Tuesday, January 22, 2013

Pascal's Triangle long before Pascal.

Blaise Pascal did not call the array of numbers he studied "Pascal's Triangle". In math, it's considered poor form to name something after yourself.

Pascal's Treatise on the Arithmetic Triangle was published posthumously. In it, he gathered together all the facts he knew about the patterns he discovered himself or had read about in other books.  It became the "go to" text for information about the array and other mathematicians started calling it "the triangle of M. Pascal" so much so that it is now the way nearly everyone in the world refers to it.

Nearly everyone. About 100 years before Pascal, the great Italian algebraist Niccolo Fontana, known by his nickname Tartaglia - which means "the stammerer" - did a lot of work with the number pattern and in Italian the array is known as Tartaglia's Triangle.

Several centuries earlier, the Chinese were discovering things about the array, and in Chinese it is known as Yang Hui's Triangle.

But the Chinese from 700 years ago are not the first people to study the numbers and leave a paper trail behind that future generations could find. There are two completely different problems from before the birth of Christ that originate in India whose answers come from the numbers in the array we call Pascal's Triangle.

Let's say we have a spice rack with six flavors: salt, pepper, garlic, nutmeg, curry and basil.  How many different combination of three spices are there? (Note: in this problem, we are not saying how much of any one spice we are using, only if it is used. Two parts salt and one part garlic would taste different from one part salt and two parts garlic, but in this problem we would say that both are salt/garlic combinations.)

If we abbreviate the spices to S, P, G, N, C and B, here are the 20 different groups of three

Salt included

SPG  SPN SPC  SPB
SGN SGC SGB
SNC SNB
SCB

Salt excluded, pepper included
PGN PGC PGB
PNC PNB
PCB

Salt and pepper excluded, ginger included
GNC GNB
GCB

No salt, pepper or ginger
NCB

The other ancient problem from India that uses the binomial coefficients deals with music and rhythm. In the musical notation developed in Europe that is used almost everywhere today, if we say a song is in a rhythm of six beats, all those beats have the same duration. In India, beats can either be short or long. For instance, let's say we had a song that has double hand claps on the second and fourth beat of every measure, so the pattern might the counted out

bump clap-clap bump clap-clap...

In Western music, we would say this is a four beat pattern. In Indian music, they would say it is a six beat pattern.

long short short long short short.

Okay so how many different six beat patterns have three long beats and three short beats? Again the answer is 20.

First beat long
LLLSSS LLSLSS LLSSLS LLSSSL
LSLLSS LSLSLS LSLSSL LSSLLS
LSSLSL LSSSLL
First beat short, second beat long
SLLLSS SLLSLS SLLSSL SLSLLS
SLSLSL SLSSLL
First two beats short, third beat long
SSLLLS SSLLSL
SSLSLL
First three beats short
SSSLLL

At first glance, these problems don't seem to be connected, but in fact they are. If we line up the spices alphabetically in English, we would get.

Basil Curry Garlic Nutmeg Pepper Salt

Take any six beat pattern.

LLSSLS

Think of L as being yes and S as being no

YesYesNoNoYesNo.

Make a spice combination where we only use the spices that correspond to the Yes positions.

Basil Yes, Curry Yes, Garlic No, Nutmeg No, Pepper Yes, Salt No.

In this way, the beat pattern LLSSLS corresponds to the recipe that uses basil, curry and pepper. If two beat patterns are different, they will correspond to different recipes. In math, this kind of matching is called a one to one correspondence, and it is one way to prove that one set of objects has the same number of things as another set.

Tomorrow, we will look at actual geometric triangles, notably triangles with one right angle, known simply enough as right triangles.