Polygonal number

In mathematics, a polygonal number is a number that can be arranged as a regular 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):

   x 
  x x 
 x x x 
x x x x 

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

x x x 
x x x 
x x x 

Some numbers, like 36, can be arranged both as a square and as a triangle (see triangular square number):

x x x x x x 
x x x x x x 
x x x x x x 
x x x x x x 
x x x x x x 
x x x x x x 

       x 
      x x 
     x x x 
    x x x x 
   x x x x x 
  x x x x x x 
 x x x x x x x 
x x x x x x x x 

The method 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 +.

Triangular numbers

1:

+               x

3:

 x               x
+ +             x x

6:

  x               x
 x x             x x
+ + +           x x x

10:

   x               x
  x x             x x
 x x x           x x x
+ + + +         x x x x

Square numbers

1:

+               x

4:

x +             x x
+ +             x x

9:

x x +           x x x
x x +           x x x
+ + +           x x x

16:

x x x +         x x x x
x x x +         x x x x
x x x +         x x x x
+ + + +         x x x x

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

Pentagonal numbers:

1:

+                   x

5:

 x                   x
+ +                 x x 
+ +                 x x 

12:

   x                   x
  x x                 x x
+ x x +             x x x x
+     +             x     x
+  +  +             x  x  x 

22:

     x                   x
    x x                 x x
  x x x x             x x x x
+ x     x +         x x     x x
+ x  x  x +         x x  x  x x
+         +         x         x
+  +   +  +         x  x   x  x

35:

       x                   x
      x x                 x x
    x x x x             x x x x
  x x     x x         x x     x x 
+ x x  x  x x +     x x x  x  x x x
+ x         x +     x x         x x
+ x  x   x  x +     x x  x   x  x x
+             +     x             x
+  +   +   +  +     x  x   x   x  x

Hexagonal numbers

1:

 x

6:

    x               x  
  +   +           x   x 
  +   +           x   x 
    +               x  

15:

     x                 x   
   x   x             x   x  
 + x   x +         x x   x x 
 +   x   +         x   x   x 
 +       +         x       x 
   +   +             x   x  
     +                 x   

28:

        x                       x    
      x   x                   x   x   
    x x   x x               x x   x x  
  + x   x   x +           x x   x   x x 
  + x       x +           x x       x x 
  +   x   x   +           x   x   x   x 
  +     x     +           x     x     x 
    +       +               x       x  
      +   +                   x   x   
        +                       x    

45:

          x                           x
        x   x                       x   x    
      x x   x x                   x x   x x   
    x x   x   x x               x x   x   x x  
  + x x       x x +           x x x       x x x 
  + x   x   x   x +           x x   x   x   x x 
  + x     x     x +           x x     x     x x 
  +   x       x   +           x   x       x   x 
  +     x   x     +           x     x   x     x 
    +     x     +               x     x     x  
      +       +                   x       x   
        +   +                       x   x    
          +                           x     

66: (which is also a triangular number and a sphenic number)

          x                             x
        x   x                         x   x
      x x   x x                     x x   x x
    x x   x   x x                 x x   x   x x
  x x x       x x x             x x x       x x x
+ x x   x   x   x x +         x x x   x   x   x x x
+ x x     x     x x +         x x x     x     x x x
+ x   x       x   x +         x x   x       x   x x
+ x     x   x     x +         x x     x   x     x x
+   x     x     x   +         x   x     x     x   x
+     x       x     +         x     x       x     x
  +     x   x     +             x     x   x     x
    +     x     +                 x     x     x
      +       +                     x       x
        +   +                         x   x
          +                             x

