Pascal s Triangle. Jean-Romain Roy. February, 2013

Transcription

1 Pascal s Triangle Jean-Romain Roy February, 2013 Abstract In this paper, I investigate the hidden beauty of the Pascals triangle. This arithmetical object as proved over the year to encompass seemingly different areas of mathematics. Despite having been studied by brilliant scientists, many believe that it still holds unknown patterns. Through unusual manipulations, I extracted new links and symmetries. I shall try to explain them, offer new perspectives on the subject and highlight peculiar occurrences. Figure 1: The Pascal s Triangle 1

2 1 Introduction 1.1 Historical context The vast science of mathematics abound peculiar objects, which can at first glance look trivial but after a deeper analysis reveal amazing behaviours. When those specific occurrences arise, the inherent but concealed beauty of geometry or algebra unveils itself. This phenomenon, when witnessed, induces the impression of being in symbiosis with the language of the universe. The symmetry dancing with the chaos, the image is aesthetically sublime. This profound emotion is the propellant of creativity which often leads to discoveries. From the many who could attest of this feeling, Blaise Pascal stands out. This French scientist of the 17th century was also one of the most ingenious mathematicians of his era. Some of his notorious contributions are the mechanical calculator, Pascals law, Pascals theorem and much more [1]. (Traite du triangle arithmetique, 1653) One of his great accomplishments is the Pascals triangle, an idea that reveals odd parallel with certain number sequences. It has been stated that the triangle could be either a gold mine or the tip of an iceberg [5]. The Pascals triangle is a triangular array of binomial coefficients [1]. Many other cultures knew about this mathematical object before him, but in the Western World he was the first to observe it. (Pascal s Triangle, World of Mathematics Summary) The numbers placed in the triangle were first studied by the Hindu, through combinatorics and the Greeks, through figurate numbers [1].The Chinese knew about it by the 11th century, as shown in Figure 1. 2

3 Figure 2: The Old Method Chart of the Seven Multiplying Squares 1.2 The original triangle and it s construction Before diving in the study of it s many properties, lets look at how the triangle was constructed by Blaise Pascal. Figure 3: The original Pascal s triangle It can be seen that the modern triangles are rotated to allow for a downward flow of numbers, while the numbers in the original Pascals Triangle of 3

4 1654 extended diagonally from left to right. Despite differences in their representations, todays arithmetic triangle contains the same numerical qualities as Pascals seventeenth century triangle[5]. The triangle was made by drawing two perpendicular lines. Then, he separated the line in identical length and thus generated a symmetric grid. In each small cell would be placed a number equal to the sum of the two previous one positioned at his left and on top of him. He then drew a line in a diagonal right and downward direction to represent the axis of symmetry. In his traite du triangle arithmetique he stated the following : We now place numbers in each cell and this is done in the following way: the number of the first cell which is in the right angle is arbitrary, but once it has been placed all the other numbers are determined, and for this reason it is called the generator of the triangle. And every one of the other numbers is specified by this sole rule: The number of each cell is equal to that of the cell preceding it in its perpendicular rank plus that of the cell which precedes it in its parallel rank(350-1)[6]. 1.3 The new avenues In this paper, I will present new patterns that I discovered through assembling the triangle in different ways and by using computer programs. For example, using the binomial (x+2) or even (x+i) to produce the triangle can provoke interesting change in behaviours. Different symmetries emerge and other disappear or get distorted. The purpose of this deeper analysis isn t clear, but the exercise of walking the path of discovery is in itself an amazing experience. The results that I will achieve to gather may not be ground-breaking, but the whole work of building this kind of paper will be useful and valuable for my future studies. 2 Patterns In the light of the precedent section, we now know how the triangle is generated. Now, one may be interested in extracting all the patterns hidden in the rows and diagonals. I will illustrate and explain most of the know one s found in the literature. 4

5 2.1 The vertical symmetry One of the triangles first mysteries to be spotted is that the numbers of the array are symmetric. Indeed by folding the triangle in half at the vertical axis, the two resulting parts are identical. Figure 4: The vertical symmetry 2.2 Horizontal sums The sum of the numbers in any row is equal to 2 to the n power or 2n, when n is the number of the row. For example: Figure 5: The powers of two 5

6 2.3 The powers of 11 If a row is made into a single number by using each element as a digit of the number (carrying over when an element itself has more than one digit), the number are equal to the powers of eleven. When we reach 11 to the power of 5, the digits start to overlap. Figure 6: The powers of eleven 2.4 Diagonal triangular number sequence Each diagonal can represent a sequence of the triangular numbers in a specific dimension. The first diagonal represent the 0 dimension where the every geometrical form is a dot. The second diagonal is the first dimension, we saw a regular expansion. The third diagonal represent the well known triangular sequence. The fourth diagonal is the tetrahedral number sequence. In short, the n diagonal equals the n-1 dimension. Figure 7: Triangular sequence 6

7 Figure 8: Tetrahedral numbers 2.5 Hockey stick pattern The diagonal of numbers of any length starting with any of the 1s bordering the sides of the triangle and ending on any number inside the triangle is equal to the number below the last number of the diagonal, which is not on the diagonal = 10 (1) = 21 (2) = 84 (3) Figure 9: The Hockey stick pattern 2.6 Changing the colors An incredible link between a well known fractal can be achieve when coloring the odd numbers (not divisible by 2) in black and leaving the evens white. The Sierpinski s Triangle is revealed. We can also color the prime or also use modular arithmetic to produce different patterns [4]. 7

8 Figure 10: The odd numbers colored, Sierpinski s Triangle Figure 11: The prime numbers colored 2.7 The Fibonacci sequence When summing the diagonals at a certain angle we can extract the Fibonacci sequence. Figure 12: The Fibonacci sequence 8

