Soccer Ball and Euler’s Law
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1. Definition
3. Using Additional Information
Figure 1. Structure of a Soccer Ball
1. Definition
⑴ A : Number of pentagons
⑵ B : Number of hexagons
2. Application of Euler’s Law
⑴ For the number of vertices V, the number of edges E, and the number of facets F, the following holds true
⑵ Relationship for the number of vertices : Utilizing the fact that three facets meet at each vertex
⑶ Relationship for the number of edges : Utilizing the fact that two facets meet at each edge
⑷ Relationship for the number of facets
⑸ Application of Euler’s Law
3. Using Additional Information
⑴ Additional information
① Each of the three sides of any hexagon is adjacent to a hexagon, while the remaining three sides are adjacent to a pentagon.
② All sides of any pentagon are adjacent to hexagons.
⑵ Calculation strategy
① (Counting from the perspective of hexagons, allowing for duplicates) Number of pentagons counted from the hexagon’s perspective ÷ Overlap count = Number of pentagons = 12
② Overlap count = 5 ( ∵ Counted once from each of the five adjacent hexagons from the pentagon’s perspective)
③ Calculation process
Input : 2019.11.05 19:03