Yang Hui Triangle, also known as Pascal's Triangle in the West, is a fascinating mathematical structure. It's a triangular arrangement where each number equals the sum of the two numbers directly above it. The triangle starts with 1 at the top, and each row begins and ends with 1.
The construction rule is simple yet elegant. Each number in the triangle is formed by adding the two numbers directly above it. For example, the 2 in the third row comes from adding 1 plus 1. The 3 in the fourth row comes from 1 plus 2. This pattern continues infinitely, creating a beautiful mathematical structure.
The numbers in Yang Hui Triangle are actually binomial coefficients. Each entry at row n and position k represents C(n,k), which is "n choose k". These coefficients appear in the binomial expansion formula. For example, row 4 gives us the coefficients for expanding (x+y) to the fourth power: 1, 4, 6, 4, 1.
Yang Hui Triangle contains many fascinating patterns. It's perfectly symmetric - each row reads the same forwards and backwards. The sum of each row equals 2 to the power of the row number. The famous Fibonacci sequence appears along the diagonals. There are also interesting even and odd number patterns throughout the triangle.
Yang Hui Triangle has a rich history spanning many centuries and cultures. It was first studied in China by Jia Xian in the 11th century, then popularized by Yang Hui in the 13th century. Centuries later, Blaise Pascal independently studied it in 17th century Europe. Today, this mathematical structure has applications in probability theory, combinatorics, algebra, and computer science, making it one of the most important discoveries in mathematics.