Download A square from similar rectangles

A square from similar rectangles
Sergei Dorichenko, Mikhail Skopenkov, Olesya Malinovskaya
Problem. When is it possible to tile a square by rectangles similar to a given one?
Leading questions
— I have a thought! — said Boa and opened his eyes. — A thought. And I think it.
— Which thought? — asked Monkey.
— It takes time to explain...
— Wow! — jumped Monkey. — What a good thought! May I also think it a bit?
G. Oster, “Boa’s grandmother”
1. Form a square from rectangles m × n, where m and n are integers.
2. A designer was offered to make square window frames. In Figures A,B it is shown how
the panes should be adjacent to each other and how they should be oriented (with the short
side or with the long one to the top). Can all panes in each frame be similar rectangles?
3. Is it possible to dissect a square into 3 similar, but not equal rectangles?
4. Is it possible to dissect a square into 5 squares?
5. All shelves in Figure C, and all scraps, from which consists the piecework in Figure D,
are squares. Are the board and the piecework also squares?
6. Is it possible to tile the plane by pairwise distinct squares with integer sides?
A
B
C
D
√
7. Is it possible
to
dissect
a
square
into
rectangles
of
side
ratio
2
+
2? The same question
√
√
√
is for 2 − 2, for 3 + 2 2, and for 3 − 2 2.
√
√
8. Is the number 1 + 2 the square of a number of the form a + b 2, where a and b are
rational?
A tiling by rectangles is drawn on a rectangular sheet of paper. One may cut the sheet of
paper along any straight line segment into two rectangles, then repeat the same operation
separately for each of the obtained parts, and so forth. If it is possible to represent the
initial tiling in this way, then the tiling is called simple. For instance, the tilings shown in
Figure A,B are simple, but those in Figure С,D are difficult. It is suggested that you solve
the following 4 problems for simple tilings first and for arbitrary tilings afterwards.
9. Which rectangles can be (simply) tiled by rectangles with one of the sides equal to 1?
10. Which rectangles may be (simply) dissected into squares?
√
11. Is it possible to√dissect a square
(simply)
into
rectangles
of
side
ratio
2? The same
√
√
3
3
question is for 1 + 2, for 1 + 2, and for 2.
√
12. All numbers, which can be represented in the form x = a + b 2 with rational a, b, are
called good. For which good x a square may be (simply) tiled by rectangles of side ratio x?
Summer Conference of the Tournament of Towns
2014
A square from similar rectangles
Sergei Dorichenko, Mikhail Skopenkov, Olesya Malinovskaya
The Freiling–Laczkovich–Rinne–Szekeres Theorem (1994).
For r > 0 the following 3 conditions are equivalent:
(1) a square can be tiled by rectangles of side ratio r;
(2) for some positive rational numbers c1 , . . . , cn we have
1
c1 r +
= 1;
1
c2 r +
1
c3 r + · · · +
cn r
(3) the number r is a root of a nonzero polynomial with integer coefficients such that all
the complex roots of the polynomial have positive real parts.
From dissections to systems of linear equations.
You road I enter upon and look around, I believe you are not all that is here,
I believe that much unseen is also here.
Walt Whitman “Song of the Open Road”.
13. Prove the implication 2) =⇒ 1) in the Freiling–Laczkovich–Rinne–Szekeres Theorem.
14. Prove that a good number√x (for the definition, see Problem 12) can be uniquely
√
represented in the form x = a + b 2, where a and b are rational. The number x̄ = a − b 2
is called conjugate to x. Prove that if x and y are good then x + y, x − y, x · y, x/y are
also good. How are conjugates to these numbers expressed through x̄ and ȳ?
15. A rectangle 1 × x is dissected into squares x1 × x1 , x2 × x2 , . . . , xn × xn . Prove that
numbers x, x1 , . . . , xn satisfy a system of linear equations with integer coefficients. Does
the following equality of areas follow from your system of equations: x = x21 + · · · + x2n ?
Try to find a system of equations implying this equality.
16. Suppose that a system of linear equations in variables x1 , . . . , xn has a unique solution.
Prove that x1 , . . . , xn can be expressed in the coefficients of the equations with the help of
operations of addition, subtraction, multiplication and division. In particular, show that if
all the coefficients are rational, then x1 , . . . , xn are rational.
17. Alice tiled a rectangle A with 7 squares A1 , . . . , A7 and a rectangle B with 7 squares
B1 , . . . , B7 . It is known that the sides of all the squares are horizontal or vertical. The
horizontal sides of rectangles A and B are equal.
It turned out that for every pair of indices i, j:
• if Ai is adjacent on the left to Aj by a part of the vertical edge, then Bi is adjacent on
the left to Bj by a part of the vertical edge;
• if Ai is adjacent from above to Aj by a part of the horizontal edge, then Bi is adjacent
from above to Bj by a part of the horizontal edge.
