Geometrography
Encyclopedia
In mathematics, in geometry, geometrography is the study of geometrical constructions. The concepts and methods of geometrography were first expounded by Émile Lemoine
(1840–1912), a French civil engineer
and a mathematician
, in a meeting of the French Association for the Advancement of the Sciences held at Oran
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 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.
constructions using ruler
s 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 nX. 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 2C2.
Every geometrical construction can be represented by an expression of the following form
Here the coefficients l1, etc. denote the number of times any
particular operation is performed.
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.
Émile Lemoine
Émile Michel Hyacinthe Lemoine 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 notably, the École Polytechnique...
(1840–1912), a French civil engineer
Civil engineer
A civil engineer is a person who practices civil engineering; the application of planning, designing, constructing, maintaining, and operating infrastructures while protecting the public and environmental health, as well as improving existing infrastructures that have been neglected.Originally, a...
and a mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, in a meeting of the French Association for the Advancement of the Sciences held at Oran
Oran
Oran is a major city on the northwestern Mediterranean coast of Algeria, and the second largest city of the country.It is the capital of the Oran Province . The city has a population of 759,645 , while the metropolitan area has a population of approximately 1,500,000, making it the second largest...
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 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.
Basic ideas
In developing the ideas of geometrography, Lemoine restricted himself to EuclideanEuclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
constructions using ruler
Ruler
A ruler, sometimes called a rule or line gauge, is an instrument used in geometry, technical drawing, printing and engineering/building to measure distances and/or to rule straight lines...
s 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
| Sl. No. | Operation | Notation for operation |
|---|---|---|
| 1 | To place the edge of the ruler Ruler A ruler, sometimes called a rule or line gauge, is an instrument used in geometry, technical drawing, printing and engineering/building to measure distances and/or to rule straight lines... in coincidence with a point |
R1 |
| 2 | To draw a straight line | R2 |
| 3 | To put a point of the compasses on a determinate point | C1 |
| 4 | To put a point of the compasses on an indeterminate point of a line | C2 |
| 5 | To describe a circle Circle A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius.... |
C3 |
In a geometrical construction the fact that an operation X is to be done n times is denoted by the expression nX. 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 2C2.
Every geometrical construction can be represented by an expression of the following form
- l1R1 + l2R2 + m1C1 + m2C2 + m3C3.
Here the coefficients l1, etc. denote the number of times any
particular operation is performed.
Coefficient of simplicity
The number l1 + l2 + m1 +m2 + m3 is called the coefficient of simplicity, or the simplicity of the construction. It denotes the total number of operations.Coefficient of exactitude
The number l1 + m1 + m2 iscalled 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 heptadecagonHeptadecagonIn geometry, a heptadecagon is a seventeen-sided polygon.-Heptadecagon construction:The regular heptadecagon is a constructible polygon, as was shown by Carl Friedrich Gauss in 1796 at the age of 19....
involving the Carlyle circles can be represented by the expression 8R1 + 4R2 + 22C1 + 11C3 and has simplicity 45.