Pythagoras and the Pythagoreans
23
The points
Q
1
, Q
2
, , Q
3
, . . .
divide the segments
AQ, AQ
1
,
|
>
AQ
2
, . . .
into extreme and mean ratio, respectively.
The Pythagorean Pentagram
And this was all connected with the construction of a pentagon. First
we need to construct the golden section. The geometric construction,
the only kind accepted
21
, is illustrated below.
Assume the square ABCE has side length
a
. Bisecting DC at E con-
struct the diagonal AE, and extend the segment ED to EF, so that
EF=AE. Construct the square DFGH. The line AHD is divided into
extrema and mean ratio.
A
B
C
D
E
F
G
H
Golden Section
Verification:
|
AE
|
2
=
|
AD
|
2
+
|
DE
|
2
=
a
2
+ (
a/
2)
2
=
5
4
a
2
.
Thus,
|
DH
|
= (
√
5
2
−
1
2
)
a
=
√
5
−
1
2
a.
The key to the compass and ruler construction of the pentagon is
the construction of the isosceles triangle with angles
36
o
,
72
o
,
and
72
o
.
We begin this construction from the line AC in the figure below.
21
In actual fact, the Greek “
Þ
xation” on geometric methods to the exclusion of algebraic
methods can be attributed to the in
ß
uence of Eudoxus