Prove that for every i = 1, . . . , 7 the sides of the squares Ai and Bi are equal.
18. The Dehn theorem. A rectangle can be tiled by squares (not necessarily equal) if and
only if the ratio of two orthogonal sides of the rectangle is rational.
Summer Conference of the Tournament of Towns
2014
From dissections to the roots of polynomials
Here is the test of wisdom,
Wisdom is not finally tested in schools. . .
Walt Whitman "Song of the Open Road".
19. A rectangle is dissected into several rectangles of side ratio r. Prove that the sides of
the initial rectangle are as P (r) : Q(r), where P (x) and Q(x) are polynomials with integer
coefficients.
20. We can choose these polynomials so that P (−x)/Q(−x) ≡ −P (x)/Q(x).
P (z)
> 0.
21. If x > 0, then P (x)/Q(x) > 0. If Re z > 0, then Re Q(z)
22. A square is dissected into several rectangles of side ratio r. Prove that r is a root of a
nonzero polynomial with integer coefficients.
23. Prove the implication 1) =⇒ 3) in the Freiling–Laczkovich–Rinne–Szekeres Theorem.
24. The Euclid Algorithm. Given a rectangle of side ratio r. Let us cut off it a square by
one straight-line cut. Then repeat the procedure with the remained rectangle, and so on.
Show that at some step we obtain a square if and only if r is rational.
25. The Euler-Lagrange Theorem. Prove that in this process two similar rectangles appear
if and only if r is irrational but is a root of a quadratic trinomial with integer coefficients.
26. The Foster-Cauer Reactance Theorem. Let R(z) = P (z)/Q(z) be a ratio of polynomials
with integer coefficients such that the degree of P is greater than the degree of Q and
R(−z) = −R(z) for all complex z. Then the following 5 conditions are equivalent:
(1)
(2)
(3)
(4)
if Re z > 0, then Re R(z) > 0;
if R(z) = 1, then Re z > 0;
if R(z) = 0, then Re z = 0 and R0 (z) > 0;
for some integer n ≥ 0 and real d1 > 0, a1 > b1 > a2 > · · · > bn ≥ 0
R(z) = d1 z
n
Y
z 2 + a2
k=1
z2
+
k
,
2
bk
(5) for some integer m ≥ 1 and rational d1 , . . . , dm > 0
R(z) = d1 z +
1
d2 z + · · · +
1
.
dm z
27. Prove the implication 3) =⇒ 2) in the Freiling–Laczkovich–Rinne–Szekeres Theorem.
28. If a square can be dissected into rectangles similar to a given one, then it can be
dissected simply (see definition in page 1).
29. Given a polynomial with integer coefficients and its root r, invent an algorithm of
dissecting a square into rectangles of side ratio r (if such a dissection exists).
30. The Yu–Chen Problem. Consider the following operation on a set of real numbers.
ab
bc
ca
Take any three elements a, b, c and add the numbers 1/a, a + b, a+b+c
, a+b+c
, a+b+c
to the
set. Then the process is repeated. Prove that if a rectangle of side ratio k can be tiled by
rectangles of side ratio r then k can be obtained from r by these operations.
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A square from similar rectangles
Sergei Dorichenko, Mikhail Skopenkov, Olesya Malinovskaya
Question. When a cube can be dissected into parallelepipeds similar to a given one?
What is next?
Faust
Where will we go, then?
Mephistopheles
Where you please.
The little world, and then the great, we’ll see.
With what profit and delight,
This term, you’ll be a parasite!
Goethe. “Faust”
Nobody knows the answer to the Question about the Cube raised above. For the following Theorems, we do not know
simple elementary proofs. We are interested even in partial progress.
31. The Prasolov-Skopenkov Theorem (2011) For r > 0 the following three conditions are
equivalent:
1) A rectangle of side ratio r can be dissected into rectangles of side ratio r in such a way
that not all of the dissection rectangles are homothetic to the initial rectangle.
2) For some positive rational numbers ci the following equality is held
1
1
= .
c1 r +
r
1
c2 r +
1
c3 r + · · · +
cn r
3) The number r2 is a root of a nonzero polynomial with integer coefficients, such that its
number of negative roots is 1 less than its degree.
32. The Laczkovich–Szegedy Theorem (1999). A square can be dissected into triangles
similar to a given one if and only if the given triangle has either angles (π/8, π/4, 5π/8), or
(π/4, π/3, 5π/12), or (π/12, π/4, 2π/3), or is right-angled with the ratio of cathetus equal
to a root of some polynomial with integer coefficients such that all its real roots are positive.
