Let $X$ be a non-empty, closed octagonal surface, in which $3$ octagons meet at every vertex. I have to show that its Euler Characteristic is $χ(X)=-V/8$.
I have solved a similar exercise stating the following:
Let $Y$ be a triangular closed surface such that $m$ triangles meet at every vertex. Find its Euler Characteristic.
In order to find it I did the following $3F=2E=mV$ and thus showed that $χ(Y)=F(3/m-1/2)$
If I am to follow the same methodology for its solution i would have something along the lines of $kF=lE=3V$.
So my question is, what are the coefficients $k,l$ of $F,E$ going to be?
Best Answer
Hint. Every octagon has eight vertices, and three octagons meet at every vertex, from which you get a relation between $F$ and $V$. Every octagon has eight edges, and two octagons meet at one edge, from which you get a relation between $F$ and $E$. This means you can express $E,F$ in terms of $V$, and you are done. In order to find the relations exactly, just think about how to count the vertices/edges correctly.
Maybe illustrating the logic for your triangle example would help. The reason that $3F=2E$, as you wrote, is because every triangle (of which there are $F$) corresponds to three edges, while every edge corresponds to two triangles. So the number of $(\text{triangle},\text{edge})$ pairs can be counted as either $3F$ or as $2E$, so they are equal. Can you extend this logic to solve your new problem?