Question #f312d

1 Answer
Cesareo R.
Apr 21, 2017

# 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#

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