33. The Su–Ding Theorem (2005). Suppose that all the vertices of a polygon (not necessarily
convex) have rational coordinates, and all its sides are parallel to the coordinate axes. Then
it can be dissected into rectangles of side ratio r if and only if a square can be dissected
into rectangles of side ratio r.
34. The Kenyon Theorem (1998) Let A1 A2 A3 A4 A5 A6 be an L-shape hexagon, and its
vertices are enumerated starting from the vertex of a non-convex angle (see Fig. E). Then
the hexagon A1 A2 A3 A4 A5 A6 can be dissected into squares if and only if the system of linear
(
equations
A3 A4 · x + A1 A2 · y = A2 A3 ,
A3
A2
A1
A5 A6 · z − A1 A2 · y
A6
= A6 A1 ;
has a solution x, y, z with nonnegative rational x, y, z.
35. The Kenyon Theorem (1998) A trapezoid is dissected into trapezoids. The bases of
all trapezoids of the dissections are parallel, and the ratio of the midline and the altitude
is rational for every trapezoid of the dissection. Then for the initial trapezoid the ratio of
the midline and the altitude is also rational.
A4
A5
E
Summer Conference of the Tournament of Towns
2014
A square from similar rectangles
Sergei Dorichenko, Mikhail Skopenkov, Olesya Malinovskaya
Where to find solutions
1. Where to find solutions
Solution of most problems are available in the text written by the Taiwan participants of
the conference Hung–Hsun Yu and Brian Chen. An elementary proof of the main Freiling–
Laczkovich–Rinne–Szekeres theorem is given in [7].
Acknowledgements
Many problems are taken from the K. Kokhas collection and the papers [9, 2]. The
authors are grateful to K. Kuyumzhiyan for help with the translation and to G. Chelnokov
and A. Zaslavsky for help with solution checking.
References
[1] R. L. Brooks, C. A. B. Smith, A. H. Stone, and W. T. Tutte, The dissection of rectangles
into squares, Duke Math. J. 7 (1940), 312–340.
[2] S. Dorichenko, M. Skopenkov, A square from similar rectangles, in preparation (in
Russian). С. Дориченко, М. Скопенков, Квадрат из подобных прямоугольников,
препринт, http://arxiv.org/abs/1305.2598.
[3] M. Gardner, Squaring the square. In: The 2nd Scientific American Book of Mathematical
Puzzles and Diversions, University of Chicago Press, 1987, 256 p. М. Гарднер, Квадрирование квадрата. В: Математические головоломки и развлечения, М.:Мир, 1999.
[4] C. Freiling and D. Rinne, Tiling a square with similar rectangles, Math. Res. Lett. 1
(1994), 547–558.
[5] M. Laczkovich and G. Szekeres, Tiling of the square with similar rectangles,
Discr. Comp. Geom. 13 (1995), 569–572.
[6] Jiří Matoušek, Thirty-three Miniatures: Mathematical and Algorithmic Applications of
Linear Algebra, Amer. Math. Soc., 2010, 182 p.
[7] M. Prasolov and M. Skopenkov, Tilings by rectangles and alternating current, J.
Combin. Theory A 118:3 (2011), 920–937, http://arxiv.org/abs/1002.1356.
[8] M. Prasolov, M. Skopenkov and B. Frenkin, Invariants of polygons, 19-th summer
conference International mathematical Tournament of towns, Belarus, Minsk, 2007. М.
Прасолов, М. Скопенков и Б. Френкин, Инварианты многоугольников, 19-я летняя конференция международного математического Турнира городов, Беларусь,
Минск, 2007. http://www.turgor.ru/lktg/2007/1/index.php
Summer Conference of the Tournament of Towns
2014
[9] M. Skopenkov, M. Prasolov, S. Dorichenko, Dissections of a metal rectangle, Kvant
3 (2011), 10-16 (in Russian). М. Скопенков, М. Прасолов, С. Дориченко, Разрезания металлического прямоугольниа, Квант 3 (2011), 10-16.http://arxiv.org/abs/
1011.3180
[10] M. Skopenkov, V. Smykalov, A. Ustinov, Random walks and electric networks, Mat.
Prosv. 3rd ser. 16 (2012), 25-47 (in Russian). М. Скопенков, В. Смыкалов, А. Устинов,
Случайные блуждания и электрические цепи, Мат. Просв., 3-я серия 16 (2012), 2547. http://www.mccme.ru/free-books/matprosh.html
[11] I.M. Yaglom, How to Dissect a Square?, Math. Bibl., Nauka, Moscow, 1968, 112 p.
(in Russian). И.М. Яглом, Как разрезать квадрат? Мат. Библ., М.: Наука, 1968, 112
стр. http://ilib.mirror1.mccme.ru/djvu/yaglom/square.htm
[12] S. Wagon, Fourteen proofs of a result about tiling a rectangle, Amer. Math. Monthly
94:7 (1987).
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