mathematics

Geometrography

Geometrography is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Geometrography rather than just read about it. In short: 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 (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.

Key takeaways

Geometrography belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Geometrography to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Geometrography from memory before moving on to harder problems.

Reference excerpt

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 (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. 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 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 2C1. 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 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 involving the Carlyle circles can be represented by the expression 8R1 + 4R2 + 22C1 + 11C3 and has simplicity 45.

References

Further reading Hess, Adrien L (Mar–Apr 1956). "Certain topics related to constructions with straight edge and compasses". Mathematics Magazine. 29 (4): 217–221. doi:10.2307/3029638. JSTOR 3029638. Newton, Guy Thornwel (1926). Geometrography with applications to the instruments of the draftsman. University of Texas. p. 190. DeTemple, Duane W. (Feb 1991). "Carlyle circles and Lemoine simplicity of polygon constructions" (PDF). The American Mathematical Monthly. 98 (2): 97–208. doi:10.2307/2323939. JSTOR 2323939. Archived from the original (PDF) on 2015-12-21. Retrieved 6 November 2011.

Illustrations

Worked examples

Example 1 — a first encounter with Geometrography

Start with the simplest possible case. Write down what Geometrography claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Geometrography before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Geometrography ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Geometrography

In research: Geometrography appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Geometrography in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Geometrography is common in secondary-school and first-year university syllabi. It links to neighbouring topics Euclidean plane geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Geometrography outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Geometrography in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Geometrography means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Geometrography out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Geometrography in simple terms?

Why does Geometrography matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Geometrography?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Geometrography.