Non-Euclidean Geometry – A New Universe
This post follows on from Non-Euclidean Geometry – An Introduction – read that one first!
The Hungarian army officer and mathematician Johan Bolyai wrote to his father in 1823 in excitement at his mathematical breakthrough with regards to the parallel postulate. “I have created a new universe from nothing.” Johan Bolyai was one of the forerunners of 19th century mathematicians who, after noting that mathematicians had spent over 2000 years trying to prove the parallel postulate, decided to see what geometry would look like if the constraint of the postulate was removed. The result was indeed, a new universe from nothing.
To recap, Euclid’s fifth postulate was as follows:
If a line crossing two other lines makes the interior angles on the same side less than two right angles, then these two lines will meet on that side when extended far enough.
It had been understood in a number of (non-equivalent) ways – that parallel lines remain equidistant from each other, that non-parallel lines intersect, that if the lines L1 and L2 in the diagram are parallel then A + B = 180 degrees, that there can only be one line through a point parallel to any given line.
Collectively these assumptions lead to the basis of numerous geometric proofs – such as the fact that angles in a triangle add up to 180 degrees and that angles in a quadrilateral add up to 360 degrees.
A geometry not based on the parallel postulate could therefore contain 3 possibilities, as outlined by the Italian mathematician Gerolamo Saccheri in 1733:
1) A quadrilateral with (say) 2 right angles A,B and two other angles C,D also both right angles. This is the hypothesis of the right angle – the “normal” geometry of Euclid.
2) A quadrilateral with (say) 2 right angles A,B and two other angles C,D both obtuse. This is the hypothesis of the obtuse angle – a geometry in which the angles in quadrilaterals add up to more than 360 degrees.
3) A quadrilateral with (say) 2 right angles A,B and two other angles C,D also both acute. This is the hypothesis of the acute angle – a geometry in which the angles in quadrilaterals add up to less than 360 degrees.
Don’t be misled by the sketch above – the top line of the quadrilateral is still “straight” in this new geometry – even if it can’t be represented in flat 2 dimensions.
Mathematicians now set about trying to prove that both the cases (2) and (3) were false – thus proving that the Euclidean system was the only valid geometry. The French mathematician Adrien Legendre, who made significant contributions to Number Theory tried to prove that the hypothesis of the obtuse angle was impossible. His argument went as follows:
1) Take a straight line and divide it into n equal segments. In the diagram these are the 4 lines A1A2, A2A3, A3A4, A4A5
2) Complete the diagram as shown above so that the lengths B1B2, B2B3, B3B4, B4B5 are all equal. From the sketch we will have lines A1B1 and A2B2 (and subsequent lines) equal.
3) Now we see what will happen if angle β is greater than α. We compare the two triangles A1B1A2 and A2B2A3. These have 2 sides the same. Therefore if β is greater than α then the length A1A2 must be larger than B1B2.
4) Now we note that the distance A1B1 + B1B2 + B2B3 + … BnBn+1 + Bn+1An+1 is greater than A1A2 + A2A3 + …AnAn+1. In other words, the distance starting at A1 then travelling around the shape missing out the bottom line (the yellow line) is longer than the bottom line (green line).
5) Therefore we can write this as
A1B1 + nB1B2 + An+1Bn+1 > nA1A2
(Here we have simplified the expression by noting that as all the distances B1B2, B2B3 etc are equal)
6) Therefore this gives
2A1B1 > n(A1A2 -B1B2)
(Here we simplify by noting that A1B1 = An+1Bn+1 and then rearranging)
7) But this then gives a contradiction – because we can make the RHS as large as we like by simply subdividing the line into more pieces (thus increasing n), but the LHS remains bounded (as it is a fixed value). Therefore as n tends to infinity, this inequality must be broken.
8) This means that β is not greater than α, so we can write β ≤ α. This will therefore mean that the angles in the triangle A1B1A2 will be ≤ 180. To see this
We can work out the angles in A1B1A2 by noting that c = (180-α)/2 . Therefore
angles in A1B1A2 = (180-α)/2 + (180-α)/2 + β
angles in A1B1A2 = 180 + β – α
But we know that β ≤ α. Therefore β – α ≤ 0
So angles in A1B1A2 = 180 + β – α ≤ 180
Adrien Legendre therefore concluded that the hypothesis of the obtuse angle was impossible. In fact, it isn’t – and the flaw wasn’t in the logic of his proof but in the underlying assumptions contained within it. This will be revealed in the next post!
If you enjoyed this you might also like:
Non Euclidean Geometry III – Breakthrough Into New Worlds – The third part of the series on non-Euclidean Geometry.
The Riemann Sphere – The Riemann Sphere is a way of mapping the entire complex plane onto the surface of a 3 dimensional sphere.
Circular Inversion – Reflecting in a Circle The hidden geometry of circular inversion allows us to begin to understand non-Euclidean geometry.