suni
On a circular island we build n straight dams going from sea to sea, so that every two intersect but no three go through the same point. use Eulers Formula to determine how many parts we get
Euler characteristic
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replied to: suni
sonurado
Replied to: On a circular island we build n straight dams going from...
Let us take n = 3.
v = 9, e = 15, f = 8.
Consider following figure with n = 4.
----Sorry Can't insert figure-----
----Try to draw it Start with n = 3------
Clearly, number of vertices (v) = 14, number of edges (e) = 24 and number of faces (f) = 12.
According to Euler’s formula, v + f = e + 2
i.e. v + f = 14 + 12 = 26 = e + 2.
In general we have, v = 2*n + n*(n-1)/2 and e = 2*n + n^2.
Hence, using Euler’s formula,
f = e + 2 – v
= 2*n + n^2 + 2 – 2*n - n*(n-1)/2
After simiplication we have,
f = n (n+1)/2 + 2
v = 9, e = 15, f = 8.
Consider following figure with n = 4.
----Sorry Can't insert figure-----
----Try to draw it Start with n = 3------
Clearly, number of vertices (v) = 14, number of edges (e) = 24 and number of faces (f) = 12.
According to Euler’s formula, v + f = e + 2
i.e. v + f = 14 + 12 = 26 = e + 2.
In general we have, v = 2*n + n*(n-1)/2 and e = 2*n + n^2.
Hence, using Euler’s formula,
f = e + 2 – v
= 2*n + n^2 + 2 – 2*n - n*(n-1)/2
After simiplication we have,
f = n (n+1)/2 + 2