The Gauss Christmath Special

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The Gauss Christmath Special : Christmas ain't over yet. The 12th day is Jan. 6th! You can get just the song here: http://vihart.com/music/gauss12days.mp3 http://vihart.com

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On the first day of Christmas
my true love gave to me
the multiplicative identity.
I always hated the song
"The Twelve Days of Christmas"
when I was younger.
Not that the tune or the words are any
worse than any other Christmas song
but it's just so long and repetitive.
Singing it sucks too
because like math class it seems
no matter how hard you try to pay attention
you lose focus out of the sheer tedium of it all
and forget where you are, unless you keep
vigorous records by drawing complicated graphs.
On the second day of Christmas
my true love gave to me the only even prime
and the absolute value of e to the i Pi.
See it doesn't have to be repetitive
but the "Twelve Days of Christmas"
is more fun to think about than
to actually sing or listen to.
I do like these kinds of visualizations
and you can think of other ways to visualize
the process of singing the "Twelve Days"
which might come in useful next time you are
at a family Christmas party and someone
insists on singing through the whole thing.
On the third day of Christmas, my true love
gave to me, the number of spatial dimensions
at least macroscopically -
don't yell at me strict theorists -
the number of points that define a line
and the limit of sine x, over x, as x goes to zero.
Unlike normal reasonable songs adding twelve
more verses wouldn't just make it twice as long
because the stupid thing just grows and grows
even ninety-nine bottles of beer on the wall
has a promise of getting to zero unless
you believe in anti beer. But "Twelve Days" is just
disheartening, because the closer you think
you're getting to the end the longer the verses get
dragging it out to the bitter end.
On the fourth day of Christmas
my true love gave to me
the smallest possible number of sides on a polyhedron
the number of points that define a plane
the divisor of even numbers and
any other number to the power of zero.
If i had a time machine and was not bitterly
anti-time travel, and yes I've actually
protested with this sign. One of the things
on my list would be to go back to year zero -
pick your era and be like "hey"" three kings
of Orient hurry it up will ya??
Also what's myrrh because it sounds like the noise
a camel makes... Really though five days
of Christmas would be more reasonable.
Even eight, I could live with.
On the fifth day of Christmas my true love
gave to me...
five golden ratio producing pentagons,
the number of sides on a square,
the number of sides on that rigid,
funtional and beautiful creature
called the triangle and
I guess a two....
and the number of sides on a Möbius strip.
Graphing the numbers like this may seem trivial
but I think it's nice to be reminded these
numbers aren't just squiggles on a page
symbols to be manipulated, but actually
represent something and I think it's nice that
it results in such a lovely triangle.
You know how I feel about triangles.
On the sixth day of Christmas
my true love gave to me...
my name in Roman numerals,
the number of feet in iambic pentameter.
My name in Roman numerals backwards,
the first Mersenne in a prime,
the number of syllables in a foot of iambic pentameter
and sine(x) squared plus cosine(x) squared.
Right, so if you want to know how many times
you're gonna have to sing a line about
whatever stuff your true love got you -
it's like counting up all the things
in one of these triangles that has
however many layers - in this case twelve
the answer will be what's called a
triangular number for obvious reasons.
You can also shift the things around
to get an equilateral triangle
which is how triangular numbers
are usually explained.
So, the first triangular number is one.
The second is three. The third is six.
The fourth is ten.
On the seventh day of Christmas
my true love gave to me the most common
lucky number, the first perfect number,
the only prime ending in five...
the number of colors sufficient
for coloring in a map
the only prime triangular number
the highest number that is its own factorial
and a half, plus a fourth, plus an eighth,
plus a sixteenth and so on...
....almost.
Here's a famous story that I've heard told
a few different ways and the actual facts are fuzzy,
but basically here is the gist of it.
Carl Gauss was bored in his math class.
I can imagine why because not only was
Gauss a pretty smart guy,
but math classes sucked
three hundred years ago
just much as the suck today and
Gauss would get himself in trouble
when he was bored.
