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Description

1. Cut out a strip of paper which measures 11 inches by 3/4of an inch.

(i) Glue, tape, or staple the two short ends together in such a way as to form

a M ̈obius band M. Take a photo of it,

(ii) How many sides does M have? How many boundary components does

M have? Convince yourself that M is non-orientable. Write a paragraph

explaining your thought process. (It is ok if you do not know the technical

definitions of these terms from topology, just make your best guess what

you think they mean.)

2. Let δ be a circle in the real hyperbolic plane of radius log 5. Calculate the

circumference of δ. (Write your answer as a rational multiple of π.)

3. Let ` be a line in the real elliptic plane. Show that there exists a point P such

that a perpendicular to ` which passes through P is not unique.

4. Let A be a triangle in the real elliptic plane. Suppose that you know the lengths

of two of its sides and the size of the two angles opposite those two sides. Show

that this is not always enough information to determine A up to congruence.