Na Copa do Mundo de 1970, a bola era um icosaedro truncado, uma figura com 12 faces pentagonais regulares (pintadas de preto) e 20 faces hexagonais regulares (pintadas de branco). Esta bela figura geométrica tem quantos vértices e quantas arestas?
(In the 1970 World Cup, the ball was a truncated icosahedron, a figure with 12 regular pentagonal faces (painted black) and 20 regular hexagonal faces (painted white). How many vertices and how many edges does this beautiful geometric figure have?)
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To find the number of edges, we find the total number of sides represented by all the faces and divide by 2 (since two faces meet at each edge):
ResponderExcluirE = (20 * 6 + 12 * 5)/2 = 180/2 = 90
So, a truncated icosahedron has 90 edges.
To find the number of vertices, we shall use Euler's formula, which states that for any polyhedron, F + V = E + 2, where F is the number of faces, V is the number of vertices, and E is the number of edges. We just showed that a truncated icosahedron has 90 edges, and since it has 12 hexagons and 20 pentagons, it has 32 faces. So, we find the number of vertices by substituting these values into Euler's formula:
32 + V = 90 + 2
32 + V = 92
V = 60
So a truncated icosahedron has 60 vertices.
My answer: 90 edges and 60 vertices
Well done, Jake!!
ExcluirHere's another way of seeing it: in the truncated icosahedron, every pentagonal face is surrounded by 5 hexagonal faces, and every hexagonal face is surrounded by 3 pentagonal faces and 3 other hexagonal faces. This means that all of the vertices are vertices of pentagonal faces.
So, in order to count the total number of vertices, we just need to count the number of vertices of the pentagonal faces: 12*5 = 60.
In order to count the number of edges, we note that there are two kinds of edges:
- edges that are sides of pentagons: 12*5 = 60;
- edges that connect two pentagons: 12*5/2 = 30.
That makes a total of 90 edges. :)
An amazing fact about the truncated icosahedron: it exists in nature! There is a molecule with this exact shape, with 60 atoms of carbon, one in each of the vertices. :)
ResponderExcluir