 000 000 1 0 0 0 00 0000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 Proof. A 1000000 00 000 0000 1 0 0 0 01 000 000 1 0 0 0 00 0000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 11
Algirdas Javtokas :: On Mathematics as Installation Art 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0
01 01 01 01 01 1 0 0 01 0 01 0 01 0 01 0 01
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 0000001111010101010
100 100 100 100
4 010 010 010 010
1 1 1 1
0 0 0 0
1 1 1 1
0 0 0 0
1 1 1 1
01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1
Lemma 3. W 10 11 111 0000 1 0 0 0 01 000 000 1 0 0 0 00 0000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 Proof. A 1000000 00 000 0000 1 0 0 0 01 000 000 1 0 0 0 00 0000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 11 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 11 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 W
ii
AA
O = f AAA = a AA
a
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 A
AA
A
O AA
≥ O12
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 1 0 01 1 0 0000001111010101010 Lemma 4. W 10 11 111 0000 1 0 0 0 01 000 000 1 0 0 0 00 0000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1
Algirdas Javtokas :: On Mathematics as Installation Art
5
Proof. A 1000000 00 000 0000 1 0 0 0 01 000 000 1 0 0 0 00 0000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 II
I
AAA = a I I
I
A
AA
R AA
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 OOO = I I
I
i
1 0 01 100 010 1 0 1 0 01 100 010 1 0000101010 A
II a AA
A
≥ AA
x
a ≥ f (x2 − 1) ≥ AA A
Rb a
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 A
O(12 ) AAA 1 0 01 1 0 1 0 01 1 0 01 1 0 01 1 0 01 1 0 01 1 0 1 0 01 1 0 01 1 0 01 1 0 01 1 0 01
3 1 1 1 1
a, b, c
100 010 1 0 1 0 01 100 010 1 0 1 0 01 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 01010111101 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 0000001111010101010
100 010 1 01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1
Proof of the main theorem 0 0 0 0
01 01 01 01 1 0
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 01 100 010 1 0 1 0
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 01 100 010 1 0 1 0
100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 01 100 010 1 0 11
Algirdas Javtokas :: On Mathematics as Installation Art 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0
01 01 01 01 01 1 0 0 01 0 01 0 01 0 01 0 01 1 0 0 01 0 01 0 01 0 01 0 01 1 0 0 01
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 01010111101 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 0000001111010101010 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 aaa I I
I
II
I
A
aaa = AA
100 100 100 100
6 010 010 010 010
1 1 1 1
0 0 0 0
1 1 1 1
0 0 0 0
1 1 1 1
01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 01 100 010 1 0 11 100 010 1 A
aa AA
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 .. I = 1 0 01 1 0 01 1 0 1 0 01 1 0 01 1 0 01 1 0 01 1 0 01 1 0 1 0 01 1 0 01 1 0 01 1 0 01 1 0 01 1 0
Pb
a Ui
.. AW A .
100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 0000001111010101010 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 010111111111111110111101 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 00000 01 100 010 1 0 1 0 01 100 010 1 0 1 0
100 010 1 0 1 0 1 01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 01 100 010 1 0 11
Algirdas Javtokas :: On Mathematics as Installation Art 1 1 1 1 1
0 0 0 0 0
01 01 01 01 01 1 0 1 0 01
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 0000001111010101010 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 I2 =
PR
100 100 100 100
7 010 010 010 010
1 1 1 1
0 0 0 0
1 1 1 1
0 0 0 0
1 1 1 1
01 100 010 1 0 11 100 010 1
.
R
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 A+ AA A = aa 1 0 01 1 0 01 1 0 1 0 01
100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 1 0 0000001111010101010 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 11 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 R
I2 = a + b + c
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 A
AA A 1 0 01 1 0 01 1 0 1 0 01 1 0 01 1 0 01 1 0 01 1 0 01
Ra a
+1
100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 0 01 1 0 0000001111010101010 01 100 010 1 0 1 0 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1 010111111111111110111101 A
AA 1 0 01 1 0 1 0 01 1 0 01
A AA A =
AA
AAA A+ a = a AA
100 010 1 0 1 0 01 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01
100 010 1 0 1 0 1 01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1 100 010 1 0 1 0 1
AAAA
100 010 1 0 1 0 01 01 100 010 1 0 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01
100 010 01 100 010 1 0 11 100 010 1 0 1 0 1 100 010 1 0 1 0 1
Algirdas Javtokas :: On Mathematics as Installation Art
8
1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 1 0 01 1 0 00000 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 11 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 1 010111111111111111111111111110111101 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 11 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 1 0 01 1 0 00001111111111111111111111111111111110 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 11 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100 010 1 0 1 0 1 1 0 01 1 0 01 1 0 00001111111111111111111111111111111110
4 1. 1 2. 1 3. 1 4. 1 5. 1 6. 1 7. 1 8. 1
References 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1000000000000000000000000000000 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100000000000 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 10100 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 1100100 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 10 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 00000000000000000010 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 100010 1 0 100 010 1 0 1 0 01 100 010 1 0 1 0 01 10001010101010101
0 01 100 010 1 0 1 0 1 0 01 100 010 1 0 1 0 1 0 01 100 010 1 0 1 0 1 0 01 100 010 1 0 1 0 1 0 01 100 010 1 0 1 0 1 0 01 100 010 1 0 1 0 1 0 01 100 010 1 0 1 0 1 0 01 100 010 1 0 1 0 1
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