On the Symmetries of Pascal’s Pyramid Ryan Petitfils Advisor: Dr. Fogel My name is Ryan Petitfils and I will be speaking to you today about the symmetries of Pascal’s Pyramid. Before I get into the Pyramid however, I would like to describe some history of the triangle and the great subject in which it is studied.

Pascal’s Triangle 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. The triangle has many symmetrical properties, in fact Pascal’s found a total of 17! The Triangle originated in China around 1200s, but it was unknown to Pascal! Pascal’s triangle has many properties that generalize to a 3-dimensional analogue known as Pascal’s pyramid. I will show you these properties and how they generalize to 3-dimensions. Pascal’s Triangle is a useful tool in the field of combinatorics. KF: make sure you point out how to add two entires to get the third (you’re arrows)

Blaise Pascal A French mathematician born in Clermont, in 1623. Made significant contributions to the fields of Geometry (studies of the cycloid), Physics (pressure experiments) and so much more… Invented one of the first calculating machines (similar to those of computers used today). Together with Pierre Fermat, he created the “calculus” of probabilities, known as Probability Theory.

Combinatorics “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.” –The Mathematical Atlas (Northern Illinois University) A Combinatorial question: How many ways can one order all the cards in a 52 card deck? In order to understand where the entries in the triangle come from we need to know something about the field of math callled Combi…

Combinatorics 52 cards, we are interested in all the different possible arrangements 52 ways to choose the first card, 51 ways to choose the second card, and so on… 52*51*50*…*1 KF: one this slide you care about order, on the next you don’t But what if we are interested in only choosing 5 cards?

52 “choose” 5 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. This is called a combination, since there are 52 ways to combine 5 cards without repetition. Notation: In general:

Uses of “Combinations” 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” Example:

The Binomial Theorem We have n boxes, in r boxes we need to place one “b” in each. Again, as with choosing the 5 cards, we are not concerned with all the possible orderings of placing b’s. There are ways to do this. In the remaining n-r boxes we place a’s, there is only one way to do this once we place the b’s. If we consider the sum of all the ways to choose a and b: We must sum all of these possibilities to obtain the total # of outcomes KF: can color coordinate a’s and b’s to stones for example (and the combination to b’s color)

The Binomial Thm. Cont. a’s and b’s “5 choose 3” 5 choose 3 is 10 These coefficients show up in a useful array known as…

Pascal’s Triangle By arranging the coefficients in the binomial expansion,Pascal actually created a triangle which could be extended indefinitely. The entries in the triangle are formed by the binomial coefficents. We know that by adding two adjacent numbers in one row we can obtain the entry in the next row between those two. Now, using binomial coefficients, I will show you how we can get from one row to the next.

Pascal’s Rule 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: Why is this statement true? I will show you two ways to prove this. One way involves using methods of combinatorics and describing how to “choose.” For example:

Algebraic vs. Combinatorial (Pascal’s Rule) For example: Imagine we are playing a game (Dr. Fogel included) and need to choose 3 out of 5 people to play. How many ways can we choose? Algebraically, we can expand the notation (nr) = n!/(r!*(n-r)!) using (53) = (42) + (43) And show that both are the same by algebraic manipulations. Combinatorically, (53) = (42) + (43) because… because if Dr. Fogel must be chosen, we can pick at most 2 others, if Dr. Fogel cannot be chosen, we must pick 3 others. Pascal’s Rule gives us a way to connect entries in the triangle and how to get from one entry to another.

Binom-> Multinom (Pyramid) The Binomial Theorem and Pascal’s Rule are just two of the properties in Pascal’s Triangle. These and many other properties of Pascal’s triangle have a 3-d analogue in an arrangement known as “Pascal’s Pyramid.” One of these analogues is the jump from the Binomial to the Multinomial Theorem. As you will notice, for convenient notation, I changed the a and b to x and y, because of the relationship to algebra. The jump from the binomial to the multinomial includes an extra variable, in this case “z,” in which the multinomial theorem jumps to 3-dimensions.

Trinomial Coefficients We have n objects, we first choose a, then we choose b from n-a, and then c from n-b-a. These coefficients form what is known as…

Pascal’s Pyramid 2+2+2=6 Pascal’s Rule for multinomial In the Pyramid, each “layer” (starting with layer zero) represents coefficients of the trinomial expansion, where the middle terms represent the sum of “three terms” above them. (by Pascal’s Rule) http://public.clunet.edu/~rpetitfi/Tetrahedron.asp Mention how we number the layers 0,1, and so on. Show that 2+2+2=(2 choose 1,1,0)+(2 choose 1,0,1) + (2 choose 0,1,1). Pascal’s Pyramid is shown here in multiple layers, however by putting together the layers and connecting lines under Pascal’s rule, one can create a 3-dimensional model much like the one I have here. Mention the pyramid is a tetrahedron enclosed in 3 copies of one row of Pascal’s triangle

The 6th Layer Explain that this layer, represents the total number of ways to choose 3 times from a total of 6 objects. Explain how to get to 6 choose 2,1,3 from 6 choose 0,0,6.

