a lgebraic solution to a geometric problem
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A lgebraic solution to a geometric problem. S quaring of a lune. Even in ancient times people have watched and studied the dependence of the moons and their daily lives. - PowerPoint PPT PresentationTRANSCRIPT
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Algebraic solution to a geometric problem
Squaring of a lune
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Even in ancient times people have watched and studied the dependence of the moons and their
daily lives
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Figure enclosed by two arcs of circles are
called Freckles (moons) because of
their similarity with the visible phases of the moon, the moon of
earth.
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The squaring of a plane figure is the
construction – using only
straightedge and compass – of a square having
area equal to that of the original plane figure.
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Since OC is a radius of the semicircle => OC=r
By the Pythagorean theorem
2 .AC r
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Construct the midpoint D of AC.
2 ,AC r
/ 2.( 2 )AD r
Construct the
Semicircle
midpoint D of AC.
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Our goal is to show that the purple lune AECF is squarable.
area of
AEC= 2( ) / 2.r
ACB= 2.r
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So, it terms of areasSo, it terms of areas
=1/2
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In terms of the first octant of our shaded figure, this says that:
Small semicircle = 1/2 large semicircle.
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=
=
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In many archaeological excavations in the
Bulgarian lands have found drawings of the
moon in different
Phases.
Excavations in Baylovo
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In 1840 T. Clausen (Danish mathematician) raises the question of finding all the freckles with a line and a compass, provided that the central angles of the ridges surrounding are equal. That means that there must be a real
positive number Q and positive mutual goals primes m, n, such that the corners are met:
= m.Q 1= n Q
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He establishes the same cases to examine and Hippocrates of Hios but expresses the hypothesis that freckles can be square in the following 5 cases:
m= 2 n= 1m= 3 n= 1m= 3 n= 2m= 5 n= 1m= 5 n= 3
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In 1902 E. Landau deals with the question of squaring a moon. He proves that the moon can be squared of the first kind Numbers
, b = sin 1а = sin
C = 122
1
SINSIN
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If the number of c= 0 is squarable moon, and if the corners are not
commensurate Moon squarable. It is believed that he used the addiction,
which is familiar and T. Clausen.
n Sin(m) = m Sin (n 1 )
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Landau considered n = 1, m = p = +
a gaussian number in which the moon is squarable. When k = 1, k = 2 are
obtained Hioski cases of Hippocrates.
1
2k
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In 1929 Chakalov Lyubomir (Bulgarian mathematician) was interested of tLandau’s work and used algebraic methods to solve geometric problems. Chakalov consider the case p = 17, making X = cos 2 and obtained equation of the eighth grade.
Х8 + Х 7 - 7Х6 + 15Х4 +10 Х3 – 10Х2 – 4Х + 1 - 17 = 0
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He proves that this equation is solvable by radicals square only when the numbers
generated by the sum of its roots are roots of the equation by Grade 4.
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Chakalov use and another equation Х = cos2 + sin 2
Then: n Xn ( Xm - 1 )2 – m Xm ( Xn – 1 )2 = 0
X = 1 is the root of the equation So he gets another equation:
22 )1
1()
1
1()(
Xm
XXP XXX n
nmm
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Chakalov consider factoring of this simple polynomial multipliers for
different values of m and n. So he found a lot of cases where the freckles are not squrable. It extends the results of Landau for non Gaussian numbers.
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In 1934 N.G. Chebotaryov (Russian mathematician) Consider a polynomial
Chakalov and proves that if the numbers m, n are odd, freckles squrable is given
only in cases of Hippocrates, and in other cases not squaring.
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In 1947 a student of Chebotaryov,
A. C. Dorodnov had proven cases in which polynomial of Chakalov is broken into simple factors and summarizes the work of mathematicians who worked on the problem before him. So the case of clauses is proven.
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Thus ended the millennial history of a geometrical problem, solved by
algebraic methods by mathematicians’ researches from different nationalities.
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Tomas Klausen
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Hippocrates of Chiosabout 470 BC - about 410 BC
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Tomas Klausen
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Edmund Landau
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Любомир Чакалов
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Чеботарьов
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Анатолий Дороднов
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References: “Bulgarian mathematicians” Sofia, avt.Ivan Chobanov, P. Roussev
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Made by:
Kalina Taneva
Ivelina Georgieva
Stella Todorova
Ioana Dineva
SOU “Zheleznik” Bulgaria