Welcome to Ryan's Math Capstone page!
This page was designed to show off my senior research at
California Lutheran University.
The Several Symmetries of Pascal's Pyramid*:
*best when viewed in Netscape 7.2 or later
Pretty much, my decision to start this project boiled down to my advisor saying "Ryan, I have found the perfect project for you and there is no way I will let you do ANYTHING ELSE!" My reaction was nonchalant since indeed it had taken me a while to decide after changing my project idea many times. Her discipline did in fact pay off however, since my interest in "The Symmetries of Pascal's Triangle" skyrocketed after a little investigation.
Background
My background going into
this project was that I had seen Pascal's
Triangle in high school and could appreciate its astounding properties. My first
college course in which I noticed an abrupt change in my abilities was Discrete
Math. I was especially proficient in seeing patterns and especially in probability
I noticed I was quick at combinatorial manipulations. More inspiration for this
project was drawn from my course in Number Theory, in which I more thoroughly
appreciated the beauty of the integers. The fact that Pascal's Triangle can
be derived from the binomial theorem is quite a fascinating property. Given
my love for puzzles and pattern-finding games of the mind, I was eager to see
what more I could discover about this mysterious arrangement of integers.
What or who is Pascal?
The triangle begins with 1 and then 1,1 and continues with 1’s on the outside.
The rest of the terms, on the other hand, are determined from the sum of the two terms directly above them.
For example, for the three terms circled in black, 1+1=2 and for the three terms circled in orange, 1+3=4.
The triangle has many symmetrical properties, in fact Pascal’s found a total of 17!
Many of the symmetries of the triangle have to do with a field of mathematics known as Combinatorics.
According to The Mathematics Atlas at Northern Illinois University, "Combinatorics is, loosely, the science of counting. This is the area of mathematics in which we study families of sets (usually finite) with certain characteristic arrangements of their elements or subsets, and ask what combinations are possible, and how many there are."
We can use this definition to distinguish between algebraic arguments and what are known as "combinatorial arguments." An example of a Combinatorial question might be:
How many ways can one order all the cards in a 52 card deck?
We have 52 cards, we are interested in all the
different possible arrangements
There exist 52 ways to choose the first card, 51 ways to choose the second card,
and so on…
These possiblities are represented by the product, 52*51*50*…*1 in which
we care about the order that we recieve cards.
This diagram represents all the possible ways to choose the next card after
choosing the first.
But what if we are interested in only choosing 5 cards?
We want five cards, so we need to stop after the fifth
card is received. 52*51*50*49*48
Also, what if we do not care about the order in which the cards are dealt?
There exist 5 ways the first card could be dealt, 4 ways the second,given the first has already been dealt, and so on. Since we must divide by these extra possibilities, we obtain the following result:
This is called a combination, since there are 52 ways to combine 5 cards without repetition. In general, given n objects, if we want to choose r from n such that repetition is not allowed, "n choose r" is equal to:
How are these combinations used?
It was Pascal, during some gambling scenarios, who realized
that the combinations could be used in describing coefficients of (x+y)^n. The
coefficients are known as the “coefficients of the binomial expansion.”
For example: 
is the expansion
of (x+y)^2.
Using binomial coefficients, we can represent the same expansion using all the possible ways to "choose from a total of 2 objects."
This expansion generates the same coefficients
but looks like: 
In order to obtain this entire formula, we can consider what is known as
The Binomial Theorem
This expansion represents the sum of all the binomial coefficients and their corresponding variables. Another way to think about this is to consider all the possible ways to place two different types of objects into n boxes, such that each box has only one object in it.
For example, say we are placing objects called "a's" and "b's" into boxes. We have n boxes, and in r of those boxes, we need to place one b in each. There are n*(n-1)*(n-2)*...*(n-r+1) ways to do this. In the remaining n-r boxes, we place the a's and there is only one way to do this (since order doesn't matter). Therefore, the result of the combination "n choose r," will correspond to how many a's and b's we placed and we may sum these up to obtain all the possibilities.
Other than using the coefficients of the Binomial Expansion,
how can we get from one row to the next?
Pascal’s Rule for the Triangle is
:
What is it that makes this true? I will show you two ways:
First algebraically, the sum is equivalent because of how the following works out:
Proof:
Second, we can prove this sum using methods of combinatorics and what is known as combinatorial reasoning. Imagine that we are playing a game. We have n total players and need to make a team of r players. The number of possible "combinations" would be the same as adding the number of ways of forcing one person not to play then choosing r from n-1 players and the number of ways of forcing one person to play then choosing r-1 from n-1 players.
For example: Say we have 5 people and need to pick 3 people out of the 5 for our first game. Note that Dr. Fogel is one of our players. The total amount of possibilities is "5 choose 3," which comes out to 10. If we consider first forcing Dr. Fogel to play, then we must choose 2 others for the team. The number of ways to do this is "4 choose 2," which comes out to 6. Next if we consider forcing Dr. Fogel not to play, then choosing a team, there are "4 choose 3" ways. The result of "4 choose 3" is 4. Therefore 4+6=10 as desired.
Now we are ready to discover the many properties of Pascal's Triangle!
The relation between the triangle and pyramid is simple. Actually all that is needed to construct the pyramid is one more dimension. Consider this jump from the Binomial Theorem to what is known as the Multinomial Theorem.
Using the coefficients of what is known as "n choose a, choose b, choose c," one obtains the entries in the pyramid.
See Pascal's Pyramid as a moviel! (*warning extremely large 46MB file, proceed with caution)
In the Pyramid, each “layer” (starting with layer zero) represents coefficients of the trinomial expansion, where using Pascal's Rule, we can sum three specific terms in one layer to obtain a specific entry in the next layer.
