000RM.dvi

328 Constructions with the golden section

11.5 Ahlburg’s parsimonious construction of the regu-

lar pentagon

Make use of a given right triangleAB′CwithAC=2B′Cto construct
a regular pentagon in the fewest number of euclidean operations. (Eu-
clidean operations include (i) setting a compass, (ii) striking an arc, (iii)
drawing a line.
Between the sideaand the diagonaldof a regular pentagon, there is

the relationa=

√

5 − 1

2 d. Here is Hayo Ahlburg’s construction.

4

A

B′ C

B P

E

D

Construction 11.3.(1) Strike an arcB′(B′C), that is, with centerB′
and radiusB′C, meetingAB′atP.
(2) Strike an arcA(AP).
(3) Strike an arcC(AP), meeting arcA(AP)atB.
(4) Strike an arcB(CA), meeting arcsA(AP)andC(AP)atDand
E.

ThenABCDEis the required regular pentagon. The construction
requires 3 compass settings, striking 4 arcs, and drawing 5 lines for the
sides of the pentagon, 12 euclidean construction in all.

(^4) Crux Math., 6 (1980) 50.