000RM.dvi

11.2 Hofstetter’s compass-only construction of the golden section 323

11.2 Hofstetter’s compass-only construction of the golden

section

Kurt Hofstetter has found the following construction of the golden sec-
tion by striking the compass only five times.^1
We denote byP(Q)the circle withPas center andPQas radius.
Figure 1 shows two circlesA(B)andB(A)intersecting atC andD.
The lineABintersects the circles again atEandF. The circlesA(F)
andB(E)intersect at two pointsXandY. It is clear thatC,D,X,
Y are on a line. It is much more interesting to note thatDdivides the
segmentCXin the golden ratio,i.e.,

CD

CX

=

√

5 − 1

2

.

This is easy to verify. If we assumeABof length 2, thenCD=2

√

3

andCX=

√

15 +

√

3. From these,

CD

CX

=

2

√

3

√

15 +

√

3

=

2

√

5+1

=

√

5 − 1

2

.

X

D

C

E AFB

Y

X

D

C

E A B F

Y

This shows that to construct three collinear points in golden section,
we need four circles and one line. It is possible, however, to replace the
lineABby a circle, sayC(D). See Figure 2. Thus,the golden section
can be constructed with compass only, in 5 steps.
Here is a simple application: to divide a given segment into golden
section.^2

Construction 11.1.Given a segmentAB, construct

(^1) K. Hofstetter, A simple construction of the golden section,Forum Geometricorum, 2 (2002) 65–66.
(^2) Communicated by K. Hofstetter on December 9, 2003.