Ruler and "Compass"

Problem Statement

Origin Q.E.D. Volume 38 Part 2 「十七」. I highly recommend this chapter.

Here's a quote from near the end of the story (no spoilers).

Japanese version

"In 1796, when the great mathematician Gauss was 18 years old, he discovered that a regular 17-gon can be drawn using only a ruler and compass."

My friend Suthee posted the Thai version on Facebook:

Thai version

The ambiguous コンパス (compass) was translated as เข็มทิศ (navigational compass), which is different from วงเวียน (the tool for drawing circles).

That's a bad translation, but can we actually draw a regular 17-gon with a markless straightedge and navigational compass? Assume the compass is a perfect circle of unknown radius with a needle that always points north.

Solution

Solution

Poncelet-Steiner theorem: Whatever can be constructed by straightedge and compass can be constructed by straightedge alone if a circle and its center are given.

There should be a simpler solution, but this is the first one I could think of.

Exported: 2017-06-10T03:05:30.397247