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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 51M15: Geometric constructions

Introduction

There is a theme in classical plane geometry in which one attempts to carry out various constructions using a ruler and compass. The simpler tasks are taught to us in primary and secondary school.

History

Applications and related fields

More general information about polygons (and polyhedra) is available on another page, as are a number of papers concerning plane geometry.

Subfields

Parent field: 51M - Real and Complex Geometry

Textbooks, reference works, and tutorials

Software and tables

Other web sites with this focus

Selected topics at this site

• Constructing the Geometric mean
• How to construct a hyperbola
• Construct the circle tangent to two lines and another circle
• For certain values of n one may construct a regular n-gon.
• Why are only certain n-gons constructible?
• Constructing a pentagon.
• Construction of the pentagon
• Constructing a heptagon
• Constructing the regular heptadecagon (17-sided polygon)
• Construction of the heptadecagon
• Summary and reference to construction of regular polygons and other classic problems
• Literature reviews: constructing the regular n-gon for n=257 or (allowing use of a trisector) for n=7, 13, 19, etc.
• Non-classical tools which allow trisection of angles
• What does it means to say one cannot square the circle?
• Non-constructibility of some extremal circle-points
• Given four points construct a square through them
• It turns out the constructions we are accustomed to can be carried out using only straightedge or only compass
• Steiner's Theorem: straightedge and compass constructions can be accomplished with straightedge, one circle, and its center; not without that center.
• Extending the Poncelet-Steiner ("no compass") theorem.
• Other variations on the Poncelet-Steiner ("no compass") theorem.
• The Mascheroni (no straightedge) theorem.
• Pointer to short proof of Mascheroni's theorem (no straightedges are needed for classical geometric constructions).
• A classic question is whether one can with compass and straightedge trisect an arbitrary angle; the answer is no, but there are ways to trisect angles with a marked straightedge
• Citation: Elementary proof that some angles cannot be trisected.
• Much easier! Trisecting a line segment
• Get your trisections here! and other unique mathematical experiences
• Trisecting an angle by motions off the plane.
• Apollonius' method of trisecting an angle.
• What can be constructed if we assume a trisector?
• Trisecting angles in the hyperbolic plane
• Using a quadratix to multisect an angle.
• Trisection is relevant if you wish to construct a regular nonagon (nine-sided polygon).
• Some things can still be done even with a short straightedge.
• Pointer to analysis of origami as an alternative to Euclidean constructions (origami allows more constructions, e.g. trisections).
• Pointer to paper on paper-folding
• Using projective geometry to perform a construction meeting incidence conditions.