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Thank you very much for a beautiful and illuminating page.
I was struggling intuitively with the concept of a crosscap today, while reading this and some similar pages. Do you have a pointer to a URL that could help me? I tried Wikipedia and others but was still left a little confused. Perhaps a crosscap is an entity more difficult to visualise than a Klein bottle?
Is it possible to show how a Klein bottle, such as one of your R^3-embedded representations here, can be "cut up into bits, two of which are crosscaps"? -In a similar manner to the way you show at the top that it can be cut into two Mobius strips and an ordinary annulus-like shape.
Thank you in advance for any help.
Dave (London, UK)