Geometrical Construction

Geometrical Construction 

the solution of certain geometry problems with the aid of auxiliary instruments (straightedge, compass, and others) that are assumed to be absolutely precise. Investigations of geometrical constructions have elucidated the range of problems that are solvable with the aid of an assigned set of instruments and have indicated the methods for solving these problems. Geometrical constructions are usually broken down into constructions on a plane and in space. Certain problems of geometrical constructions on a plane had been considered even in antiquity (for example, the famous problems of trisecting an angle, duplicating a cube, and squaring a circle). Like many others, they belong to problems of geometrical constructions with the aid of a compass and a straightedge. Geometrical constructions on a plane have a rich history. The theory of these constructions was worked out by the Dutch geometer G. Mohr (1672) and by the Italian engineer L. Mascheroni (1797). A considerable contribution to the theory of geometrical constructions was made by the Swiss scientist J. Steiner (1833). Only in the 19th century was the range of problems that are solvable with the aid of the aforementioned instruments ascertained. Specifically, the foregoing famous problems of antiquity are not solvable with the use of a compass and a straightedge.

Geometrical constructions on a Lobachevskii plane were studied by N. I. Lobachevskii himself. The general theory of such constructions and constructions on a sphere was developed by the Soviet geometer D. D. Mordukhai-Boltovskii.

Geometrical constructions in space are associated with the methods of descriptive geometry. The theory of geometrical constructions is of interest in that aspect related to practical applications in descriptive geometry.

REFERENCES

Adler, A. Teoriia geometricheskikh postroenii, 3rd ed. Leningrad, 1940. (Translated from German.)
Chetverukhin, N. F. Metody geometricheskikh postroenii. Moscow, 1938.
Steiner, J. Geometricheskie postroeniia, vypolniaemye s pomoshch’iu priamoi linii i nepodvizhnogo kruga. Moscow, 1939. (Translated from German.)
Aleksandrov, I. I. Sbornik geometricheskikh zadach na postroenie s resheniiami, 18th ed. Moscow, 1950.

E. G. POZNIAK