Chaos and GraphicsDimensions and the probability of finding odd numbers in Pascal's triangle and its relatives
Introduction
What is the probability of finding an odd number among the positive integers? Surely the answer is 1/2. A careful formulation of this result requires a bit of thought: selected uniformly randomly from , the probability of finding an odd number is 1/2. An appropriate limit as gives the result.
We can ask a similar question of Pascal's triangle. In Pascal's triangle with n rows, what fraction of the entries are odd? What is the limit of this fraction as ? Odd numbers are those not divisible by 2, so we can ask similar questions about the limit of the fraction of Pascal's triangle entries not divisible by 3, not divisible by not divisible by p for each prime p. What about the limit of the fraction not divisible by prime powers, or by products of prime powers? Refined in the right way, we see that all these questions have the same answer. How can this be?
Our approach is to relate each of these limits to a specific fractal and show all these fractals have dimensions . From this, we deduce the answers to the questions posed above.
Often Pascal's triangle is encountered in precalculus classes as a simple method for finding the coefficients of the terms in . Taking the top to be the 0th row and the left edge the 0th column, the coefficient of in is the j th entry of the n th row. The left side of Fig. 1 shows rows 0 through 8 of Pascal's triangle, the right side shows the row and column numbers.
Almost every introduction to fractals includes the observation that darkening the Pascal's triangle squares containing numbers that are not multiples of 2 gives a pattern like the familiar Sierpinski gasket. In fact, appropriately rescaling the sequence of Pascal's triangle with rows gives a sequence of sets that converges to the Sierpinski gasket. In addition, Pascal's triangle can be formulated using cellular automata and iterated unction systems. See Chapter 8 of [1] for an excellent exposition. Many interesting applications of Pascal's triangle for visualizing constructions in group theory are given in [2]. An early study of the aesthetics of Sierpinski gaskets found in Pascal's triangles is found in [3].
This appearance of the Sierpinski gasket suggests the natural question: for every integer , what pattern results from darkening the Pascal's triangle squares containing numbers that are not multiples of n? Fig. 2 shows the patterns in the first 200 rows for . Some of these are simple fractals, relatives of the Sierpinski gasket. Others are more complicated, still fractal, but not built by a simple iterated function system in any obvious way. A moment's observation suggests how to differentiate these two categories: for n prime, the pattern is a relative of the gasket; for n composite, the pattern is more complicated. More careful inspection differentiates the composites. Prime powers appear to give simpler patterns than do products of distinct primes.
One very interesting consequence of the proliferation of computer graphics is that many people have seen that a few lines of code can generate images whose complex structure contains features unanticipated by the programmer. Manual drawings rarely revealed such surprises; with computers, mathematics has blossomed as an experimental science. Excellent examples can be found in [4], [5], and of course in the Journal of Experimental Mathematics and in the Chaos and Graphics section of Computers & Graphics.
Here our goal is modest. What else can we deduce by looking at the images in Fig. 2?
In the top row of Fig. 3 we see the squares of Pascal's triangle colored green if the number of the square is divisible by 6, blue if divisible by 3 but not by 2, red if divisible by 2 but not by 3, and black if not divisible by 2, 3, or 6. Left to right we see Pascal's triangle with 64, 128, 256, and 512 rows. The amount of green appears to increase with the number of rows in Pascal's triangle. This suggested to us that as Pascal's triangle grows, the fraction of entries not divisible by 2 and 3 goes to 0.
In the bottom row of Fig. 3 we use a different coloring scheme, based on divisibility by 2, 3, or 5. Specifically, squares are colored red if their Pascal's triangle entry is divisible by 2 but not 3 or 5, green if by 3 but not 2 or 5, and blue if by 5 but not 2 or 3; yellow if divisible by 2 and 3 but not by 5, purple if divisible by 2 and 5 but not by 3, and aqua if divisible by 3 and 5, but not by 2; white if divisible by 2, 3, and 5; black if divisible by none of 2, 3, and 5. More white area appears in larger Pascal's triangles, so we expect the fraction of entries not divisible by 2, 3, and 5 goes to 0 as Pascal's triangle grows.
