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**Non Euclidean Geometry – Spherical Geometry **

*This article follow on from Non Euclidean Geometry – An Introduction – read that first!*

Most geometers up until the 19th century had focused on trying to prove that Euclid’s 5th (parallel) postulate was true. The underlying assumption was that Euclidean geometry was *true* and therefore the 5th postulate must also be true.

The German mathematician Franz Taurinus made huge strides towards developing non-Euclidean geometries when in 1826 he published his work on spherical trigonometry.

Spherical trigonometry is a method of working out the sides and angles of triangles which are drawn on the surface of spheres.

One of the fundamental formula for spherical trigonometry, for a sphere of radius k is:

cos(a/k) = cos(b/k).cos(c/k) + sin(b/k).sin(c/k).cosA

So, say for example we have a triangle as sketched above. We know the radius of the sphere is 1, that the angle A = 60 degrees, the length b = 1, the length c =1, we can use this formula to find out what the length a is:

cos(a) = cos(1).cos(1) + sin(1).sin(1).cos60

a = 0.99996

We can note that for the same triangle sketched on a flat surface we would be able to use the formula:

a^{2} = b^{2} + c^{2} – 2bc.cosA

a^{2}= 1 + 1 – 2cos60

a = 1

Taurinus however wanted to investigate what would happen if the sphere had an *imaginary radius* (i). Without worrying too much about what a sphere with an imaginary radius would look like, let’s see what this does to the previous spherical trigonometric equations:

The sphere now has a radius of ik where i = √-1, so:

cos(a/ik) = cos(b/ik).cos(c/ik) + sin(b/ik).sin(c/ik).cosA

But cos(ix) = cosh(x) and sin(ix) = (-1/i)sinh(x) – where cosh(x) and sinh(x) are the hyperbolic trig functions. So we can convert the above equation into:

cosh(a/k) = cosh(b/k)cosh(c/k) – sinh(b/k).sinh(c/k).cosA

This equation will give us the relationship between angles and sides on a triangle drawn on a sphere with an imaginary radius.

Now, here’s the incredible part – this new geometry based on an imaginary sphere (which Taurinus called Log-Spherical Geometry) actually agreed with the *hypothesis of the acute angle * (the idea that triangles could have an angle sum less than 180 degrees).

Even more incredible, if you take the limit as k approaches infinity of this new equation, you are left with:

a^{2} = b^{2} + c^{2} – 2bc.cosA

What does this mean? Well, if we have a sphere of infinite imaginary radius it stretches and flattens to be indistinguishable from a flat plane – and this is where our normal Euclidean geometry works. So, Taurinus had created a geometry for which our own Euclidean geometry is simply a special case.

So what other remarkable things happen in this new geometric world? Well we have triangles that look like this:

This triangle has angle A = 0, angle C = 90 and lines AB and AC are parallel, (they never meet). This sketch introduces a whole new concept of parallelism far removed from anything Euclid had imagined. The angle β is called the angle of parallelism – and measures the angle between a perpendicular and parallel line. Unlike in Euclidean geometry this angle does not have to be 90 degrees. Indeed the angle β will now change as we move the perpendicular along AC – as it is dependent on the length of the line a.

So, we are now into some genuinely weird and wonderful realms where normal geometry no longer makes sense. Be warned – it gets even stranger! More on that in the next post.

If you enjoyed this post you might also like:

Non Euclidean Geometry IV – New Universes – The fourth part in the non-Euclidean Geometry series.

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.