On the Symmetries of Pascal’s Pyramid

Ryan PetitfilsAdvisor: Dr. Fogel

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!

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?

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*…*1But 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:

5)!*5!(52

52!

12345

4849505152

****

****

552

r!(n-r)!

n!rn

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:222 2)( yxyxyx

222

12

022 2)( yxyxyx

The Binomial TheoremWe 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:

rrnn

r

n baba

)(

0rn

rn

The Binomial Thm. Cont.a’s and b’s

“5 choose 3”

555

0r55 )( baba r

r

These coefficients show up in a useful array known as…

505541

4532

3523

2514

1505

05 babababababa

Pascal’s Triangle By arranging the

coefficients in the binomial expansion,Pascal actually created a triangle which could be extended indefinitely.

00

01 1

1

02 1

2 22

Pascal’s RuleOther 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:

111

kn

kn

kn

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 (n

r) = n!/(r!*(n-r)!)

using (53)

= (42) + (4

3) And show that both are the

same by algebraic manipulations.

Combinatorically, (53)

= (42) + (4

3) because…

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.

rrnn

r

n yxyx

)(

0rn

ncba

zyxcbann cbazyx ,,)(

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…

!!!

!

,, cba

n

cba

n

Pascal’s Pyramid 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

enclosed in 3 copies of one row of Pascal’s triangle

Pascal’s Rule for multinomial 1,,

1,1,

1,,1

1,,

cba

ncba

ncba

ncba

n

The 6th Layer

6,0,06 0,0,6

6

0,6,06

2,2,26

060 zyx

006 zyx 600 zyx

222 zyx 3,1,26 312 zyx

ncba where, cba zyx

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

kn

kn

kn

kn

kn

kn 1

*11

*111

*1

*1

24

3-Petals

30*90*20=60*15*60

2,3,16

3*3*3=3*3*3

1,1,13

“Layer 6”

1,1,

*,1,1

*1,,1,1,1

*1,,1

*1,1, cba

n

cba

n

cba

n

cba

n

cba

n

cba

n

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:

21 nnn fff

Fibonacci Diagonals

11

1+1=22+1=3

1+3+1=5

If we sum the entries along specific diagonals, we obtain the fibonacci #s!

Fibonacci Diagonals cont.

00

01 1

1

02 1

2 22

03 1

3 23 3

3

123

1

3 to the left, down 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?

1

3

2+1=3

For n=5…

1+3+(1+2)=7

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!

n

r

nrnn

r

nn

0rn

0rn 11 2)11(

Sum of rows = 2n

1+1=21+2+1=4

1+3+3+1=8

and so on…

1

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.

ncba

zyxcbann cbazyx ,,)(

ncba

cbannn cba 111,,3)111(

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)

n

kk

kk

n

nn

nnf

0 2

12

11

2

12

11

01 ...

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)

n

k

n

i3k,i,k2in2kin1nf

0 0

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? (petalsspheres?)

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.