1

––

ϕ

AB

––

AP

1

––

ϕ

1

––

2

golden rectangle 231

1

and radius AD to find this point.) Then point

Pdivides the segment AB in the golden ratio.

(To see why this works, suppose that the line segment

AB is of unit length, and the length of AP is x. Then

triangle ABC is a right triangle with hypotenuse

+ x. P

YTHAGORAS

’

S THEOREM

now shows that x=

ϕ– 1 = , giving = ϕ.)

Noting that the construction of midpoints, circles,

and perpendicular lines can all be accomplished with a

straight edge (that is, a ruler with no markings) and a

compass alone, the above procedure shows that the

golden ratio is a

CONSTRUCTIBLE

number.

Dividing the relation ϕ2= ϕ+ 1 through by ϕyields

ϕ= 1 + . Substituting this formula into itself multiple

times establishes:

\\

Repeating this process indefinitely shows that the

golden ratio has the following simple

CONTINUED

FRACTION

expansion:

If one terminates this expansion after a finite number

of steps, then ratios of the F

IBONACCI NUMBERS

appear:

, and so forth. (That this pattern persists can

be proved by

INDUCTION

.) We have:

An induction argument also proves that:

ϕn= Fnϕ+ Fn–1

Substituting the formula ϕ= √

–

1+ϕinto itself multiple

times gives an expression for ϕas a sequence of nested

radicals akin to V

IÈTE

’

S FORMULA

for π. We have:

The golden ratio also occurs in

TRIGONOMETRY

as

, for

instance.

The number ϕalso appears in a number of unex-

pected places in nature and throughout mankind’s

artistic pursuits. Since the golden ratio is well approxi-

mated by the fraction 16/10, some scholars suggest that

the ancient Egyptians of 3000

B

.

C

.

E

. used the golden

ratio repeatedly in the construction of their tombs. The

“golden chamber” of the tomb of Rameses IV measures

16 ells by 16 ells by 10 ells, that is, approximately the

ratio ϕ: ϕ: 1; other tombs are found in the approxi-

mate ratio ϕ2: ϕ: 1; and Egyptian furniture found in

those tombs often had overall shape based on the ratio

ϕ: 1 : 1. German artist A

LBRECHT

D

ÜRER

(1471–1528)

wrote a four-volume text, Treatise on Human Propor-

tions, detailing occurrences of the ratio ϕin the human

body. (He claimed, for instance, that ratio of the length

of the human face to its width is approximately ϕ, and

also that the elbow divides the human arm, shoulder to

fingertip, in the golden ratio.) Artists of that time came

to view the golden ratio as a “divine

PROPORTION

” and

used it in all forms of artistic work. The

GOLDEN RECT

-

ANGLE

was deemed the rectangular shape most pleasing

to the eye.

golden rectangle Any rectangle whose sides are in

the ratio 1 to ϕ, where ϕ= is the

GOLDEN

RATIO

, is called a golden rectangle. Such a rectangle has

the property that excising the largest square possible

from one end of the figure leaves another rectangle in

the same proportion. (The remaining rectangle has pro-

portions ϕ– 1 to 1. Since the golden ratio satisfies the

equation ϕ2= ϕ+ 1, we have: = .) By this

method, new golden rectangles can be constructed from

ϕ– 1

––––

1

1

––

ϕ

1 + √

–

5

––––

2