History of the non-Euclidean geometry intrigues pupils

Iris van Gulik                                      Gerard Lankwarden

University of Groningen                       Van der Capellen comprehensive school Zwolle

The Netherlands                                 The Netherlands        

gulikgulikers@home.nl                        g.lankwarden@planet.nl   

Pupil: “When you hear for the first time that the three angles of a triangle do not add up to 180⁰ your first thought is, that’s impossible! But it is possible. This runs counter to everything you know. And that is difficult to accept. But it is interesting.”

Introduction

The first systematic dissertation on geometry dates back to approximately 300 BC. The Greek mathematician Euclid gives in his Elements an axiomatic construction of plane geometry on the basis of 23 definitions, 5 axioms and 5 postulates. From there further propositions are logically deduced. Euclid’s book has served as a model textbook for the study of geometry for ages.

Still the geometry of more than 2000 years ago in Greece was not finalised. It has been an active research area for many centuries, because the construction of the Elements has been subject to many doubts. This is because many mathematicians were of the opinion that one of Euclid’s postulates, the parallel postulate, can be derived from the first four postulates. Not until the 19^th century did Lobačevskiĭ and Bolyai discover that this is impossible. Subsequently they construct a geometry in which they accept the negation of the parallel postulate. This is how the non-Euclidean geometry originated.

I have incorporated the development of geometry and the creation of the non-Euclidean geometry in a booklet written for 17/18 years old VWO pupils (pre-university level). It offers the opportunity for pupils to enlarge their knowledge of geometry and to familiarize themselves with the history of a field in mathematics. The booklet is part of a larger research project. The central question of the research is to what extent the history of geometry can be used, by both pupil and teacher, in “rediscovering” geometrical knowledge.

Important arguments to use the history of mathematics in class are[i]:

In this article I will give a survey of the history of non-Euclidean mathematics[ii] and I will demonstrate a few assignments for pupils.

The Elements of Euclid

Euclid’s postulates[iii] form the “rules” of reasoning within the mathematics of the Elements.

The five postulates of the Elements from the first printed version in 1482:

The fifth postulate, also called the parallel postulate, runs in modern terms as follows:  

When two straight lines m and l are intersected by a third straight line t, and the interior angles a and b in one side of t are together less than 180⁰, then m and l intersect each other at the same side of t.

A well-known proposition from the Elements that is equivalent to the parallel postulate is proposition I.32:

In any triangle, if one of the sides is produced, then the exterior angle equals the sum of the two interior and opposite angles, and the sum of the three interior angles of the triangle equals 180⁰.

In 1795 the Scotsman John Playfair reformulated the fifth postulate:

Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.

Here Playfair makes use of  Euclid’s definition of parallel lines:

Parallel lines are straight lines that are in the same plane and that in both directions extended infinitely do not meet in either direction.

The advantage of Playfair’s way of formulating is that it is easier to formulate its negation. Furthermore Playfair’s postulate appears to be equivalent to Euclid’s fifth postulate.

It is striking that Euclid himself refrains from using the parallel postulate as long as possible (until proposition 29). Therefore it was suspected that the fifth postulate is superfluous. Mathematicians investigated whether the parallel postulate is a proposition that can be deduced from the other axioms or can be replaced by another, simpler postulate that would result in the same mathematics.

‘Proofs’ of the parallel postulate

For centuries mathematicians continued studying the parallel postulate and one of them was Girolamo Saccheri. Since his predecessors had not succeeded in deducing the fifth postulate from the other postulates, the Italian Saccheri tried a new approach. In his Euclides ab Omni Naevo Vindicatus (‘Euclid exonerated from all blame’), a work published in 1733, he makes use of a ‘reductio ad absurdum’ proof. Saccheri accepts the first 28 propositions of Euclid, that can be proved without using the parallel postulate. Subsequently he rejects the fifth postulate and examines the conclusions that can be drawn. He hopes to find the impossible, because he is convinced of the truth of the fifth postulate.

Saccheri studies quadrilaterals, of which the basic angles are right angles and the upright sides are equal to each other.

Without using the parallel postulate it is easy to demonstrate that the summit angles C and D are equal to each other. For the angles in C and D there are now three hypotheses:

·         Hypothesis 1: the angles in C and D are right angles

·         Hypothesis 2: the angles in C and D are obtuse angles

·         Hypothesis 3: the angles in C and D are acute angles

Saccheri accepts the hypothesis of the obtuse angle and the hypothesis of the acute angle. He tries to demonstrate that, together with the first four postulates, this results in a contradiction. From this it might be concluded that the hypothesis of the acute angle is valid and that the fifth postulate can be deduced from this.

