Polygonal numbers in ancient Greek mathematics 315
He proceeds to show in the same way that:
Th e n th square number is the sum of the fi rst n odd numbers and its side is n.
but in this case the odd numbers are added so as to preserve the square
shape ( Figure 9.2 ).
Th e formulation corresponding to the presentation of the pentagonal
numbers is:
Th e n th pentagonal number is the sum of the fi rst n numbers x 1 , x 2 ,... , x (^) n which
are such that x (^) i + 1 = x (^) i + 3, and its side is n.
Th e fi rst three are represented below ( Figure 9.3 ).
We are not given a graphic representation of the the next pentagonal num-
ber 22, but its representation would certainly be the following ( Figure 9.4 ):
Figure 9.3 Th e generation of the fi rst three pentagonal numbers.
αα α
1 αα α α
αα α α α
1 + 4 ααα
α α α
1 + 4 + 7
Figure 9.2 Th e generation of square numbers.
ααα
1αα
1 + 3
ααα
ααα
ααα
1 + 3 + 5
αααα
αααα
αααα
αααα
1 + 3 + 5 + 7.