Interesting facts: Surfaces

Here are more examples:

$$\LARGE{g = 1 + \frac{N}{2} \left( k - 2 - \sum_{i=1}^{k}\frac{1}{n_i}\right)}$$

Legend:

• \(g\) is the ‘genus’ of the surface, denoting the number of ‘holes’ in the pretzel.
• \(k\) is the number of sides of the polygon.
• \(n_i\) is based on the formula for the internal angels between the sides of a polygon. Each internal angle in a polygon can be written as a reduced fraction \(\pi\frac{m_i}{n_i}\), where \(i=1,...,k\). Reduced means, that numerator and denominator have no common divisors. We see that \(n_i\) are the denominators of the fractions representing the internal angles.
• \(N\) is the lowest common multiple (LCM) of the denominators \(n_i,\) i.e. the smallest number which is divided by all \(n_i\).
• The expression \(\sum_{i=1}^{k}\frac{1}{n_i}\) is a sum in which the index \(i\) takes all values from \(1\) to \(k\). E.g. for \(k=3\) 
  $$\sum_{i=1}^{3}\frac{1}{n_i} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3}.$$

Example 1: Rectangle

For a rectangle it holds: \(k=4\); all angles are the same, namely \(\pi\frac{m_i}{n_i}=\pi\frac{1}{2}\) or \(90^{\circ}\). We see all \(n_i=2\) and therefore the LCM is \(N=2\). Inserting in the above formula we get

$$g=1+\frac{2}{2}\left(4-2-\sum_{i=1}^{4}\frac{1}{2}\right)=1+\frac{4}{2}\left(4-2-2\right)=1.$$

The genus is 1 - we get a donut!

Example 2: Pentagon

For a pentagon it holds: \(k=5\); all angles are the same, namely \(\pi\frac{m_i}{n_i}=\pi\frac{3}{5}\) or \(108^{\circ}\). We see all \(n_i=5\) and therefore the LCM is \(N=5\). Inserting in the above formula we get

$$g=1+\frac{5}{2}\left(5-2-\sum_{i=1}^{5}\frac{1}{5}\right)=1+\frac{5}{2}\left(5-2-1\right)=6.$$

The result is a pretzel with 6 holes.