NEW GEOMETRIES, NEW WORLDS


Euclid's geometrical system was used and developed century after century and no one doubted its truth.

However, a few thinkers including Euclid himself were disturbed by 2 of his axioms.

a) A line segment can be extended as far as one pleases in either direction.

b) Parallel axiom - says that through a point P not on a line L there passes one and only one line M (in the plane of P and L) and does not meet L no matter how far M and L are extended.


These axioms are open to doubt because we cannot say we know what happens in physical space - it may be true around us but what about 10 miles up in space? Even limited space in which we move they might not be true - projective geometry.

Euclid doesn't use the parallel axiom unless he has to. Also he uses line segments not lines.

Many mathematicians tried to work with the parallel axiom - they sought to either deduce it from other axioms or find a more acceptable substitute.

1800 - parallel axiom labelled the scandal of Geometry.

Saccheri had a brand new idea - he argued given a line and a point P then either:

a) There is exactly one parallel to L through P, or

b) there are no parallels to L through P, or

c) there are at least 2 parallels to L through P.

Alternative (a) was Euclid's parallel axiom. Suppose it was replaced by (b) and the latter together with the other 9 axioms of Euclid were shown to lead to contradictory theorems. Then (b) couldn't be correct, similarly with (c) - then it would follow that Euclid's axiom is the only one, - so Saccheri said.

Saccheri produced contradictions using (b) but not (c). With (c) he produced strange theorems for his axioms and because they were so strange he decided Euclid must be right.

One explanation of Saccheri's and other mathematician's failure was that they couldn't reject a habit of thought, 3 men in the nineteenth century Gauss, Lobatchevsky and Bolyai did just that.

Gaus

Gauss adopted Saccheri's third alternative and like Saccheri he developed strange theorems - he decided this meant there could be other geometries as valid as Euclid's. His work wasn't published until after his death.

Lobatchevsky

Lobatchevsky did the same as Gauss and concluded like him that there were other geometries.

Bolyai

John's father had worked on the parallel axiom but John in 1825 suddenly saw the light - published his work in 1833.

Works of Lobatchevsky and Bolyai were completely neglected - Lobvatchevsky was dismissed from Kazan University.

About 30 years after Lobatchevsky and Bolyai published, Gauss' work was published. More people then read Lovatchevsky and Bolyai.

Lobatchevsky and Bolyai's Theory


Euclid's parallel axiom asserts that there is one and only one line K through P which does not meet L.

Let Q be any point on L. As Q moves to the right the line PQ revolves counterclockwise about P and seems to approach the line K. In like manner as Q moves to the left along L the line PQ rotates clockwise about P and again approaches K. In each case PQ approaches one and the same limiting line K.

Bolyai and Lobatchevsky assumed however that the two limiting positions of PQ are not the same line K but 2 different lines through P and these limiting lines M and N do not meet L.


Moreover they assured that every line through P and between M and N such as J does not meet L. Hence Bolyai's and Lobatchevsky's parallel axiom affirms the existence of an infinite set of parallels to L through P. (These men reserved the word parallel for just the limiting lines M and N but we shall use it and denote any line through P that does not meet L.

Diagrams failure to correspond to visual sensations is irrelevant.

All theorems of Euclid that don't use parallel postulate remain the same - eg. vertical angles are equal, in a triangle with equal sides the angles opposite these sides are equal.

Theorems Lobatchevsky and Bolyai proved by deduction - (not much use of figures) which weren't the same are:

a) the sum of the angles of any triangle is always less than 180

b) of two triangles the one with a larger area has a smaller angle sum

c) two similar triangles must be congruent

d) the distance between 2 parallel lines approaches zero in one direction along the lines and becomes infinite in the other direction.

Proved hundred of theorems - none contradicted each other. This meant that the old parallel axiom could not be deducted for the other Euclidean axioms.

Other development for Bolyai and Lobatchevsky was that we could not hope to establish the incontrovertible truth of the Euclidean parallel axioms by showing that alternatives produced contradiction.

Main point - there are geometries different from Euclid's.

How can new geometry be correct? What about physical space - is Euclidean geometry really true for all space? Gauss tried an experiment to see if the angles in a triangle are 180. He had to choose a large triangle (in Lobatchevsky and Bolyai's theory sum 180 as triangle shrinks) Gauss stationed an observer on each of 3 mountain peaks. Each observer measured the angle found by his lines of sight to other two observers - angle sum was 180 to within 2" - experiment not decisive. But assumptions had been made - triangle was large enough, and light rays are straight lines.

For investigations like this it was found that the 2 geometries fit physical space equally well.

Geodesics and Physical Represntation of Lobatchevsky and Bolyai

Experiment tells us that if we take the straight line to be a streched string then Euclidean geometry applies very well.

However for people who live in a mountaneous country and are interested in the geometry of the surface of their country, a straight line would be better defined as geodesic - the curve of shortest distance between 2 points. What axioms do these straight lines obey?


The figure shows a pseudosphere and the straight line on the surface have all the properties of Lobatchevsky and Bolyai straight lines eg. the axiom that 2 points determine one and only one straight line applies to these geodesics, eg. 2 points on the psuedosphere (C and D) determine one and only one geodesic or shortest path between them. eg. the parallel axiom which says that through a point P not on a line L there is an infinite number of lines that do not meet L applies to the geodesics of the pseudosphere.

Riemann's Theories and Geometry

Riemann worked at infiniteness of straight line - he thought we must distinguish between endlessness and infiniteness eg. the Equator is endless but finite so his alternative to Euclid axiom was that all lines are finite in length but endless.

He looked at parallel axiom


As R moves to the left along L and as Q makes to the right both points must ultimately meet for Riemann supposed the line L to be finite. The line PR will as a result rotate around

into PQ without ever losing contact with L. So Riemann suggested that there were no parallel lines.

Also suggested unlike Euclid that 2 points determine one and only one line, he said 2 points may determine more than one line.


Questions 2

Non-Euclidean Geometry Resources

Projective Geometry

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