Maybe he also like to escape via the window.
So, the teacher got fed up and was like
"Gauss" I mean he probably
called him "Carl"...but that's not the point.
"Gauss if you're so freaking bored with my class
how 'bout you go sit in the corner and add up
all the numbers between one and a hundred.
That ought to keep you busy for awhile."
So, Gauss goes to the corner, but he's just
sitting there and his teacher gets all mad and
is like "Hey Gauss, I guess that means
you've already added up all those numbers right?"
And Gauss is like "Sure, it's five thousand and fifty."
On the eighth day of Christmas, my true
love gave to me, the only perfect cubed
positive Fibonacci number besides one,
the number of frieze patterns,
the number of sides on a cube,
the number of platonic solids!
The first composite number,
the number of regular polytopes
in all dimensions greater than four
the Euler characteristic
of polyhedral homeomorphic to a sphere
and the number where the indivisive
phase of a logarithm is undefined....
His teacher of course didn't believe him.
I think the teacher spent the next 10 minutes
adding up the numbers by hand in an effort to
catch Gauss in a lie and when he saw that Gauss
was right, he probably gave him detention anyway.
Or, more likely whacked him with a ruler,
a few times for having the gall
to be smarter than him, or it could be
that the story is mostly made up.
I don't know, nevertheless
here's how he did it.
Instead of adding up the numbers
individually like his teacher did,
which would have been super boring,
he realized this fact.
The numbers one through a hundred, come in pairs
that add up to a hundred and one.
One plus a hundred, two plus ninety-nine,
three plus ninety-eight, four plus ninety seven,
and so on. There's fifty such pairs of numbers.
And, fifty times a hundred and one
is real easy to do in your head
'cause it's fifty times a hundred plus fifty.
Or, five thousand and fifty.
On the ninth day of Christmas, my true love
gave to me... an upside down six,
infinity sideways, flipped over "L",
an upside down 9, a funny looking "S",
a sail for a boat, backwards "E", half a heart on a
plate and a simple boring short and straight line...
So, yeah if you want to add up all the numbers
between one and anything, this trick works.
For example all the numbers between
one and twelve. One plus twelve is thirteen.
Two plus eleven is thirteen. All the way in
to six plus seven. That's six times thirteen,
which I do in my head as
sixty plus eighteen equals seventy-eight.
So, seventy-eight is the twelfth triangular number.
While five thousand and fifty is the
one hundredth triangular number.
At least it's not "One Hundred Days of Christmas".
I'll stick with bottles of beer.
On the tenth day of Christmas,
my true love gave to me...
The base of our Arabic numeral system!
The base of the nonary numeral system.
The base used in octal,
and the base of septenary,
and the base of senary
and a number five...
quaternary's base, tenary's base also
and binary too...
and the base of unary....
Here's my favorite way of visualizing
the triangular numbers.
Say you've got them in this configuration
where they make a nice right equilateral triangle.
Or, half a square. Finding the area of a square
is easy because you just square the length of it.
In this case, twelve times twelve.
And the triangle is half of that. Only, not really
because half the square means you only get
half of this diagonal. So, you've got to
add back in the other half.
But, that's easy because there's twelve things
in the diagonal, and twelve over 2 is six.
So, to get the nth triangular number,
just take n squared, over 2, plus n, over 2.
Or, n squared, plus n, over 2.
On the elventh day of Christmas
my true love gave to me...
the number my amp goes up to,
the number of fingers on my hand,
the German word for no,
what I did after I eat,
the number of heads on a hydra,
at least until you start cutting them off...
The number of strings on a guitar.
The number I like to do high....
The amount of horsemen of the Apocolypse.
The number of notes in a triad,
the number of pears, in a pair of pears...
and the number of Partridges
in a pear tree...
On the twelfth day of Christmas,
my true love gave to me..
Actually enough is enough...
Merry Christmath!
Epsilon greater-than, VI HART