Jacquerie27

A polygonal number is a number that can be arranged in the shape of a polygon. Ancient mathematicians discovered that numbers could be arranged in certain ways when they were represented by pebbles or seeds. The number 10, for example, can be arranged as a triangle (see triangular_number):

l
ll
lll
llll

But 10 cannot be arranged as a square. The number 9, on the other hand, can be (see square_number):

lll
lll
lll

Some numbers, like 36, can be arranged both as a square and as a triangle:

llllll
llllll
llllll
llllll
llllll
llllll

l
ll
lll
llll
lllll
llllll
lllllll
llllllll

Polygons with higher numbers of sides, such as pentagons and hexagons, can be represented also be represented as arrangements of dots (by convention 1 is the first polygonal number for any number of sides):

Pentagonal numbers

1:

l

5:

 l
l l
l l

12:

  l
 l l
l l l
l l l
l l l

22:

   l
  l l
 l l l
l l l l
l l l l
l l l l
l l l l

Hexagonal numbers

1:

l

6:

 l l 
l   l
 l l

15:

  l l l
 l     l
l       l
l       l
l       l
 l     l
  l l l

28:

   l l l l
  l       l
 l         l
l           l
l           l
l           l
l           l
 l         l
  l       l
   l l l l

If s is the number of sides in a polygon, the formula for the nth s-polygonal number is n((s-2) - (4-s)).

References

• The Penguin Dictionary of Curious and Interesting Numbers, David Wells (Penguin Books, 1997) [ISBN 0140261494].

• [Polygonal numbers at MathWorld (http://mathworld.wolfram.com/PolygonalNumber.html)]