9 3 Applications Pascals Triangle is not only an interesting mathematical object because of its hidden patterns, it is also interesting because of all the applications extending in calculus, trigonometry, plane geometry, and solid geometry [5]. 3.1 Binomial expansions As one familiar with algebra may notice, the numbers in each row of the triangle are precisely the same numbers that are the coefficients of binomial expansions. Indeed, we can use Pascal s triangle to generate their coefficient. For example, when one expands the binomial : (x + y) n = x 2 + 2xy + y 2 (4) The coefficients of this binomial expansion, 1 2 1, correspond exactly to the numbers in the second row of Pascals Triangle. (x + y) n = a 0 x n + a 1 x n 1 y + a 2 x n 2 y a n 1 xy n 1 + a n y n (5) where the coefficients a in this expansion are precisely the numbers on row n of Pascal s triangle. 3.2 Combination The triangle also shows you how many Combinations of objects are possible. For example, you have 8 pool balls and you want to know how many different ways you could you choose just 4 of them. To find it using Pascal s triangle, go down to row 8, and then along 4 places and the value there is your answer, 70 [4]. The number of combinations of n things taken k at a time (called n choose k) can be found by the equation : C(n, k) = C n k = n C k = ( n k) = n! k!(n k)! (6) 9

10 Figure 13: The combinations representation This is also the formula for any cell of the triangle. Rather than performing the calculation, one can simply look up the appropriate entry in the triangle. We can link the two precedent applications with the general binomial theorem : n (x + y) n = ( n k)x n k y k (7) 4 New results k=0 In the light of all the patterns I presented, we can understand how the Pascal s triangle is at the junction of many mathematical concepts. Many variations of the triangle exists, for example we can change the binomial for the following: Figure 14: The (x+2) triangle 10

11 Figure 15: The (x+3) triangle We can see in the two precedent figures how the Fibonacci sequence change because of the different y variable. These expansions of the original Pascal s triangle reveals amazing behaviours. Some patterns previously explained still hold true, others are not present anymore, but more interestingly some are modified in a way that we can predict. I shall evaluate each of the pattern explored for those new triangles. The vertical symmetry is broken, we right away see that if we fold the new triangle in two, the numbers wouldn t correspond. The horizontal sums are quite interesting because we now understand that for (x+2) they will give the powers of 3 and for (x+3) they will give the powers of 4 and so on. The general statement would be, for an (x+n) triangle, the horizontal sum would equal (n+1) of the same power then the binomial. The original triangle had an amazing link with the powers of 11, now for the (x+2) triangle we see that the same pattern exist for the powers of 12 and the (x+3) triangle has the same pattern for the power of 13. The triangular sequences are still present but only if we follow the diagonal starting in the upper left to the down right. We also see that the sequence has been multiplied by the powers of two in the case of the (x+2) triangle and by the power of three in the case of the (x+3) triangle. The Hockey stick pattern is not applicable anymore because it needed the vertical symmetry For applying colors to the odd and even number, when the y component of the binomial is an odd number it doesn t change the pattern by when it is an even number it changes completely the mosaic, because every number is even except the one s on the left side. The change in the Fibonacci sequence is explained on the figures. 11

12 5 Conclusion In conclusion, my study of the patterns when expanding the triangles to higher binomials allowed me to predict how they would evolve when changing the y variable. Some patterns didn t apply to the change I made in the original triangle but other patterns evolved with the change. I encountered some problem with the applying color pattern, because I didn t have the computational tool to illustrate the changes, I couldn t find any ressource in the mathematical literature to help me with this. This in fact means that it would thus be an interesting area of research. 5.1 Potential path of investigation Through my investigation, I stumbled on many potential paths of study that I didn t cover in this paper. The first one is the use of computational power to illustrate different patterns with modular algebra. For example, one may want to cover in blue all the multiples of 2 and in red all the multiples of 3, the resultant drawing could lead him to interesting conclusions. I was thinking about using this process to detect prime numbers. The second one would consist in extrapolating the procedure by which we extract the Fibonacci sequence. One may change the angle of the line even more to see if he gets interesting sequences. The study of higher dimension of the Pascal s triangle, would also be interesting. The Pascal s pyramid or even the Pascal s simplex could lead to interesting discoveries. Generating triangles with not only the powers of 11, could also be interesting. Figure 16: Modulo Triangle 12

13 6 Acknowledgements I would like to thank my teacher, Mr. St-Amant for the support he gave me and his precious guiding advices. This work wouldn t of been possible without him. References [1] Pascal s Triangle. Wikipedia. Wikimedia Foundation, 02 Nov Web. 16 Feb [2] Pascal s Triangle. All You Ever Wanted to Know About and More. N.p., n.d. Web. 16 Feb [3] Experimental Feature. Wolfram Alpha: Computational Knowledge Engine. N.p., n.d. Web. 16 Feb [4] Pascals Triangle. Phi 1618 The Golden Number RSS. N.p., n.d. Web. 16 Feb [5] Pascal s Triangle. Pascal s Triangle. N.p., n.d. Web. 16 Feb kazimir/index.html. [6] Pascal, Blaise. Trait Du Triangle Arithmetique, Avec Quelques Autres Petits Traitez Sur La Mesme Matire. Paris: Guillaume Desprez, Print. [7] Pascal s Triangle. Pascal s Triangle. N.p., n.d. Web. 17 Feb [8] Fibonacci Numbers and the Pascal Triangle. Fibonacci Numbers and the Pascal Triangle. N.p., n.d. Web. 17 Feb [9] Hemenway, Priya. Le Code Secret. Lugano: Evergreen, Print. 13