Pascal’s Triangle has been studied for centuries—and for good reason. This simple-looking triangle hides a world of fascinating patterns, from triangular numbers to the famous Fibonacci sequence.
It’s a key tool in algebra and a powerful ally in probability and combinatorics.
Though named after Blaise Pascal (1653), the triangle was known long before—in India as Meru-prastaara (मेरुप्रस्तार), in China as Yang Hui’s triangle (杨辉三角形), in Iran as Khayyám triangle (مثلث خیام), and in Italy as Triangolo di Tartaglia.
In Pascal’s Triangle, each number is the sum of the two numbers directly above it. This additive rule builds the triangle and connects it to binomial coefficients, as seen in the expansion of (a + b)ⁿ.
In the second diagonal of Pascal’s Triangle, the square of any number equals the sum of the numbers directly beside it and beneath those in the next row.
Each diagonal of Pascal’s Triangle, from right to left, contains famous number sequences—repunits, natural numbers, triangular, tetrahedral, pentatope numbers, and higher-dimensional figurate numbers.
The sum of the numbers in any row of Pascal’s Triangle equals a power of 2.
The sum of the numbers along any shallow diagonal of Pascal’s Triangle gives a Fibonacci number.
In some rows of Pascal’s Triangle, concatenating the numbers forms prime numbers (highlighted in orange in the picture)—though rare, these patterns highlight curious numerical symmetries.
Here’s a final curiosity: by shading the cells containing odd numbers in Pascal’s Triangle, a fractal pattern appears—strikingly similar to the Sierpiński Triangle.
The same effect occurs when shading numbers not divisible by a given prime p (i.e., those not congruent to 0 mod p).
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Curious to explore a little more?
For those who want to dive into its history and deeper meanings, this classic book is worth a look.
