Geometric Construtions

Some (mostly on-line) resources

Introduction:

• R. Courant, H. Robbins, "What is Mathematics?", Oxford University Press, 1941. Chapter III.
• Surprising Geometric Constructions by Moti Ben-Ari.

Variation of rules:

• On the impossibility of ruler-only constructions, by V.J.Baston, F.A.Bostock
• Two Circles and only a straightedge, by Arseniy Akopyan and Roman Fedorov
• Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries, by Marvin Jay Greenberg

Origami:

• The Combinatorics of Flat Folds: a Survey, by Thomas C. Hull
Origami and Geometric Constructions, by Robert Lang
• The Folding Mathematics, by Archana S. Morye
• Folding Questions – A Paper about Problems about Paper, by Robert Geretschlaeger
• Roger C. Alperin, Barry Hayes, Robert J. Lang, Folding the hyperbolic crane. See p.14 for the picture!
• Paper folding geometry on cut-the-knot portal

Fold and one cut theorem:

• Fold and one cut theorem. Webpage by Erik Demaine on Folding and Unfolding
• Geometric Folding Algorithms: Linkages, Origami, Polyhedra , book by Erik Demaine and Joseph O'Rourke (available through the library)
• Fold-and-Cut Magic by Erik D. Demaine Martin L. Demaine

Mechanical Constructions (linkages):

• Geometry and topology of polygonal linkages, by Robert Connelly and Erik D. Demaine (Chapter 9 in Handbook of Discrete and Computational Geometry)
• Spaces Of Polygons In The Plane And Morse Theory, by Don Shimamoto and Catherine Vanderwaart
• Topology of robot motion planning, by Michael Farber
• Kempe’s Linkages and the Universality Theorem, by Anupam Saxena
• Generalizations of Kempe’s Universality Theorem, Timothy Good Abbott
• On the geometric definition of hinge mechanisms and Kempe's theorem., talk by Mikhail Kovalev
• Universality theorems for configuration spaces of planar linkages, by Michael Kapovich and John J. Millson

Illumination problem:

• Wikipedia page on Illumination problem, see also the references therein
• Art Gallery and Illumination Problems, book by Jorge Urrutia
• A Lighthouse Illumination problem, by Erhan Tezcan

Tensegrities:

• A Primer on the Mechanics of Tensegrity Structures, by W. O. Williams
• Open Problems on Configuration Spaces of Tensegrities, by Oleg Karpenkov
• Seventy years of tensegrities (and counting), by Andrea Micheletti and Paolo Podio-Guidugli

Flexagons:

• Flexagons, webpage containing papers by Harold V. McIntosh and his students (scroll down).
• FLEXAGON.NET, flexagon portal.

Constructions in Spherical Geometry:

• Straightedge and Compass Constructions in Spherical Geometry, by Daniel J.Heath

Constructions in Hyperbolic Geometry:

• Constructions in hyperbolic geometry, by Frans Hansdest
• An instrument in hyperbolic geometry, by M.W.Al-Dhahir
• Compass and Straightedge in the Poincare Disk, by Chaim Goodman-Strauss
• Squaring circles in the hyperbolic plane, by C. Jagy [in The Mathematical Intelligencer 17 (2) (1995) p.31–36]
• Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries,, by M. Greenberg

Miscellaneous:

• Algorithms and Geometric Constructions, by Vladimir Uspenskiy and Alexander Shen
• Error Propagation in Geometric Constructions, by J.Wallnera, R.Krasauskasb, H.Pottmann
• An Introduction to Medieval Spherical Geometry for Artists and Artisans, by Reza Sarhangi

Some webpages:

• Paradoxes of Impossibility: Classic Geometric Constructions, by John D. Norton
Approximate Construction of Regular Pentagon by A. Dürer, on Cut-the-Knot portal
• Construction of dodecahedron , in Euclid's Elements

Lectures:

• Geometric Folding Algorithms: Linkages, Origami, Polyhedra, by Erik Demaine

Art:

• Unterweysung der Messung mit dem Zyrkel und Rychtscheyd, by A. Dürer