Euler characteristic of the projective plane (using embedding diagram)
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You can see four squares in the figure. The two diagonally opposite squares form a face, as you can see from the gluing. That is why it has two faces in total.
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Author by
thinker
Updated on August 01, 2022Comments
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thinker over 1 year
Make the square into the projective plane $\mathbb{P}$ by identifying edges and compute the Euler characteristic by embedding the following graph onto the surface:
Here is my diagram of the identified square of the projective plane with the embedded graph on it:
Now I can see that $v=3$ and $e=4$ but I am unsure how to calculate the number of faces from this. According to my lecturer the answer is $2$ leading to an euler characteristic of $\chi(\mathbb{P})=v-e+f=3-4+2=1$.
Could you explain why $f=2$ in this case?
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Riccardo over 7 yearsOh man, just read something else:) my bad!
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Gerard over 6 yearse=4 because: A) there are 4 coloured line pieces? Or B) there are 2 couloured lines + 2 border lines?
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thinker over 7 yearsthank you. I suppose for torus and klein bottle will have $f=1$ since the space 'wraps around' itself
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Henry over 7 years@thinker exactly!