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Figurative Numbers
Pythagoreans discovered the Figurative Number. The Greeks were deeply interested in numbers especially to those connected with the geometric shapes, and given the name therefore figurative numbers. Since Pythagoreans as the early custom of Greeks used to play with the pebbles to form the different shapes, so they were more fascinated with the relationship that emerged with the different shapes of pebble-like Triangular, Square, Cubic, Pyramid, Hexagonal …etc
The Greek word for pebble was pséphoi, meant to calculate. The pebbles made it possible for Pythagoreans to identify different shapes, the simplest being the two dimensional figure the triangle and simplest three dimensional figures was the tetrahedron.
Aristotle in his Metaphysics writes “They (the Pythagoreans) supposed the elements of numbers to be the elements of all things and the whole heaven to be a musical scale and a number …Evidently then these thinkers also consider that number as the principal both as a matter for things and as forming both their modification end their permanent states.”
This part of the chapter deals with only the figurative numbers and its different properties.
Triangular Numbers:-  This is a kind of Polygonal number. It is the number of dots required to draw a triangle. The triangular numbers are formed by the partial sum of the series 1+ 2 + 3+ … + n.
The Greeks also noted that these triangular numbers are the sum of consecutive natural numbers, as they appear in the number sequence. If the process continues till n th array then numbers of pebbles in the nth array is 1+2+3+…+n=n* (n+1)/2
1                                  first triangular number
1+2=3                          second triangular number
1+2+3=6                     third triangular number
1+2+3+4=10               fourth triangular number
1+2+3+4+5=15           fifth triangular number
And so on…
Here is a picture of first few triangular numbers.

Properties of Triangular Numbers:-
v  A triangular number can never end with 2, 4, 7, or 9.
v  The sum of the two consecutive triangular numbers is always a square number.                             T1 +T2 = 1 + 3 = 4 = 22                                                                                                                             T2 + T3 = 3 + 6 = 9 = 32                                                                                                           T3 + T4 = 6 + 10 = 16 = 42
v  All perfect numbers are triangular numbers.
v  A triangular number greater than 1 can not be a cube, a fourth Power or a fifth Power.
v  The only triangular number which is also a prime is 3.
v  The only triangular number which is also a Fermat number is 3
v  The only Fibonacci numbers which are also triangular are 1, 3, 21, and 55.
v  Some triangular numbers are the product of three consecutive numbers.                                                  T3 = 6 = 1* 2 * 3                                                                                                                       T15 = 120 = 4* 5* 6                                                                                                                      T20 = 210 = 5 * 6 * 7                                                                                                                        T44 = 990 =9 * 10 * 11                                                                                                              T608 = 185136 = 56 * 57 * 58                                                                                                  ———————- ———————
v
 1   1   1          1   2   1   1   3   3   1      1   4   6   4   1  1  5  10   10   5   1 1  6    15    20  15 6  1
A triangular number can be seen in Pascal’s triangle. Look at the Pascal’s Triangle and you will find that the third diagonal is all triangular numbers.
Square Numbers:- The number 1, 4, 9,16,25,36…  are called the square numbers. It is the numbers of dots arranged in such a way that it represents a square shape. These are the square of the natural numbers 1, 2, 3, 4, 5, 6…..  respectively.
The Greeks also have discovered that if consecutive odd numbers are added they become square numbers. 1=1*1
1+3=4=22
1+3+5=9=32
1+3+5+7=16=42
1+3+5+7+9=25=52

More interestingly each higher square number is formed by adding L shaped set of pebbles to the previous number. The L-shape was called gnomon by the Greeks which referred to an instrument imported to Greece from Babylon for measuring time.
Note that the square number can be found by addition of all triangular number in the following manner—
1       3        6          10        15        21        28        36…
1    3       6       10         15        21        28        36    …
1     4       9       16         25        36        49        64….
Properties of Square Numbers:-
o   Every square number can end with 00, 1, 4, 5, 6, or9.
o   No square number ends in 2, 3, 7, or 8.
o   Look at the following pattern                                                                                                                                           12 = 1                                                                                                                                112 = 121        and      1 + 2 + 1 = 4 = 22                                                                   1112 = 12321     and     1 + 2 + 3 + 2 + 1 =9 = 32
11112 = 1234321    and     1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 =  42
111112 = 123454321  and     1 + 2 + 3 + 4 + 5 + 4 + 3 +2 +1 = 25 = 52                       ——————————————————————————————————————————————————————–
Cube Numbers:-  The numbers which can be represented by three dimensional cubes are called cubic number. 1,8,27,64,125…are cubic numbers which are obviously the cubes of 1,2,3,4,5,….

