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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