Constructions Using Conics

Conclusion  After recalling definitions and results about the constructibility of a geometric object, we have shown by more and more efficient methods how the works of Gauss, computer algebra systems (Maple), and dynamic geometry software (Cabri-Geometry, distributed by Texas Instruments) could be used together to construct regular polygons, using ruler, compass, and simple conics. In particular, we have given the list of small ₂-constructible polygons, and presented new ₂-constructions of the regular polygons with 19, 37, 73, and 97 sides. The ancient Greeks gave precedence to constructions using only ruler and compass, not because they did not know about the other curves (they invented a number of mechanical devices drawing some algebraic curves of degrees 2, 3, 4, and more), but for the neatness, perfection of reasoning, and the simplicity of the shapes involved (circle and straight line). Today’s tools such as Cabri-Geometry enlarge the notion of geometric simplicity by allowing the manipulation of algebraic expressions (the sequences defined by Gauss) and complex geometric objects (the conic sections). Some generalizations of the questions treated here may be considered:

1.	What does the set of constructible numbers become if we consider algebraic curves of higher degrees?
2.	What is the asymptotic distribution of the primes of the form 2a3b + 1?
3.	Can the 2-constructions of the regular polygons be fully automated?
4.	Givenn, what is the most efficient way of 2-constructingR n, in terms of number of steps and in terms of precision of the intersections involved (avoiding intersection between near-tangent curves)?

An erratum to this article is available at .