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
- Weisstein, Eric W. "Möbius Strip." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MoebiusStrip.html