Pythagoras and the Pythagoreans
22
A line AC divided into
extreme and mean ratio
is defined to mean
that it is divided into two parts, AP and PC so that AP:AC=PC:AP,
where AP is the longer part.
A
Q
P
C
Golden Section
AP : AC = PC : AP
Let
AP
=
x
and
AC
=
a
. Then the golden section is
x
a
=
a
−
x
x
,
and this gives the quadratic equation
x
2
+
ax
−
a
2
.
The solution is
x
=
−
1
±
√
5
2
a.
The
golden section
20
is the positive root:
x
=
√
5
−
1
2
∼
.
62
The point
Q
in the diagram above is positioned at a distance from
A
so that
|
AQ
|
=
|
P C
|
. As such the segment
AP
is divided into mean
and extreme ratio by
Q
. Can you prove this? Of course, this idea can
be applied recursively, to successive refinements of the segment all into
such sections.
In the figure to the right
Q
1
, Q
2
, Q
3
, . . .
are selected so
that
|
AQ
1
|
=
|
QP
|
,
|
AQ
2
|
=
|
Q
1
Q
|
,
|
AQ
3
|
=
|
Q
2
Q
1
|
, . . .
respectively.
A
Q
P
C
Golden Section
| AP | : | AC | = | PC | : | AP |
Q
1
Q
3
Q
2
20
...now called the
Golden ratio.
Curiously, this number has recurred throughout the devel-
opment of mathematics. We will see it again and again.