(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 


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 


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 



 Why are there only six?
 What about combining transformations to give new symmetries: