000RM.dvi

310 The golden section

10.1 The golden sectionφ

A segmentABis said to be divided into the golden ratio by a pointPif

AP^2 =AB·PB.

A B

M

P

1

φ

AB

AP

=φ=

√

5+1

2

.

Some basic properties ofφ

1.φ^2 =1+φ;φ^1 =φ− 1.

2.The diagonal and the side of a regular pentagon are in the golden

ratio.

3.The ratio of successive Fibonacci numbers tends toφ.

4.The simple continued fraction expansion ofφis

1+

1

1+

1

1+

1

1+

1

...

.