Nicomedes's Geometric Construction of Principal Cube Root

Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
This Demonstration shows Nicomedes's method for constructing a cube root using a marked ruler. (Nicomedes was born about 270 BC.) This is actually the real principal cube root. Every nonzero number also has a pair of complex conjugate cube roots.
Contributed by: Izidor Hafner (September 2017)
Open content licensed under CC BY-NC-SA
Snapshots
Details
This Demonstration is based on [1, pp. 128, 129]. The proof from [1] follows.
Let the line parallel to that passes through
intersect
at
. So
. Then, since
bisects
,
. Also, since
, we have
. With
, by two applications of the Pythagorean theorem,
,
which reduces to the quartic equation
.
Fortunately, this quartic easily factors as . Since
,
.
Reference
[1] G. E. Martin, Geometric Constructions, New York: Springer, 1998.
Permanent Citation