Fractals in Pascal's Triangle

Fascinating patterns can arise out of arrays of numbers defined by simple rules.

For example, start with the number 1, and make it the apex of what will become a triangle of numbers. In the second row, write two 1s. For each subsequent line, add together adjacent numbers of the previous row and write the sums in the new row, then place 1s at both ends of the line.

Here's what you get for the first eight rows:

This set of numbers is now widely known as Pascal's triangle, named for French philosopher and mathematician Blaise Pascal (1623-1662), who studied it intensively. Pascal, however, was not the first to identify the pattern.

The mathematician, astronomer, and poet Omar Khayyam (1048-1122) described this number triangle in his writings. It was also well-known to the Chinese. A nine-row version was featured prominently in the introduction to the book Precious Mirror of the Four Elements," which appeared in 1303. The book's author refers to the triangle as a "diagram of the old method for finding eighth and lower powers."

MAA Mathematical Treasures

Indeed, the triangle represents a simple way to determine, for example, that (x + 1)⁴ = 1x⁴ + 4x³ + 6x² + 4x¹ + 1x⁰. In other words, the rows represent the binomial coefficients—the multipliers of the powers of x that occur when you multiply out expressions of the type (x + 1)ⁿ.

Notice also that the numbers along the diagonals follow certain patterns. The second diagonal running from 1 to 7, for example, consists of consecutive whole numbers. The numbers along the third diagonal are known as triangular numbers.

It's possible to convert this triangle into eye-catching geometric forms. For example, you can replace the odd coefficients with 1 and even coefficients with 0 to get the following array (for up to eight rows):

Continuing the pattern for many more rows reveals an ever-enlarging host of triangles, of varying size, within the initial triangle. In fact, the pattern qualifies as a fractal. The even coefficients occupy triangles much like the holes in a fractal known as the Sierpinski gasket (or triangle).

In other words, the pattern inside any triangle of 1s is similar in design to that of any subtriangle of 1s, though larger in size, Andrew Granville noted in a paper on the arithmetic properties of binomial coefficients, titled "Zaphad Beeblebrox's Brain and the Fifty-ninth Row of Pascal's Triangle."

"If we extend Pascal's triangle to infinitely many rows, and reduce the scale of our picture in half each time that we double the number of rows, then the resulting design is called self-similar—that is, our picture can be reproduced by taking any subtriangle and magnifying it," Granville wrote.

The pattern becomes more evident if the numbers are put in cells and the cells colored according to whether the number is 1 or 0.

Similar, though more complicated, designs appear if you replace each number of the triangle with the remainder after dividing that number by 3. Thus, you get:

This time, you would need three different colors to reveal the patterns of triangles embedded in the array. You can also try other prime numbers as the divisor (or modulus), again writing down only the remainders in each position.

In this example, the modulus is 5.

Originally posted February 10, 1997