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Example text
They won't bump into each other, they will be on opposite sides of the disk, but they may or may not be in the same universe. On the other hand, nipping back smartly where you have just come from, walking around the disk, and then going in the same side would get them together again. On this model. If someone ever does invent a gateway for travelling between universes, the mathematicians are ready for talking about it3. The reason for thinking about multidimensional car parks, sliders and bizarre topologies, is that it has everything to do with the Riemann surface for p w = f (z) = z2 + 1 We need to take both f and f , and we have quite a large region in which we can have each branch of f single valued and 1-1, namely the whole plane with the slit from i to i removed.
The more ways you have of talking and thinking about things, the easier it is to draw conclusions, and the harder it is to be led astray. It is also a lot more fun. The converse is also true: the inversion of a straight line is a circle through the origin. To see this, let ax + by + c = 0 be the equation of a straight line. Turn this into polars to get ar cos + br sin + c = 0 Now put r = 1=s to get the inversion: (a=s) cos + (b=s) sin + c = 0 and rearrange to get s2 + (as=c) cos + (bs=c) sin = 0 a= 2 c which is a circle passing through the origin with centre at b=2c .
CONCLUSIONS 29 then this means that we must have uu0 + vv0 = 0 We need to show that zw1 and zw10 are also at right angles. if z = x + iy , then we need to show uu0 + vv0 = 0 ) (xu yv)(xu0 yv0) + (xv + yu)(xv0 + yu0) = 0 The right hand side simpli es to (x2 + y2)(uu0 + vv0) so it is true. The above problem and the two solutions that go with it carry an important moral. It is this: If you can see what is going on, you can solve some problems instantly just by looking at them. And if you can't, then you just have to plug away doing algebra, with a serious risk of making a slip and wasting hours of your time as well as getting the wrong answer.