The Wizard of Oz by Frank L' Baum is a well known American childrens story first published in 1900'

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The Wizard of Oz by Frank L. Baum is a well
known American childrens story first published
in 1900. The story follows the adventures of a
young Kansas farm girl who carried away by a
cyclone to the magical Land of Oz. The girl,
Dorothy, and her little dog, Toto, journey along
the Yellow Brick Road to Emerald City. There,
she will ask the Wizard of Oz to help her get
back home. Along the way she meets a Scarecrow,
a Tin Woodsman, and a Lion, who join the quest in
order to ask the Wizard for a brain, a heart, and
courage. MGM Studios made a musical version of
the story in 1939 that is considered one of the
best films ever made. An equation is stated near
the end of the film which has become known as the
Scarecrows Conjecture.
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The Problem as presented in 1939
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The Pythagorean Theorem The square of the
hypotenuse of a right triangle is equal to the
sum of the squares of the remaining sides.

• What the Scarecrow said
• The sum of the square roots of any two sides of
  an isosceles triangle is equal to the square root
  of the remaining side.

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Case I (AB)1/2 (BC)1/2 (AC)1/2 (x)1/2
(x)1/2 (y)1/2 2(x)1/2 (y)1/22 4x
y Case II (AB)1/2 (AC)1/2 (BC)1/2
B
(x)1/2 (y)1/2 (x)1/2 (y)1/2 (x)1/2 -
(x)1/2 y 0
A
C
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Many have assumed that this proves that the
Scarecrow didnt have a brain in his head
y 4x
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• The scarecrow represents the idealistic
  Kansas farmer of the late 1800s. Though these
  farmers would have a lot of common sense, they
  may not feel intelligent because of their lack of
  formal education. The wizard offers a diploma,
  the symbol of formal education. To demonstrate
  the impact of artificial education, the scarecrow
  tries to state some mathematics, the quintessence
  of knowledge.
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  Yick, Rafiee, and Beaseley
• The Scarecrow Conjecture Activity

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The Crow Theorem

• The sum of the square roots of sides of an
  isosceles triangle is NOT equal to the square
  root of the remaining side.
• Proof by contradiction
• Case I y 4x gt 2x gt y, y gt y is
  false.
• Case II 2x 2x y, 0 y is a
  contradiction

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Is Oz Euclidean?
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What do we know about Oz?
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In Emerald City, things are not always as they
seem
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• Oz is an American Fairyland
• Small (little girl can walk across it)
• Surrounded by a deadly desert
• Divided into 4 kingdoms
• 2 good witches, 2 bad witches
• Yellow Brick Road
• Capital city located at x-y intersection
• Evidence of an underworld
• Repeated imagery of magic spheres
• Many of its inhabitants, real, imaginary, or
  somewhere in between, are aware that they would
  not exist in other places (like Kansas)

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• ABC as a closed shape on a curved surface, which
  satisfies case I y4x

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To start thinking of a solution, put the pacman
shape on a giant ball with point B stuck to the
top and start collapsing the ball under it.
As it turns out, when AB is above the equator, AC
is less than the corresponding circumference it
is riding on, and greater than the circumference
when AB has stretched below the equator.
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At the equator, AC2pr and
AB 1/2(pr) So just before this instant, AC is
approaching 2pr, And CA is approaching
0. Satisfying Case II,
y 0


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In Riemannian geometry, there is a Reimannian
Sphere, on which lines are great circles on a
sphere. The sphere is balanced on a coordinate
plane of made of real and imaginary numbers.
Lines on the imaginary are shadows of the great
circles. Infinity is an actual point in space
on the north pole, opposite of the origin on the
south pole (shown reversed here out of respect
to Australia).
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Suppose there are 2 such triangles, one positive
and one negative.
If the triangles are allowed to continue to curve
past the equator as the sphere collapses, the
non-isosceles legs will begin to wind up
somewhere in the interior
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• They could get quite tangled, depending on how
  long this is allowed to go on

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• If nothing stops this action, the number of
  scarecrow instants will approach infinity as
  the vertices approach infinty.

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• Fortunately, the vertices positive and negative
  triangles create opposing forces which prevent
  the isosceles sides from reaching the infinite
  pole, and equilibrium is achieved.

Simplified map of Oz
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The snowglobe model of Oz.
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Our research reveals the true and actual story on
which the Wizard of Oz is based
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• Dorothy arrived in Oz at when the cyclone dropped
  her house near the western negative pole.

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• The non-isosceles leg of east-west triangle
  began to deteriorate...

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• ...and disappear all together

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With the eastern (negative) pole annihilated,
attraction by the northern and southern
(positive) poles would bring the western
(negative) pole crashing down on Emerald City.
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Dorothy and company reached Emerald City and
obtained and audience with the Wizard. The
Wizard, is his wisdom, refused to grant the
partys demands until they had averted the
impending disaster (which Dorothy had started in
the first place). The party set out on
another quest to retrieve the Wicked Witch of the
Wests broomstick, and in the process,
accidentally killed her.
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Now that the crisis was over, the Wizard could
consider their requests.
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The Wizard advised Scarecrow that wisdom is
gained from experience, not brains. But, if he
still wanted a brain, he could come back the next
day.
Dorothy said she liked him better the way he was,
but he went through with the operation anyway.
(which you would only know if youd read the book)
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But why did he say such a thing?
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Dorothys mental condition and the psychological
consequences of the scarecrow instant
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• The Scarecrow told them there were wonderful
  thoughts in his head but he would not say what
  they were because he knew no one could understand
  them but himself.
• -L. Frank Baum

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• Baum, Frank L. The Wonderful Wizard of Oz. 1900.
  Illustrated by W. W. Denslow.
• Baum, Frank L. Ozma of Oz. Illustrated by
• MGM Studios. The Wizard of Oz. 1939.
• Yick, Rafiee, and Beasley. The Scarecrow
  Conjecture Activity. Augusta State University,
  March 2000.
• Pancari and Pace. Two Views of Oz.
  Mathematics Teacher Vol. 80 No. 2, February
  1987.
• Osserman, Robert. The Poetry of the Universe.
• Special thanks to Mr. John Gishe who taught a
  unit on non-euclidean geometry to a bunch of very
  confused tenth graders at Northgate High in the
  spring of 1979, and still does. Euclid, Greek,
  The Elements, 300 BC, University of Alexandria.