Question

euler’s formula for planar graphs also applies to platonic solids (like dice)!
let’s see that in action. a dodecahedron (12 - sided die) has 12 faces, each of which is a regular pentagon.
that means for each face, facesize =
we can use this with the facesize lemma to find that
$2|e| = \sigma$ facesizes =
that means that $|e| = $
then euler’s formula applies, so without counting we could find $|v| = $
note. this is the formula $\sigma deg(r_i) = 2q$ in class. euler’s formula: $r = q - p + 2$.
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Explanation:

Step1: Determine facesize

A regular pentagon has 5 sides, so facesize (number of edges per face) is 5.

Step2: Calculate Σ facesizes

There are 12 faces, each with facesize 5. So \( \Sigma \text{facesizes} = 12 \times 5 = 60 \).

Step3: Find |E|

Using \( 2|E| = \Sigma \text{facesizes} = 60 \), we solve for \( |E| \): \( |E| = \frac{60}{2} = 30 \).

Step4: Apply Euler's Formula

Euler's Formula for polyhedra (Platonic solids) is \( F - E + V = 2 \), where \( F \) is number of faces, \( E \) is number of edges, \( V \) is number of vertices. We know \( F = 12 \), \( E = 30 \). Plugging in: \( 12 - 30 + V = 2 \). Solving for \( V \): \( V = 2 + 30 - 12 = 20 \).

Answer:

facesize = 5; \( \Sigma \) facesizes = 60; \( |E| = 30 \); \( |V| = 20 \)