Why Is Euler's Formula For Polyhedron?
link in notes *insert pictures + make report format* In my mathematical investigation, I have chosen the topic geometry, focusing on 2 generalisations and conjectures. The aim of this investigation is to further understand the topic I have chosen and to be able to apply it in my daily life. I decided on the topic geometry as I feel that I am more interested in this topic as the school has not fully covered this topic. *historical background of polyhedrons and triangles* Euler’s Formula for polyhedron is named after Leonhard Euler (1707 - 1783) , a swiss mathematician as he was the first person who discovered F + V − E = in 1750. *historical background* Euler’s Formula for polyhedron proves that for any convex polyhedron which includes platonic solids, the number of faces (F) and vertices (V) is 2 more than the number of edges (E). In other words, Euler’s Formula for polyhedron helps identify solids that are convex polyhedrons and platonic solids. Polyhedrons are solids in which each face is a polygon which is a shape with straight sides, for …show more content…
After calculating F + V − E, each is 2. To prove any convex polyhedrons will get = 2, take a cube and add a edge from one corner to the opposite. After adding the edge, the number of faces is 7, number of vertices is 8 and number of edges is 13. As adding an edge does not make the solid intersect itself, should equal to 2. So, F + V − E = 7 + 8 − 13 and the answer would be 2 which is the expected outcome. Another way to prove the formula does not work for all solids is to take a Icosahedron and connect 2 opposite corners. Joining both corners, the solid will be a Icosahedron but it is not a convex polyhedron. As the soild is not a convex polyhedron should not be 2. So, F + V − E = 20 + 11 − 30, getting the answer as 1 which proves the generalisation F + V − E = 2 does not work. Therefore, the 2 is always replaced with
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