What can be Constructed with the Compass Alone ?

Jen-chung Chuan
Tsing Hua University
Hsinchu, Taiwan 300
jcchuan@math.nthu.edu.tw

Abstract

Mascheroni dedicated one of his books Geometria del compasso (1797) to Napoleon in verse in which he proved that all Euclidean constructions can be made with compasses alone, so a straight edge in not needed. This theorem was (unknown to Mascheroni) proved in 1672 by a little known Danish mathematician Georg Mohr. In the setting of dynamic geometry, the Mohr-Mascheroni construction asks for specific procedures in which the figure is constructed using the compass alone. In this tutorial the participants are guided through this sort of constructions step-by-step using Cabri Java Applets as the tool. We shall concentrate the constructions of
1) the conics: hyperbola, parabola and ellipse.
2) the epicycloids (the cardioid and the nephroid), hypocycloids (the deltoid and the astroid) and their osculating circles.
3) the Lemniscate of Bernoulli.
4) the Bowditch curve.

For the preparation of the lecture, you are invited to visit our website at:
http://poncelet.math.nthu.edu.tw/disk3/cabrijava/compass.html