Question

Question #f312d

Answer 1

`4/(1 - sqrt[5]) + 4/(1 + sqrt[5])=-2`

Explanation:

Given `z_1` we can calculate `z_2` according to

`4z_2^2+z_1^2-2iz_1z_2=0` or

`z_2 =i/4(1pm sqrt(5)) z_1`

which means that given `z_1` to obtain `z_2` we need two transformations
1) Scaling by `1/4(1pm sqrt5)`
2) Rotating `pi/2` counterclockwise as associated to the product by `i`

so with `z_0 = 0` we can build two triangles

`[z_0,z_1,z_2^a]` and `[z_0,z_1,z_2^b]`

with

`z_2^a=i/4(1+ sqrt(5)) z_1`
`z_2^b=i/4(1- sqrt(5)) z_1`

so `z_2^a,z_0,z_2^b` are aligned and

`[z_2^a,z_2^b]` is perpendicular to `[z_0,z_1]`

and given `(z_1)_k` all triangles `[z_2^a,z_1,z_2^b]_k` are similar.

The least angles at `[z_2^a,z_0,z_1]` and `[z_2^b,z_0,z_1]` are respectively

`alpha = arctan((1-sqrt(5))/4), beta = arctan((1+sqrt(5))/4)`

and

`cot(alpha)+cot(beta) = 4/(1 - sqrt[5]) + 4/(1 + sqrt[5])=-2`