91:

            x                             x
          x   x                         x   x
        x x   x x                     x x   x x
      x x   x   x x                 x x   x   x x
    x x x       x x x             x x x       x x x
  x x x   x   x   x x x         x x x   x   x   x x x
+ x x x     x     x x x +     x x x x     x     x x x x
+ x x   x       x   x x +     x x x   x       x   x x x
+ x x     x   x     x x +     x x x     x   x     x x x
+ x   x     x     x   x +     x x   x     x     x   x x
+ x     x       x     x +     x x     x       x     x x
+   x     x   x     x   +     x   x     x   x     x   x
+     x     x     x     +     x     x     x     x     x
  +     x       x     +         x     x       x     x
    +     x   x     +             x     x   x     x
      +     x     +                 x     x     x
        +       +                     x       x
          +   +                         x   x
            +                             x

If s is the number of sides in a polygon, the formula for the nth s-polygonal number is ½n((s-2)n - (4-s)).

Name	Formula	n=1	2	3	4	5	6	7	8	9	10	11	12	13
Triangular	½n(1n + 1)	1	3	6	10	15	21	28	36	45	55	66	78	91
Square	½n(2n - 0)	1	4	9	16	25	36	49	64	81	100	121	144	169
Pentagonal	½n(3n - 1)	1	5	12	22	35	51	70	92	117	145	176	210	247
Hexagonal	½n(4n - 2)	1	6	15	28	45	66	91	120	153	190	231	276	325
Heptagonal	½n(5n - 3)	1	7	18	34	55	81	112	148	189	235	286	342	403
Octagonal	½n(6n - 4)	1	8	21	40	65	96	133	176	225	280	341	408	481
Nonagonal	½n(7n - 5)	1	9	24	46	75	111	154	204	261	325	396	474	559
Decagonal	½n(8n - 6)	1	10	27	52	85	126	175	232	297	370	451	540	637
11-agonal	½n(9n - 7)	1	11	30	58	95	141	196	260	333	415	506	606	715
12-agonal	½n(10n - 8)	1	12	33	64	105	156	217	288	369	460	561	672	793
13-agonal	½n(11n - 9)	1	13	36	70	115	171	238	316	405	505	616	738	871
14-agonal	½n(12n - 10)	1	14	39	76	125	186	259	344	441	550	671	804	949
15-agonal	½n(13n - 11)	1	15	42	82	135	201	280	372	477	595	726	870	1027
16-agonal	½n(14n - 12)	1	16	45	88	145	216	301	400	513	640	781	936	1105
17-agonal	½n(15n - 13)	1	17	48	94	155	231	322	428	549	685	836	1002	1183
18-agonal	½n(16n - 14)	1	18	51	100	165	246	343	456	585	730	891	1068	1261
19-agonal	½n(17n - 15)	1	19	54	106	175	261	364	484	621	775	946	1134	1339
20-agonal	½n(18n - 16)	1	20	57	112	185	276	385	512	657	820	1001	1200	1417
21-agonal	½n(19n - 17)	1	21	60	118	195	291	406	540	693	865	1056	1266	1495
22-agonal	½n(20n - 18)	1	22	63	124	205	306	427	568	729	910	1111	1332	1573
23-agonal	½n(21n - 19)	1	23	66	130	215	321	448	596	765	955	1166	1398	1651
24-agonal	½n(22n - 20)	1	24	69	136	225	336	469	624	801	1000	1221	1464	1729
25-agonal	½n(23n - 21)	1	25	72	142	235	351	490	652	837	1045	1276	1530	1807
26-agonal	½n(24n - 22)	1	26	75	148	245	366	511	680	873	1090	1331	1596	1885
27-agonal	½n(25n - 23)	1	27	78	154	255	381	532	708	909	1135	1386	1662	1963
28-agonal	½n(26n - 24)	1	28	81	160	265	396	553	736	945	1180	1441	1728	2041
29-agonal	½n(27n - 25)	1	29	84	166	275	411	574	764	981	1225	1496	1794	2119
30-agonal	½n(28n - 26)	1	30	87	172	285	426	595	792	1017	1270	1551	1860	2197

References

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

• [Polygonal numbers at MathWorld]