The Wizard of Oz by Frank L. Baum is a well known American children’s 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 Scarecrow’s Conjecture.

The Problem as presented

in 1939

• 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.”

• The Pythagorean Theorem:

– “The square of the hypotenuse of a right triangle is equal to the sum of the squares of the remaining sides.”

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)1/2]2

4x = y

Case II: (AB)1/2 + (AC)1/2 = (BC)1/2

A

B

C

(x)1/2 + (y)1/2 = (x)1/2

(y)1/2 = (x)1/2 - (x)1/2

y = 0

Many have assumed that this

proves that the Scarecrow didn’t

have a brain in his head

y = 4x

“The scarecrow represents the idealistic Kansas farmer of the late 1800’s. 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.”

Yick, Rafiee, and Beaseley

The Scarecrow Conjecture Activity

The Crow TheoremThe 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 > 2x > y, y > y is false.

Case II: 2x = 2x + y, 0 = y is a contradiction

Is Oz Euclidean?

What do we know about Oz?

In Emerald City,

things are not

always as they

seem…

• 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)

ABC as a closed shape on a curved surface, which satisfies case I: y=4x

B C

A

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.B C

A

At the equator, AC=2πr and AB= 1/2(πr)

So just before this instant,

AC is approaching 2πr,

And CA is approaching 0.

Satisfying Case II, y = 0

B C

A

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).

0

Infinity

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

They could get quite

tangled, depending

on how long this is

allowed to go on

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

• 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

The “snowglobe” model of Oz.

Our research reveals the true and actual story on which the Wizard of Oz is based

• Dorothy arrived in Oz at when the cyclone dropped her house near the western negative pole.

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

...and disappear all together

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.

Dorothy and company reached Emerald City and obtained and audience with the Wizard.

The Wizard, is his wisdom, refused to grant the party’s 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 West’s broomstick, and in the process, accidentally killed her.

Now that the crisis was over, the Wizard could consider their requests.

Dorothy said she liked him better the way he was, but he went through with the operation anyway.

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.

(which you would only know if you’d read the book)

But why did he say such a thing?

Dorothy’s mental condition and the psychological consequences of the ‘scarecrow instant’

“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

• 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.