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Geometric Problems of Antiquity

The Greek problems of antiquity were a set of geometric problems whose solution was sought using only Compass and Straightedge:

1. Circle Squaring.

2. Cube Duplication.

3. Trisection of an Angle.

Only in modern times, more than 2,000 years after they were formulated, were all three ancient problems proved insoluble using only Compass and Straightedge.

Another ancient geometric problem not proved impossible until 1997 is Alhazen's Billiard Problem. As Ogilvy (1990) points out, constructing the general Regular Polyhedron was really a ``fourth'' unsolved problem of antiquity.

See also Alhazen's Billiard Problem, Circle Squaring, Compass, Constructible Number, Constructible Polygon, Cube Duplication, Geometric Construction, Regular Polyhedron, Straightedge, Trisection


Conway, J. H. and Guy, R. K. ``Three Greek Problems.'' In The Book of Numbers. New York: Springer-Verlag, pp. 190-191, 1996.

Courant, R. and Robbins, H. ``The Unsolvability of the Three Greek Problems.'' §3.3 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 117-118 and 134-140, 1996.

Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 135-138, 1990.

Pappas, T. ``The Impossible Trio.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 130-132, 1989.

Jones, A.; Morris, S.; and Pearson, K. Abstract Algebra and Famous Impossibilities. New York: Springer-Verlag, 1991.

© 1996-9 Eric W. Weisstein