George Gamow: Turning Double Apples into Doughnuts & Inverse Universes

[daniel_i_l]

In George Gamow's book "One Two Three Infinity" he explains on page 56 how to turn a "double apple" with a free surface into a doughnut:
http://books.google.com/books?id=EZ...=X&oi=book_result&ct=result&resnum=2#PPA57,M1
He then (page 58) goes on to explain how the reverse process can be used to make an "inverse universe" where a human is somehow tured inside out.
I'm pretty sure that I understand how the double apple is changed to a doughnut but I don't understand how the reverse process make an inverse universe. It seems as if we just have to change the material of the doughnut itself and not the surrounding space.
Thanks.

 

[slider142]

Have a look at the surface he demarcates "FREE". What is it bounding in each step? :)