I cannot make head or tail of fig 8.2. What's more, it seams to me impossible that the spiral ramp figured in 8.1 has the topology of a sphere! (I did a topology course 30 years ago as a maths undergraduate, but I have since been very successful at eliminating the small amount I understood from my mind)
It seems to me that section 8.2 is not explained in enough detail, or are we supposed to just accept these findings and understand them later on?

I struggled for a long time with this diagram myself. I think I now understand it. Have you tried reading my document which contains some explanation of this diagram? It's in the exercise discussion section here:
http://www.roadtoreality.info/viewtopic.php?f=4&t=281 ?
Try it and let me know what you think or ask me some questions.

Also see solution to Exercise 8.3 here:
http://www.roadtoreality.info/viewtopic.php?f=19&t=280
Vasco

Thanks Vasco for your help! I will take your advice and will get back to you on it. It might take a few days as I'm pretty busy with my day job, but its good to know this forum is alive! Its nice to know that i'm not alone in my battle with this book ('the Sunday Times Top Ten Bestseller' it says on the cover of my edition! What are they using it for, as door stops?)

Tony

vasco,
I began reading your article, but I haven't got to the bit in RTR about stereographic projections. However perhaps I have found a minor error in page 1 , where you consider a circle in the z plane of radius r, and then write:
w = r + iÞ
(sorry, the Þ is the closest I can get to phi on my keyboard)

Surely if the radius in the z plane is r then that transforms to logr in the w plane

Yes you are right Tony. This doesn't alter the argument of course, so bearing this fact in mind, that r should be log r, you can still follow the proof through.
If you find any other errors let me know as soon as you can so that I can make the changes.
I will post an amended version of the pdf as soon as I can.

Thanks for the feedback - it's good to know that people are reading your stuff.