Polygonal number

In mathematics, a polygonal number is a number that counts dots arranged in the shape of a regular polygon. These are one type of 2-dimensional figurate numbers. Polygonal numbers were first studied during the 6th century BC by the Ancient Greeks, who investigated and discussed properties of oblong, triangular, and square numbers. 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. The triangular number sequence is the representation of the numbers in the form of equilateral triangle arranged in a series or sequence. These numbers are in a sequence of 1, 3, 6, 10, 15, 21, 28, 36, 45, and so on. 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.