Pascal’s “Petals” in the Triangle Another property is the following product: Given any “n choose k,” 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 Given any n choose k, if we arrange the entries most immediately around that combination, we obtain a circle in which the product of three alternating entries equals the product of the remaining three alternating entries. I do not yet have the combinatorial argument yet, but algebraically it works out.

3-Petals “Layer 6” 3*3*3=3*3*3 30*90*20=60*15*60 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 of the other 3 entries. See if we choose the entry 6 in plane 3, 3*3*3=3*3*3? If you don’t believe me, look at plane 6 and the entry 60 (6 choose 1,3,2) 30*90*20=60*15*60

Fibonacci Numbers An astounding property of the triangle is that it generates the Fibonacci Sequence along certain diagonals. The Fibonacci Sequence 1,1,2,3,5,… is an important sequence in nature and 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 as: The fibonacci numbers arise in the triangle! After explaining how it is an important sequence in nature, explain how it was used to estimate rabbit populations. Assume the rabbits never die. We start with 1 rabbit and 1 rabbit (total of 2 rabbits), then after a while we have 2+1 rabbits, then 3+2 rabbits and so on…

Fibonacci Diagonals If we sum the entries along specific diagonals, we obtain the fibonacci #s! 1 1 1+1=2 2+1=3 1+3+1=5 How are these diagonals drawn?

Fibonacci Diagonals cont. 3 to the left, down 1 3 2 1 Well if we start with any combination and count the horizontal space between combos as one unit, we create diagonals that have a slope of 1/3. While diagonals generate the fibonacci sequence in the triangle, certain planes in the pyramid generate the fibonacci 3-sequence. These planes are obtained using a similar process. 1

Fib 3-seq fn=fn-1+fn-2+fn-3 1,1,2,4,7,13… fn= the sum of the multinomial coefficients cut by “fibonacci planes,” which correspond to cutting (n-1) fibonacci diagonals, starting with the (n-1)st plane and finding the sum of the entries of: The first diagonal in the (n-1)st plane, the second diagonal in the (n-2)nd plane… the rth diagonal in the (n-r)th plane We do not care about zero or negative results. (when r>n) For n=5? For n=5… 1+3+(1+2)=7 2+1=3 give an example of the 3-seq: 1,1,2,4,7,etc: make it the first bullet add one sentence verballly about how you prove this. Or add another slide with the 'proof' for small n: f5=sum sum:=layout f4=layout f3=layout f2=layout 3 1

Sum of rows Another property is the sum of the rows in the triangle For those combinations that have the same “n” value, their sum is equal to a power of 2. In fact this is equivalent to letting a=b=1 in the binomial theorem! Change “unique” to interesting Could show what happens when a=b=c=1 (show both sides are equal). Counting up all the coefficients (How many ways are there to but an a or b in each box?)

Sum of rows = 2n 1 1+1=2 1+2+1=4 1+3+3+1=8 and so on…

Sum of Planes For the Pyramid, each plane (or “layer”) sums to 3n. 1=30 1+1+1=3 1+2+2+1+2+1=9 and so on… This relationship comes about when letting a=b=c=1 in the Multinomial Thm.

Sum of the planes How many ways to put an x, y, or z in n boxes, given we can only put one in each box? 3 possible ways for first box, 3 for second… 3 for nth.

Conclusion 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.” Challenge: Can you find all 17 properties in the triangle and pyramid?

Acknowledgements Thank you Math Department of CLU Dr. Wyels, Prof. Wiers, Dr. Garcia (CSUCI), Dr. King and my advisor Dr. Karrolyne Fogel. Kevin Aguirre (creator of model tetrahedron) YOU! Math (because it’s cool)

Sources Putz, John F. “The Pascal Polytope: An Extention of Pascal’s Triangle to N Dimensions,” The College Mathematics Journal, Vol. 17, No. 2 (Mar., 1986), pp. 144-155.   Rosen, Kenneth. Elementary Number Theory, Addison Wesley Longman, 1999. Smith, Karl J. “Pascal’s Triangle,” The Two-Year College Mathematics Journal, Vol. 4, No. 1 (Winter, 1973), 1-13. The Mathematical Atlas. “Combinatorics.” http://www.math.niu.edu/~rusin/known-math/index/05-XX.html,

Fibonacci Numbers (Adapted from John Putz) Method of Proof: Using Strong Math Induction. List all the possibilities for k, given fn-1 and fn-2. Next, show that when added, these two produce all the entries in fn+1.

Fibonacci 3-Sequence (Adapted from John Putz) Method of Proof: Using Strong Math Induction. List all the possibilities for k and i, given fn,fn-1 and fn-2. Next, show that when added, these three produce all the entries in fn+1.

Open ?s Is there a 3-d equivalent of the petals in the pyramid? (petalsspheres?) Is there a way to model the pyramid using Cartesian coordinates and generate the fibonacci planes?

Background/Inspiration High School Algebra introduced me to Pascal’s Triangle In College, while taking Discrete Math and Number Theory, I worked with combinatorial ideas and learned more about the triangle. In College Geometry, I described 2-dimensional geometric figures and their relationships to those in 3-dimensions.