 |
"Pascal's Triangle" is an arithmetic table, ususlly written in the shape of a triangle, which has great significance in mathematics. Starting with "1's" on the edges, the numbers inside the triangle are formed by adding up the 2 numbers above. For example, the "5" in the bottom row shown is 1 + 4 from the row above. The "10" is 4 + 6 from the row above. Can you find the next row? |
One interesting property of Pascal's triangle is that if you add the numbers in the rows you get 1, 2, 4, 8, 16, ..., which are all powers of 2. This property, along with some other important algebraic properties were known in India and China in the 11th century. Blaise Pascal was a French mathematician in the 17th century who studied the values in this triangle and was able to prove many other important facts about it, including how the values in the triangle are related to probability. For example, if you flip a coin 3 times, there is 1 way to get no heads (tails-tails-tails), 3 ways to get exactly one head (heads-tails-tails), (tails-heads-tails), and (tails-tails-heads), 3 ways to get exactly two heads (heads-heads-tails), (tails-heads-heads), and (heads-tails-heads), and 1 way to get three heads (heads-heads-heads). The numbers 1,3,3,1 which count these combinations, are the fourth row in Pascal's triangle.
Another, perhaps more important property of Pascal's triangle is that each row contains the coefficients of terms in a binomial expansion. For example
Note that the coeficients (in red) are precisely the values in the fifth row of Pascal's triangle. Using this property, together with a direct formlula for determining the coeficients in the triangle (for example, the combinatorial formula) Isaac Newton generalized the formula for binary series to fractional and negative powers. This in turn, led to great advances advances in his Calculus, and for example, improved the approximation of pi.
The paterns drawn in Pascal's Patterns come from dividing the numbers in the triangle by a fixed number, and coloring in the locations depending on the remainder. For example, In the numbers listed above, the even numbers are colored red, while the odd numbers are black. So the fixed divisor in this case is 2, and red is the color for remainder 0, while black is for remainder 1.