What is Pascal's triangle?
A triangle of numbers where each entry is the sum of the two directly above it, starting from a single 1 at the top.
// maths › Surfaces & Shapes
Pascal Terrain
Pascal's triangle as a 3D terrain, revealed row by row - each number the sum of the two above it.
binomial coefficients as a 3D terrain
A mind behind this: Pingala c. 300 BCE
Drag to rotate · scroll to zoom · right-drag to pan
Frequently asked questions
What do the rows mean?
Row n gives the coefficients of (a + b)^n, and also the chances of getting 0, 1, 2... heads when you flip n coins.
Why is the height log10?
The middle numbers grow huge quickly, so plotting their number of digits keeps the terrain readable.
How does it connect to probability?
Each entry is 'n choose k', the number of ways to pick k items from n - the heart of probability and statistics.
Who really discovered Pascal's triangle?
It was found independently across many cultures, long before Pascal. In India, Pingala (c. 3rd-2nd century BCE) studied the combinatorics in his work on poetic metre, and the explicit triangle - the Meru-prastara, or 'staircase of Mount Meru' - was set out by the commentator Halayudha (10th century CE). In Persia, al-Karaji (c. 1000 CE) and Omar Khayyam (1048-1131) described it independently. In China, Jia Xian (11th century) devised it and Yang Hui (13th century) preserved it, so it is known there as 'Yang Hui's triangle'. Blaise Pascal (France, 1654) tied it to probability theory, which is why the West named it after him. You can read about each of them in the Scientists & Scholars directory.