On the Symmetries of Pascal’s Pyramid

1. Ryan Petitfils Advisor: Dr. Fogel On the Symmetries of Pascal’s Pyramid

2. 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! Pascal’s Triangle

3. 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.

4. 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?

5. 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 Combinatorics But what if we are interested in only choosing 5 cards?

6. 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: 52 “choose” 5

7. 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: Uses of “Combinations”

8. 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:

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

10. Pascal’s Triangle • By arranging the coefficients in the binomial expansion,Pascal actually created a triangle which could be extended indefinitely.

11. 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:

12. 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… Algebraic vs. Combinatorial(Pascal’s Rule)

13. 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.

14. 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…

15. 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 2+2+2=6 Pascal’s Rule for multinomial Pascal’s Pyramid enclosed in 3 copies of one row of Pascal’s triangle

16. The 6th Layer

17. 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

18. 3*3*3=3*3*3 30*90*20=60*15*60 3-Petals “Layer 6”

19. 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 f1is the first term, f2 is the second term, the sequence is defined as:

20. 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

21. Fibonacci Diagonals cont. 3 to the left, down 1 3 2 1 1

22. 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? Fib 3-seq fn=fn-1+fn-2+fn-3 For n=5… 1+3+(1+2)=7 2+1=3 3 1

23. 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!

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

25. 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.

26. 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.

27. 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?

28. 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)

29. 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,

30. 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.

31. 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.

32. 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?

33. 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.