Layers 0 through 4 are shown in the following:
Using Pascal's Rule for the Pyramid, we can get from one layer to the next. In the below example the 3 twos in layer 2 add together to get 6.
In general, given any entry in the nth layer:
What are some properties of this peculiar configuration?
An interesting property that arises in both the triangle and pyramid is what I call "Pascal's Petals." Given any "n choose r" or any "n choose a, choose b, choose c," the 6 entries surrounding that combination have the following special relationship.
For "4 choose 2," this property works since 4*3*10=3*4*10.
In general, given any "n choose r," if we take the 6 entries immediately adjacent to it and find the product of three alternating entries, we obtain the same product as that of the other 3 entries.
For the pyramid, the following relationship must
be satisfied:
These coefficients show up in each layer of the pyramid.
As always if n,a,b, or c is negative, the combination
is zero.
A not so obvious property of Pascal's Triangle
is it's ability to produce what is known as the Fibonacci Sequence. The Fibonacci
Sequence begins 1,1,2,3,5,… and is an important sequence in nature. It
can be obtained by starting with 1, 1 then adding the previous two terms to
get the next term.
If f1 is the first term, f2 is the second term, the sequence is
defined recursively as: fn=fn-1+fn-2.
Where do these numbers show up?
?:)
The Diagonals of Course!
These diagonals seem quite specific and are in fact parallel! In order to find all the entries in the nth diagonal we must know the slope. Given any "n choose r," the nth fibonacci diagonal can be obtained using a slope of -1/3 (right to left).
Check it out---> Say we picked "2 choose 1." If I define the single horizontal space between combinations as one unit, we can see that by going down the diagonal, "3 choose 0" will also be included as an entry in the 4th fibonacci diagonal.
During my investigation of a paper entitled "The Pascal Polytope: An Extention of Pascal's Triangle to N Dimensions," by John Putz, I discovered the formula to generate these diagonals and manipulated it to make use of combinations.
The following property was revealed (Note the bracketed notation inside the combination refers to the floor function of (k-1)/2. This function merely reduces the result of (k-1)/2 to the integer value immediately less than it. We don't like fractional combinations :-P).
A short outline of the proof is necessary:
Split up fn to even and odd cases. Using Pascal's Rule, we can show that when n is even, that is n=2j for some integer j, f 2j=f 2j-1+f 2j-2. This proof amounts to collapsing the sums using Pascal's Rule and then showing that every term in f 2j is produced, which can be done using math induction. The same process must then be shown for an odd integer, where n=2m+1, for some integer m. Proof is complete.
In the Pyramid, things get more interesting..
PASCAL'S PYRAMID: The Fibonacci 3-Sequence
The Fibonacci 3-sequence is another recursive sequence, in which the next entry in obtained by summing the previous 3. The way in which the entries of this sequence are obtained from the pyramid is similar to that of the triangle, except we are finding the diagonals in specific layers. Also, we add up the resulting entries in more than just one diagonal.
The process is explained below:
As demonstrated, in order to find the entries that make up the 5th fibonacci 3-sequence number, we find the entries of the 1st diagonal in the (5-1)th =4th layer, add that to the entries of the 2nd diagonal in the 3rd layer, add that to the entries in the 3rd diagonal in the 2nd layer, and since the 1st layer does not have a 4th diagonal, we stop. (Remember the topmost layer is known as the 0th layer) These distinct diagonal entries unusually produce the nth entry in the fibonacci 3-sequence.
A similar manipulation of John Putz's formula gave the following double sum of combinations:
A simpler outline of proof is necessary:
First we expand all the entries of fn, fn-1 and fn-2 for all entries of i and k. Next, we show, using Pascal's Rule for the pyramid, that the entries collapse into fn+1. By induction we can show this property holds.
Another property in the triangle is that the sum
of the entries in the nth row is equal to 2^n. In other words, for those combinations
that have the same “n” value, their sum is equal to a power of 2.
This property is equivalent to letting a=b=1 in the Binomial Theorem!
We can use the "box" analogy as described in the proof the binomial theorem to show why this is true. Given n boxes and two sets, a's and b's, the number of ways to put an "a" or "b" in these boxes, given we can only put one in each box is 2 ways. Either an "a" goes in the box, or a "b" goes in the box. In our n boxes, there are 2 ways we can do this for the first box, 2 ways for the second box... 2 ways for the nth box. The product of these is the total number of possibilities.
Of course, the pyramid exhibits a similar property, which is that the sum of the entries in the nth layer is 3^n. Equivalently, we can set x=y=z=1 in the multinomial theorem.
Using the box analogy again, the number of ways to put an "a," "b," or "c" in n boxes, given we can only put one in each box is equal to 3^n since we can either put an "a," "b," or "c" in box 1, same for box 2, ... same for box n. By multiplying the number of ways for each box, we obtain all the possibilities.
I would like to conclude by mentioning that this project was extremely worthwhile to me. Overall, I realized that Pascal’s Triangle has many interesting properties, most of which show up as a 3-dimensional analogue in Pascal’s Pyramid. These properties have significance in describing combinatorial arguments, i.e. “playing games and choosing people.”
I believe that a study of Pascal's Triangle is an appropriate way to analyze and piece together some amazing higher forms of mathematics. This study would be a good way to get any high school student interested in college mathematics. The symmetries seem endless!
My challenge for you is: Can you find
all original 17 properties in the triangle and pyramid?
I would like to thank the math department of CLU and all of my professors. My success on this project is shared with you all!
A very special thanks to my advisor and mentor, Dr. Karrolyne Fogel.