<p>If require that all faces have the same number of sides <span class="math-container">$s$</span> and require that all vertices also have a certain valency <span class="math-container">$t$</span>. We see that the following relation between edges, and faces hold for a regular mesh:</p>
<p><span class="math-container">$$s\cdot f = 2e,$$</span>
<span class="math-container">$$t\cdot v = 2e.$$</span> Substitution in the Euler-Poincare formula yields:</p>
<p><span class="math-container">$$\left(\frac{1}{s} + \frac{1}{t} - \frac{1}{2}\right)e = 1 - g$$</span></p>
<p>If we then take for instance a regular plane which can be said to have the topology of a torus with genus <span class="math-container">$g = 1$</span>. We then set the valency of faces <span class="math-container">$s = 3$</span> then</p>
<p><span class="math-container">$$\left(\frac{1}{3} + \frac{1}{t} - \frac{1}{2}\right)e = 0,$$</span>
<span class="math-container">$$\left(-\frac{1}{6} + \frac{1}{t}\right)e = 0.$$</span></p>
<p>The solution for <span class="math-container">$t$</span> is <span class="math-container">$6$</span>, which says the valency of a vertex in a regular triangulation is 6.</p>

CC BY-SA 4.0
2020-09-24T10:00:21.543Z
2020-09-25T18:12:22.583Z
2020-09-25T18:12:22.583Z
Reynolds
<p>So I recently learnt that supposedly for any mesh, and pretty much any scheme, the valence of regular vertices must be 6. It seems to be related to the Euler-Poincare formula but I have not been able to find the justification.</p>

CC BY-SA 4.0
2020-09-24T06:13:24.623Z
This does not really apply to a tetradedral outcropping but it does apply most quatrilateral turned to triangles meshes where the repetition of triangulation is uniform
joojaa
<computational-geometry><geometry><mesh>
Why is the valence of regular vertices 6?
Makogan
# Why is the valence of regular vertices 6?

(Asked by: username_0 on 2020-09-24)

So I recently learnt that supposedly for any mesh, and pretty much any scheme, the valence of regular vertices must be 6. It seems to be related to the Euler-Poincare formula but I have not been able to find the justification.

**Tags:** computational-geometry, geometry, mesh

> **Comments:**
> **username_1**: This does not really apply to a tetradedral outcropping but it does apply most quatrilateral turned to triangles meshes where the repetition of triangulation is uniform  

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## 1 answers

### Answer by username_2 on 2020-09-24. Score: 4 | Accepted answer

If require that all faces have the same number of sides $s$ and require that all vertices also have a certain valency $t$. We see that the following relation between edges, and faces hold for a regular mesh:

$$s\cdot f = 2e,$$
$$t\cdot v = 2e.$$ Substitution in the Euler-Poincare formula yields:

$$\left(\frac{1}{s} + \frac{1}{t} - \frac{1}{2}\right)e = 1 - g$$

If we then take for instance a regular plane which can be said to have the topology of a torus with genus $g = 1$. We then set the valency of faces $s = 3$ then

$$\left(\frac{1}{3} + \frac{1}{t} - \frac{1}{2}\right)e = 0,$$
$$\left(-\frac{1}{6} + \frac{1}{t}\right)e = 0.$$

The solution for $t$ is $6$, which says the valency of a vertex in a regular triangulation is 6.


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