Properties of Cubic Numbers:-
• 13=1                                                     first odd number
23=8=3+5                                            sum of next two odd numbers
33=27=7+9+11                                    sum of the next three odd numbers
43=64=13+15+17+19              sum of next four odd numbers
53=125=21+23+25+27+29                  sum of next five odd numbers
• Between 1 and 100 there are only two numbers 1 and64 that are also square numbers.
• If C1, C2, C3 ….are the first, second, third… cubic number then they exhibit a unique property:-                                                                                                                                                                            C1 = ( T1)2                                                                                                                                C1 + C2 = 1 + 8 =( T2) 2                                                                                                       C1 + C2 +C3 = 1 + 8 + 27 = 36 = ( T3) 2                                                                               C1 + C2 +C3+C4 = 1 + 8 + 27 + 64 = 100 = (T4)2
• Tetrahedral Numbers:- The numbers that can be represented by the layers of triangles forming a tetrahedron shape are called tetrahedral numbers. It is a figurative numbers of the form  Tn =   nC3 where n = 3, 4, 5,….4, 10, 20…are the example of tetrahedral numbers.
Properties of Tetrahedral Numbers:-
1. The tetrahedral numbers are the sums of the consecutive triangular numbers beginning from 1.                                                                                                                                                                  T1= 1                                                                                                           T2 = 1 + 3 = 4                                                                                                                    T3 = 1 + 3 + 6 = 10                                                                                                               T4 = 1 + 3 + 6 + 10 = 20                                                                                                        T5 =1 + 3 + 6+ 10 + 15 = 35                                                                                       T6 = 1 + 3 + 6 + 10 + 15 + 21 = 56                                                                            ————————————————-
1. The sum of two consecutive numbers is a Pyramidal number.                                                                                                T1 + T2 = 1 + 4 = 5                                                                                                       T2 + T3  = 4 + 10 = 14                                                                                                           T3 + T4  = 10 + 20 = 35                                                                                                        T4 + T5 = 20 + 35 = 55                                                                                                                 T5 + T6 = 35 + 56 = 91
2.  1   1   1          1   2   1   1   3   3   1      1   4   6    4   1      1   5   10    10   5   1  1   6   15    20   15   6   1

The tetrahedral numbers can be seen in the fourth diagonal of a Pascal’s triangle

Pentagonal Numbers:- Those numbers which represent the shape of a pentagon are called a pentagonal number. In the pentagonal numbers, the lower base is a square with a triangle on the top. 1, 5,12,22,35…are its example. The nth pentagonal number Pn is given by the formula:-
Pn = n ( 3n – 1 )
If we represent the pentagonal numbers by P1, P2,…. then the n th number Pn =n(n-1)/2+n2
Properties:-
1. Every nth pentagonal number is one-third of the 3n – 1 th triangular number.
Hexagonal Number:- Those numbers which form a shape of a hexagon are called hexagonal numbers.  1, 6, 15, 28, 45…. are the few examples of hexagonal numbers.
Hexagonal numbers are of the form n (2n-1).

Properties:-
·         Every hexagonal number is a triangular number.
·
•  1,7,19,37,61,91… are the centered hexagonal numbers.

·             11 and 26 are the only numbers that can be represented by the sum using the maximum possible of six hexagonal numbers.

11 = 1 + 1 + 1 + 1 + 1 + 6                                                                                                                     26 = 1 + 1 + 6 + 6 + 6 + 6

Pyramidal Number:-    Those numbers which can be represented as layers of squares forming a pyramid are called pyramidal numbers.  The pyramid class can be formed by adding successive layers of which the next above the nth is the (n-1)th member of the same figurative number series.

35                                                                                        55
There are many more figurative numbers which are not discussed here but one thing is clear that they are really very- very interesting. Though in the initial phase; the study of such numbers produced no immediate results but certainly they are important as it led to the study of series, which provided the clue to an understanding of numbers which are not full grown. The credit certainly goes to the Pythagoreans who dealt with such numbers. Even in the history, triangular numbers played an important role in suggesting rules for forming and adding the terms of series. A relic of such numbers is seen in the problems relating to the pilling of round shot, still to be found in algebras. Ovid in his poem De Nuce talks about the pyramidal number.  So the journey which Pythagoreans began with pebbles has now reached many milestones in the mathematics and mathematicians are also looking for other figurative numbers making their journey endless.