In the mathematical field of geometry, geometrography is the study of geometrical constructions. The concepts and methods of geometrography were first expounded by

Émile Lemoine Émile Michel Hyacinthe Lemoine (; 22 November 1840 – 21 February 1912) was a French civil engineer and a mathematician, a geometer in particular. He was educated at a variety of institutions, including the Prytanée National Militaire and, most ...

(1840–1912), a French civil engineer and a

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

, in a meeting of the French Association for the Advancement of the Sciences held at

Oran Oran ( ar, وَهران, Wahrān) is a major coastal city located in the north-west of Algeria. It is considered the second most important city of Algeria after the capital Algiers, due to its population and commercial, industrial, and cultural ...

in 1888. Lemoine later expanded his ideas in another memoir read at the Pau meeting of the same Association held in 1892. It is well known in

elementary geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...

that certain geometrical constructions are simpler than certain others. But in many case it turns out that the apparent simplicity of a construction does not consist in the practical execution of the construction, but in the brevity of the statement of what has to be done. Can then any objective criterion be laid down by which an estimate may be formed of the relative simplicity of several different constructions for attaining the same end? Lemoine developed the ideas of geometrography to answer this question. The question of the ubiquity of a construction is also raised. Whether or not a construction, regardless of simplicity, can be applied in all or most conditions, or just in the special cases, is an important consideration.

Basic ideas

In developing the ideas of geometrography, Lemoine restricted himself to Euclidean constructions using rulers and compasses alone. According to the analysis of Lemoine, all such constructions can be executed, as a sequence of operations selected form a fixed set of five elementary operations. The five elementary operations identified by Lemoine are the following: Elementary operations in a geometrical construction In a geometrical construction the fact that an operation X is to be done ''n'' times is denoted by the expression ''n''X. The operation of placing a ruler in coincidence with two points is indicated by 2R₁. The operation of putting one point of the compasses on a determinate point and the other point of the compasses on another determinate point is 2C₁. Every geometrical construction can be represented by an expression of the following form :''l''₁R₁ + ''l''₂R₂ + ''m''₁C₁ + ''m''₂C₂ + ''m''₃C₃. Here the coefficients ''l''₁, etc. denote the number of times any particular operation is performed.

Coefficient of simplicity

The number ''l''₁ + ''l''₂ + ''m''₁ +''m''₂ + ''m''₃ is called the ''coefficient of simplicity'', or the ''simplicity of the construction''. It denotes the total number of operations.

Coefficient of exactitude

The number ''l''₁ + ''m''₁ + ''m''₂ is called the ''coefficient of exactitude'', or ''the exactitude of the construction''; it denotes the number of preparatory operations, on which the exactitude of the construction depends.

Examples

Lemoine applied his scheme to analyze more than sixty problems in elementary geometry. *The construction of a triangle given the three vertices can be represented by the expression 4R₁ + 3R₂. *A certain construction of the regular

heptadecagon In geometry, a heptadecagon, septadecagon or 17-gon is a seventeen-sided polygon. Regular heptadecagon A '' regular heptadecagon'' is represented by the Schläfli symbol . Construction As 17 is a Fermat prime, the regular heptadecagon is a ...

involving the

Carlyle circle In mathematics, a Carlyle circle (named for Thomas Carlyle) is a certain circle in a coordinate plane associated with a quadratic equation. The circle has the property that the solutions of the quadratic equation are the horizontal coordinates of ...

s can be represented by the expression 8R₁ + 4R₂ + 22C₁ + 11C₃ and has simplicity 45.Weisstein, Eric W. "Heptadecagon." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Heptadecagon.html

Basic ideas

Coefficient of simplicity

Coefficient of exactitude

Examples

References

Further reading