If
I Only Had A Brain
I could while away the
hours
Conferrin' with the flowers,
Consulting with the rain;
And my head I'd be a scratchin'
While my thoughts are busy hatchin'
If I only had a brain.
Does the Scarecrow already
have a brain?
In the movie The Wizard of Oz, the Scarecrow demonstrates
that he has a brain, but he is given a diploma to prove that he has a
brain.
"Oh No, Scarecrow! Math from the Wizard of Oz by JW Gaberdiel
At the end of The Wizard of Oz,
the Scarecrow receives a diploma and then immediately says,
“The sum of the square roots of
any two sides of an isosceles triangle is equal to the square root of the
remaining side.” https://www.youtube.com/watch?v=DUCZXn9RZ9s
This is unfortunate. It sounds a lot like the Pythagorean
Theorem:
“The sum of the squares of the
legs of a right triangle is equal to the square of the hypotenuse.”
However, Scarecrow’s version is
wildly and devastatingly different from Pythagoras’ version. "
Actually neither is correct. Gaberdiel assumes that the Scarecrow was talking
about a triangle on a Euclidean, flat, surface.
What the Scarecrow should be
saying if the surface is hyperbolic is
“The product of the hyperbolic cosines
of the legs of a right triangle is equal to the hyperbolic cosine of the remaining
side."
The squares, square roots, and the Pythagoras’ Theorem for
right triangles only apply on Euclidean, flat, surfaces. It is true because on a flat surface, cos(c)=cos(a)*cos(b),
where a, b, and c are the sides of a right triangle, and this is equivalent
to Pythagoras Theorem. On a spherical surface,
such as the Earth, the formula is cos(c/R)=cos(a/R)*cos(b/R) where R is the Radius
of the special surface. When a, b, and c
are very small compared to R , i.e. as R goes to infinity, the limt is cos(c)=cos(a)*
cos(b), which is Pythagoras’ Theorem. But as any airplane pilot will confirm, a Great Circle
Distance is not solved using Pythagoras’ Theorem. If the surface is hyperbolic, not spherical, then Pythagoras’
Theorem is also not correct. The correct formula is cosh(c)=cosh(a)cosh(b). The Scarecrow may have been given a diploma, but
to be correct he also needed to get an imagination, the component that
makes a function, and the surface it is on, hyperbolic.
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