Möbius Strip has Euler Characteristic Zero

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Theorem

Let $M$ be a Möbius Strip.


Then:

$\map \chi M = 0$

where $\map \chi M$ denotes the Euler characteristic of the graph $M$.


Proof

Let the number of vertices, edges and faces of $M$ be $V$, $E$ and $F$ respectively.

From Möbius Strip has no Vertices:

$V = 0$

From Möbius Strip has 1 Edge:

$E = 1$

From Möbius Strip has 1 Face:

$F = 1$

By definition of the Euler characteristic:

\(\ds \map \chi M\) \(=\) \(\ds V - E + F\) Definition of Euler Characteristic
\(\ds \) \(=\) \(\ds 0 - 1 + 1\)
\(\ds \) \(=\) \(\ds 0\)

$\blacksquare$


Source

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