by iampunha | 7/10/2008 08:00:00 AM
1+2=3.

I assume this is not news to anyone.

But 10+15+55=80.

Why should you care?

Because 1+2+3+4=10, 1+2+3+4+5=15 and 1+2+3+4+5+6+7+8+9+10=55.

100 is nothing more (or less) than 55+45 (1+2+...9).

1,000 is just 990 (1+2+...44) and 10.

In short (and it gets long), every positive integer is the sum of no more than three triangular numbers. A triangular number is the sum of n and every number between n and 0.

So today, on a triangular number day (10), we're going to have some fun with Carl Friedrich Gauss, who discovered this lovely little rule on July 10, 1796.



For the victims of the Jedwabne Pogrom, which took place 67 years ago today.

One of the benefits of being allowed to ask a lot of questions of people when you are a child is that you get to play around in your mind a lot.

Sometimes this leads to interesting stories, like lighting rockets indoors. And sometimes it leads to comfort with subjects that unnecessarily frighten people, like math.

Forget about all the math that confuses you. Toss square roots if they tie you in knots. Think of imaginary numbers as just that: imaginary.

And walk with me down triangular number boulevard.

Every positive number is the sum of at most three other numbers (themselves the sums of a varying number of numbers). (By the time I'm done with this entry, number will look weird to you.)

Does this much daunt you? Do you wish you hadn't started reading it?

Then think of it a different way.

What if your age were the product of at most three years of your life? What if, at 50, your life had been shaped by what you did at 1, 21 and 28?

Come to think of it, that isn't an untenable position. You could strike up a pretty good argument that your life at 50 is defined largely by when your physical and social development started speeding up, life in your last year of college and life after your second promotion.

I'm 26. According to this conjecture (which is so far from scientific as to be the Intelligent Design of the math world), my life might be defined by life at 15, 10 and 1.

At 1, I was a slow social developer. (I'm told I still lack social clue-intake skills.)

At 10 and 15, I was being punished socially for knowing a lot academically but not a lot outside of books. I fled to the Internet pretty much as soon as that was viable.



Anyone who remembers Simon from Alvin and the Chipmunks can imagine reading a book titled "Incredibly Complex Geometry Constructions for a Rainy Day."

How about this one?

All Gauss did for that was figure out how to get an angle of 16cos(2pi/17) with just two compasses and a straight edge.



If that construction (not a drawing but a construction) isn't enough to make you dizzy, consider this gift from Gauss:

5=3 (mod 2)

How does this work? Very simply:

If your counting mechanism repeats (as it does on a clock, on a track, that sort of thing), then equal positions are equal. So if you are alternating between two choices, your fifth switch is equal to your third switch. So 5=3. For a more complicated discussion of this concept, have fun here. (And note that the concept of a number system base non-10 is definitely not a Gauss invention.)



From here, things get a little more complicated. My schooling stopped, for our purposes, at calculus, with some statistics and matrix algebra thrown in just to see what I could handle. So I'm not going to show you the math behind how Gauss ... reinvented a system of calculations in his 20s:

To recover the object once it emerged from the sun's glare several months later, astronomers needed to know its orbit. Piazzi's observations, however, covered a period of just 41 days, during which time the object had moved through an arc of only 3 degrees across the sky. Any attempt to compute the orbit of such an inconspicuous object from this meager set of data appeared futile.

To Carl Friedrich Gauss (1777-1855), a 24-year-old mathematician who early in life had displayed a prodigious talent for mathematics and a remarkable facility for highly involved mental arithmetic, this problem presented an enticing challenge. Having completed his studies at the University of Göttingen, Gauss was living on a small allowance granted by his patron, the Duke of Brunswick.

With a major mathematical work just published and little else to occupy his time during the latter part of 1801, Gauss brought his formidable powers to bear on celestial mechanics. Like a skillful mechanic, he systematically disassembled the creaky, ponderous engine that had long been used for determining approximate orbits and rebuilt it into an efficient, streamlined machine that could function reliably given even minimal data.


It gets cooler.

Way cooler.

Remember having to solve for x?

Now imagine solving for x, x1 and x2.

And you have an a1, a b1, etc. Bolding mine:

Historical accounts typically omit the mathematical details of how Gauss solved the problem of determining the orbit of Ceres. In an illuminating article in the April Mathematics Magazine, Donald Teets and Karen Whitehead of the South Dakota School of Mines and Technology in Rapid City fill in that gap.

"Gauss's work offers a rare instance of solving an historically great problem in applied mathematics using only the most modest mathematical tools," Teets and Whitehead remark. "It is a complicated problem, involving over 80 variables in three different coordinate systems, yet the tools that Gauss uses are largely high school algebra and trigonometry!"

"Gauss achieves greatness in this work not through deep, abstract mathematical thinking, but rather through an incredible vision of how the various quantities in the problem are related, a vision that guides him through extraordinary computations that others would likely abandon as futile," they add. Indeed, it's often difficult to see how the various computational steps Gauss undertook might reasonably lead to the final goal.

In essence, Gauss confronted the problem of how to determine two sun-centered vectors approximating the planet's position at two different times, given three Earth-centered observations of the object's latitude and longitude. Solving that problem then allowed him to determine the planet's orbital plane and the shape and orientation of its elliptical orbit within this plane.


Want more detail? Have fun (PDF).

Want more on other things Gauss did? (The list of Gauss' accomplishments is pretty nontrivial.) Use this as a jumping-off point.

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