(180 - 360/n) + (180 - 360/k) + (180 - 360/p) = 360
3*180 -360( 1/n+1/k+1/p)= 2*180
1*180 = 360( 1/n+1/k+1/p)
So, for example, n=3, k=4 and p=
5 is not possible since
Number of polygons around a vertex |
Equation for angle sum = 360 | Equivalent Arithmetic equation | Solutions to the arithmetic equations. | |||||||||||||||||||||||||||||||
3: n , k, p | 180 - 360/n+180 - 360/k+180 - 360/p = 360 | 1/n+1/k+1/p =1/2 |
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4: n, k, p, z | 180 - 360/n+180 - 360/k+180 - 360/p 180 - 360/z = 360 | 1/n+1/k+1/p +1/z =2/2 =1 |
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5: n, k, p, z, w | 180 - 360/n+180 - 360/k+180 - 360/p+180 - 360/z+180 - 360/w = 360 | 1/n+1/k+1/p +1/z+1/w =3/2 |
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- Why are there only six?
- What about combining transformations to give new symmetries: