Pythagoras and the Pythagoreans
17
quences have sums given by
1 + 4 + 7 +
. . .
+ (3
n
−
2) =
3
2
n
2
−
1
2
n
and
1 + 5 + 9 +
. . .
+ (4
n
−
3) = 2
n
2
−
n.
Similarly, polygonal numbers of all orders are designated; this
process can be extended to three dimensional space, where there results
the
polyhedral numbers
. Philolaus is reported to have said:
All things which can be known have number; for it is not
possible that without number anything can be either con-
ceived or known.
6
Pythagorean Geometry
6.1
Pythagorean Triples and The Pythagorean Theorem
Whether Pythagoras learned about the 3, 4, 5 right triangle while he
studied in Egypt or not, he was certainly aware of it. This fact though
could not but strengthen his conviction that
all is number
. It would
also have led to his attempt to find other forms, i.e. triples. How might
he have done this?
One place to start would be with the square numbers, and arrange
that three consecutive numbers be a Pythagorean triple! Consider for
any odd number
m
,
m
2
+ (
m
2
−
1
2
)
2
= (
m
2
+ 1
2
)
2
which is the same as
m
2
+
m
4
4
−
m
2
2
+
1
4
=
m
4
4
+
m
2
2
+
1
4
or
m
2
=
m
2