Unfortunately Saccheri draws the wrong conclusions at the critical moment, prejudiced by his Euclidean convictions and intuitions. Wrongly he rejects his results as impossible, as a result of which he believes to have proven the parallel postulate.

Founders of the non-Euclidean geometry

In the nineteenth century two unknown mathematicians suddenly find themselves in the limelight of attention, the Russian Nicolai Lobačevskiĭ and the Hungarian Janos Bolyai.

János Bolyai (1802-1860)  Nicolai Lobačevskiĭ (1792-1856)

They try, independent of each other, a completely new approach. They investigate whether it is possible to come to a different, non-Euclidean geometry with the help of the first four postulates and the negation of the fifth postulate. They manage to do so indeed and so they come to the conviction that it is impossible to prove the parallel postulate from the first four postulates.

There are two variants within the non-Euclidean geometry. In the first case, the hyperbolic geometry, the fifth postulate is replaced by the following postulate:

There are several lines parallel to a line l through a point P that is not situated on l.

In the other possibility, the elliptic geometry, the parallel postulate is replaced by:

There are no lines parallel to a line l through a point P that is not situated on l.

A model of the hyperbolic geometry: The Poincaré-disk

In 1906 the French mathematician Jules Henri Poincaré gives a model for the hyperbolic geometry within a Euclidean circle γ. Within this Poincaré-disk the following rules apply:

With the help of the computer programme Cabri[iv] results within the Poincaré-disk can be examined. For that purpose the file hyperbol.men can be downloaded (from http://mcs.open.ac.uk/tcl2/nonE/intro.html). Now it can be demonstrated that the hyperbolic postulate applies within the Poincaré model:

There are several lines through a point P outside a line l that do not intersect l.

The non-intersecting lines are separated from the intersecting lines by the so-called ‘limiting-parallels’. The limiting-parallels each have a certain ‘direction of parallelism’. The angle made by the line from P perpendicular to l with one of the parallels is called the ‘parallel angle’.

Now that the fifth postulate in the hyperbolic geometry is not valid anymore, the proposition that the sum of the angles in a triangle is equivalent to 180⁰, does not apply any longer. This is because this proposition is equivalent to the parallel postulate.

With Cabri we have the possibility to examine that within the hyperbolic geometry the sum of the angles of a triangle is smaller than 180⁰.

Classroom experiences

A 6^th form (pre-university level) of the Meridiaan College from Amersfoort worked with the booklet 2 lessons a week for a period of six weeks. Teacher Klaske Blom makes her pupils look at geometry with a critical view by means of lively class discussions. The non-Euclidean geometry turns the world upside down for these pupils. Pictures on geometry that are stored in their minds have to be discarded and that is difficult.

A pupils uses the blackboard to explain to his fellow pupils why the summit angles of a Saccheri quadrilateral are equal to each other.

Blom finds the non-Euclidean geometry a fascinating subject for her pupils:

“It is aiming very high, they do not really fully understand it, I notice. You see they are making the same leaps that were made in the history of geometry. They really find it fascinating. And that is exactly what these pupils need.”

The pupils themselves also react in a very positive way. After this series of lessons they have all written their own evaluation. Some reactions are:

“We were completely unfamiliar with the material: the non-Euclidean geometry, which I found very confusing and incomprehensible in the beginning. It was completely different from what we were used to. But by applying it yourself with the help of the programme Cabri it became increasingly clear.”

“It is nice to prove. And also to get to know more about the history of mathematics. Because you know more about its background, it really comes to life. You get the feeling that you step into the life of such a mathematician, because you are going to imitate his proof and so going to ‘prove’ again.”

“When we started with this, I was a bit pessimistic about the subject. This is because I am not so good at mathematics and especially not at the non-Euclidean geometry. But that changed rapidly. What I also enjoyed was the fact that we could philosophise about the fact that there may be another kind of geometry. This always resulted in nice discussions.”

In the booklet there are also research assignments for pupils, among others about the meaning of non-Euclidean geometry in the work of Escher. Pupil:

“I really think that Escher’s drawings are superb”. Right from childhood I have been able to look at a drawing of his for a long time, trying to figure out how he had drawn it. I really enjoy it that I know now at least how his drawings fit into a circle, whereas it seems that it goes on infinitely.”

M.C. Eschers "Cirkellimiet IV" (c) 2003 Cordon Art - Baarn. All rights reserved.