Definition and Examples
The number 10, for example, can be arranged as a triangle (see triangular number):
But 10 cannot be arranged as a square. The number 9, on the other hand, can be (see square number):
Some numbers, like 36, can be arranged both as a square and as a triangle (see square triangular number):
By convention, 1 is the first polygonal number for any number of sides. The rule for enlarging the polygon to the next size is to extend two adjacent arms by one point and to then add the required extra sides between those points. In the following diagrams, each extra layer is shown as in red.
- Triangular numbers
- Square numbers
Polygons with higher numbers of sides, such as pentagons and hexagons, can also be constructed according to this rule, although the dots will no longer form a perfectly regular lattice like above.
- Pentagonal numbers
- Hexagonal numbers
Read more about this topic: Polygonal Number
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