These pictures led to the conjecture that for every integer , as Pascal's triangle grows, the fraction of entries not divisible by r goes to 0. After completing our analysis, we found that this result had been obtained in [6] using algebraic methods. Our approach was motivated by detail found in the images, and is straightforward, given some elementary properties of dimension and measure, reviewed in Section 2.
Section snippets
Review of dimension and measure
For a self-similar set G with similarity transformations satisfying if the are sufficiently disjoint, the Hausdorff dimension of G is the unique solution d of the Moran equationwhere is the contraction factor of . That the are sufficiently disjoint is guaranteed by the open set condition: there is a nonempty bounded open set O for which with the unions disjoint. For the gasket, O can be taken to be the interior of its convex hull.
Rescaling Pascal's triangles
We consider a sequence of Pascal's triangles, those having rows, for example, ask what fraction of the entries of these Pascal's triangles are odd, and look for limiting behavior of this fraction as . Can we use this to deduce the general behavior of the proportion of odd entries in Pascal's triangles? Thinking of odd numbers as those , we can ask what proportion of the entries of Pascal's triangles are ? What proportion are ? . For every prime p, what
Why is the limit the gasket?
That filling in the Pascal's triangle squares containing odd numbers has limiting shape the Sierpinski gasket is a straightforward consequence of a result of Kummer. See Chapter 8 of [1]. With the row and column indexing introduced in Fig. 1, the Pascal's triangle entry in row n and column k is . Kummer [11] showed that for each prime p, the number of factors of p in is the number of carries that occur when the base p representations of and k are added. For example, and 2 have base
The probability of finding odd numbers, and other surprises
We have seen that is the Sierpinski gasket of dimension , hence having area 0. From this, we deduce that the probability of finding an odd number in goes to as . In this section we address how to extend this result to other primes, and how to replace the limit with the limit.
Odd numbers are just those . For every prime number p, we can look for similar patterns in , the squares in the rescaled m-row Pascal's triangle that contain numbers
Prime powers and composite numbers
We have seen that in the infinite rescaled Pascal's triangle, for each prime p the subset corresponding to those numbers not divisible by p has dimension less than 2. Based on this, we say that the probability of finding a number not divisible by p is 0. What about the probability of finding a number divisible by p but not by , or divisible by but not by , and so on? Or for that matter, what about the probability of finding a number not divisible by ? Fig. 3 suggests that all
Some generalizations: other polynomials
One familiar motivation for studying Pascal's triangle is that the coefficients of in are the entries of the th row of Pascal's triangle, taking the apex of the triangle to be the 0th row. Motivated by [15], similar arguments can be carried out for powers of any polynomial. For example, in Fig. 12 we see the first 200 rows of the Pascal's triangle for , shading the squares containing numbers (left) and (right). To adapt our arguments to these
Conclusion
While some effort is required to compute the dimensions of the for any integer using the hIFS formalism, that computing the dimension is worthwhile is immediately apparent from looking at the pictures of Pascal's triangle, filling all squares containing numbers that are not multiples of r. Common experience with summing the areas of the empty triangles in the base 2 Sierpinski gasket suggests that the empty triangles in sum to 1/2, so the numbers form a subset of
References (16)
Computing fractal dimensions for additive cellular automata
Physica D
(1987)- et al.
Chaos and fractals. New frontiers of science
(2004) - Shannon K, Bardzell M, Bergner J, Evans T, Fisher M, Ludwick K, et al. PascGalois project....
On the aesthetics of Sierpinski gaskets formed from large Pascal's triangles
Leonardo
(1990)- Borwein J, Bailey D. Mathematics by experiment. Plausible reasoning in the 21st century. Natic: A. K. Peters;...
- Borwein J, Bailey D, Girgensohn R. Experimentation in mathematics. Computational paths to discovery. Natic: A. K....
Binomial coefficients modulo a prime
Amer Math Monthly
(1947)Fractal geometry. Mathematical foundations and applications
(1990)
Cited by (0)
- 1
Partially supported by NSF DMS 0203203.