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Math-Tans present....
"Plane Geometry"
(*aka "Winter Wonderland")
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Original 1934 by Richard Bernhard Smith (lyrics), Felix Bernard (music), Richard Himber (song).
Updated in 1947 ("circus clown"), sung by Johnny Mathis with Bridge Verses.
Johnny Mathis Version: https://www.youtube.com/watch?v=HAOI-iiHydU
Eurythmics Version (starts with Bridge): https://www.youtube.com/watch?v=4xVLeW9UmjE
Most other versions simply run Chorus1, Chorus2a, Chorus2a (so no elliptic space).
*
CHORUS 1:
In our schools, we're agreein'.
What's the plane? Euclidean.
It's a vector space,
Five rules are in place,
Working in this plane geometry.
Euclid states: Points, lines, circles.
Elements, from his journals.
Right angles equate,
Plus fifth postulate,
Working in his plane geometry.
CHORUS 2a ("Parson Brown"):
But let's redefine that parallel line.
Hyperbolic space is the outcome.
We'll check, got rectangles?
This shows no sign.
And triangles give smaller angle sums.
Later on, we'll see drawn,
Regular apeirogons.
Relativity,
Fits naturally,
Working in NEW plane geometry.
*BRIDGE VERSES:
How this is found takes too long to recite,
Parallel postulate? Was needed outright.
First Lobachevsky,
Then Riemann did set free,
Euclid's grid.
If we go 3-D, if we add height,
Now there's eight models. Thurston sheds light.
Look up at the sky,
For those paths, where planes fly,
Aren't Euclid!
(Not Euclid, aren't Euclid, not Euclid.)
CHORUS 1:
In our schools, we're agreein'.
What's the plane? Euclidean.
It's a vector space,
Five rules are in place,
Working in this plane geometry.
Euclid states: Points, lines, circles.
Elements, from his journals.
Right angles equate,
Plus fifth postulate,
Working in his plane geometry.
CHORUS 2b ("circus clown"):
But let's redefine that parallel line.
An elliptic space this could become.
Tweak a second, sphere
Comes from that design.
Now triangles give larger angle sums.
Three systems, have us reeling,
Though our world, we're revealing,
So frolic and play,
With space every way,
Working in NEW plane geometry.
OPTIONAL REPEATS:
Planes fly curves in plane geometry.
Taxicabs constrain geometry.
Working in some plane geometry.
Quirks don't work in plain geometry.
Workin' work in plain geometry.
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Previous 11 XMas Specials:
2021: "Walkin' Around Unit Circles" ; 2020: "Solving for Unknowns" ; 2019: "Three Things of Math" ; 2018: "Sequenced" ; 2017: "Triangles Rock" ; 2016: "Do We Have a Square Here?" ; 2015: "Like a Bell" ; 2014: "Polar Plot"; 2013: "O Factor Tree" ; 2012: "A Squared Song" ; 2011: "Decimal"
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