Figurate numbers take the shape of regular twodimensional polygons and threedimensional polyhedra.
Each type of figurate number has a formula associated with it. For instance, n² generates square numbers. The first few square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. . . .
The sixth square: 6² = 36
(n² +n)/2 generates triangle numbers.
The fifth triangle: (5² +5)/2 = 15
(6n² –6n +1) generates star (hexagram) numbers.
The third star, (6 × 3² –6 × 3 +1) = 37
There are also threedimensional figurate numbers, such as truncated octahedrons, (16n³ –33n² + 24n –6).
The fourth truncated octahedron: (16 × 4³ –33 × 4² + 24 × 4 –6) = 586
